# Partial Identification and Inference in Duration Models with Endogenous
Censoring††thanks: I would like to thank the managing editor Elie Tamer, an
anonymous associate editor, and an anonymous referee for their constructive
suggestions and comments. This paper is based on a chapter of my dissertation
at Kyoto University. I am grateful to Yoshihiko Nishiyama and Ryo Okui for
their guidance and helpful suggestions. Parts of this paper were written while
I was visiting Penn State as a visiting student. I would like to thank Keisuke
Hirano for his hospitality and helpful advice. I would also like to thank
Hidehiko Ichimura, Toru Kitagawa, Andrew Chesher, Francesca Molinari, Daniel
Wilhelm, Jeff Rowley, and participants at various seminars and conferences for
their comments and suggestions. I acknowledge financial support from JSPS
KAKENHI Grant (number 18J00173) and ERC Grant (number 715940). Supplementary
material is available here

Shosei Sakaguchi  Department of Economics, University College London, Gower
Street, London WC1E 6BT, United Kingdom. Email<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

This paper studies identification and inference in transformation models with
endogenous censoring. Many kinds of duration models, such as the accelerated
failure time model, proportional hazard model, and mixed proportional hazard
model, can be viewed as transformation models. We allow the censoring of a
duration outcome to be arbitrarily correlated with observed covariates and
unobserved heterogeneity. We impose no parametric restrictions on either the
transformation function or the distribution function of the unobserved
heterogeneity. In this setting, we develop bounds on the regression parameters
and the transformation function, which are characterized by conditional moment
inequalities involving U-statistics. We provide inference methods for them by
constructing an inference approach for conditional moment inequality models in
which the sample analogs of moments are U-statistics. We apply the proposed
inference methods to evaluate the effect of heart transplants on patients’
survival time using data from the Stanford Heart Transplant Study. Keywords:
Partial identification, duration models, transformation models, censoring,
conditional moment inequality.
JEL codes: C14, C24, C41.

###  1 Introduction

Duration models are widely used in various empirical studies in economics and
biomedical sciences, where outcomes of interest are durations up to the
occurrence of some event. Durations of interest in economics include
unemployment duration, strike duration, insurance claim duration, and the
duration until the purchase of a durable good.111van den Berg (2001) surveys
many applications of duration models.

In practice, duration data are often censored. For example, unemployment
duration is likely to be censored due to some individuals dropping out of the
survey, due to attrition say. Dealing with censoring has been a substantial
challenge in duration analysis, and various methods have been proposed. A
standard approach is to assume that censoring is independent of unobserved
heterogeneity (conditional or unconditional on observed characteristics).
Studies employing this approach include Cox (1972), Powell (1984), Ying et al.
(1995), Yang (1999), Honoré et al. (2002), Hong and Tamer (2003), and Khan and
Tamer (2007), among others. However, in many cases, justifying this
independence assumption is difficult. For example, in unemployment duration
analysis, unemployed individuals with low motivation to find a job may tend to
drop out of the survey at an early stage. Szydłowski (2019) presents a number
of examples where censoring is correlated with unobserved heterogeneity (i.e.,
censoring is endogenous).

In this paper, we study identification and inference in transformation models
in the presence of endogenous censoring. The transformation model is expressed
as

$\displaystyle Y^{\ast}=\Lambda(X^{\prime}\beta_{0}+U),$ (1.1)

where $\Lambda$ is a non-degenerate monotone function; $Y^{\ast}$ is a
dependent variable, which represents a duration outcome in this paper; $X$ is
a $k$-dimensional vector of observed covariates, where $k\geq 2$; $\beta_{0}$
denotes a $k$-vector of regression parameters; and $U$ is unobserved
heterogeneity that is independent of $X$. Many kinds of duration models, such
as the accelerated failure time model, proportional hazard model, and mixed
proportional hazard (MPH) model, can be viewed as transformation
models.222Aside from duration models, a class of transformation models
contains other important kinds of models, for example, the linear index model
and Box-Cox transformation model. In this paper, we consider a nonparametric
transformation model in which neither the transformation function nor the
distribution function of the unobserved heterogeneity is parametrically
specified. One important model represented by the nonparametric transformation
model is the nonparametric MPH model in which neither a baseline hazard
function nor the distribution function of the unobserved heterogeneity is
parametrically specified.

Allowing for endogenous censoring, we develop bounds on the regression
parameters $\beta_{0}$ and transformation function $\Lambda$ in model (1.1).
To the best of our knowledge, this is the first work to derive bounds on
$\beta_{0}$ and $\Lambda$ in the nonparametric transformation model with
endogenous censoring. The construction of the bounds is built on the rank
properties of the nonparametric transformation model studied by Han (1987) and
Chen (2002). Han (1987) shows that if there is no censoring, at least one
element of $X$ has full-support on the real line, and $X$ is full-rank, the
regression parameters are point-identified up to scale by looking at the rank
correlation between the outcomes and covariates. In the presence of endogenous
censoring, we develop bounds on $\beta_{0}$ by supposing that, in the rank
property studied by Han (1987), each censored outcome takes an infinitely
large value or is equal to the censoring time. This reflects the fact that
concerning each censored outcome, all we know is that it may take any value
larger than censoring time. When $\Lambda$ is strictly increasing, a lower
bound on $\Lambda$ is also attained by incorporating endogenous censoring into
the rank property studied by Chen (2002). Once the bounds on the regression
parameters and the transformation function are obtained, we can also derive
bounds on the distribution function of the unobserved heterogeneity $U$.

The bounds on $\beta_{0}$ and $\Lambda$ are characterized by conditional
moment inequalities whose sample moments are U-statistics. Based on these
conditional moment inequalities, we construct inference methods for these
parameters by extending the inference approach of Andrews and Shi (2013) for
conditional moment inequality models to the case of U-statistics. The proposed
inference approach can be applied not only to this work but also to other
works involving conditional moment inequalities and U-statistics. In this
sense, this paper also contributes to the literature on inference for
conditional moment inequality models.333Various inference methods for
conditional moment inequality models have been proposed, for example, by
Andrews and Shi (2013, 2014, 2017), Chernozhukov et al. (2013), Armstrong
(2014, 2015), Menzel (2014), Chernozhukov et al. (2019), and so on. But none
of them can be applied to sample moment functions of U-statistics.

The bounded sets of the parameters proposed in this paper are not necessarily
sharp identified sets. On the other hand, using concepts from random set
theory (e.g., Beresteanu et al. (2011, 2012)), we also characterize the sharp
identified set of the regression parameters. However, constructing a feasible
inference method based on it is difficult, whereas the proposed sets are
tractable to construct feasible inference methods. In the paper, we also
discuss conditions under which the proposed set of $\beta_{0}$ approaches the
sharp identified set.

This paper is mostly related to works that study endogenous censoring. Khan
and Tamer (2009), Khan et al. (2011, 2016), Li and Oka (2015), and Fan and Liu
(2018) study identification and estimation of parameters in quantile
regression models with endogenous censoring. For cross-sectional linear
quantile regression models, Khan and Tamer (2009) provide a point
identification result for the linear coefficients under a certain support
condition, while Khan et al. (2011) provide a partial identification result
without this support condition. Under censoring characterized by a certain
copula, Fan and Liu (2018) partially identify the linear coefficients of the
same model. Li and Oka (2015) and Khan et al. (2016) consider panel quantile
regression models with endogenous censoring and provide partial identification
results. In contrast to these works, the identification result in this paper
does not rely on quantile modeling, copula characterization of censoring, or
panel data. Aside from quantile models, Szydłowski (2019) considers the
parametric MPH model and proposes a sharp identified set and inference method
for its parameters. While Szydłowski (2019) considers the parametric MPH
model, we consider the nonparametric one, which is robust to misspecification
of the hazard function or the distribution function of unobserved
heterogeneity.

For competing risks models, Honoré and Lleras-Muney (2006) partially identify
the parameters in the accelerated failure time model, and Kim (2018) derives
computationally tractable bounds on distributions of latent durations by
exploiting the discreteness of observed durations. In this paper, we allow for
continuous observed durations and do not specify competing risks. In a sample
selection model, Honoré and Hu (2020) obtain the sharp identified set of
linear regression coefficients by imposing a particular structure on the
sample selection.

The remainder of this paper is structured as follows. Section 2 describes the
setup and assumptions and then provides the main results to develop bounds on
the regression parameters and the transformation function. We also
characterize the sharp identified set of the regression parameters and compare
it to our proposed superset. Section 3 provides an inference method for the
regression parameters and derives its asymptotic properties. A joint inference
method for the regression parameters and the transformation function is
presented in Appendix C. Section 4 presents numerical examples and Monte Carlo
simulation results. The numerical examples show how the bounds on the
regression parameters and transformation function vary depending on the degree
of censoring and the support of covariates. The Monte Carlo simulation results
show the finite sample properties of the proposed inference method for the
regression parameters. Section 5 presents an empirical illustration, where we
apply our proposed inference methods to evaluate the effect of heart
transplants on patients’ survival duration using data from the Stanford Heart
Transplant Study. We conclude this paper with some remarks in Section 6. All
proofs are presented in Appendices A and B. Some additional numerical examples
and Monte Carlo simulation results are presented in the supplementary material
to this paper.

###  2 Model and Identification

We first describe the setting of the paper and provide conditions to develop
bounds on the regression parameters in Section 2.1. Subsequently, in Section
2.2, we present the main result to construct bounds on the regression
parameters. In Section 2.3, we characterize the sharp identified set of the
regression parameters using concepts from random set theory, and compare this
with our proposed superset. Section 2.4 derives bounds on the transformation
function and the distribution of the unobserved heterogeneity.

####  2.1 Model

We consider the transformation model in the form of (1.1). In the model, we do
not specify the transformation function $\Lambda$ or the distribution function
of the unobserved heterogeneity, which we denote by $F_{U}$. Because of this,
we impose location and scale normalizations. For the location normalization,
we suppose that the constant term is equal to zero (i.e., $X$ does not contain
a constant term). For the scale normalization, we suppose that the absolute
value of the first component of $\beta_{0}$ is equal to one (i.e.,
$\left|\beta_{0,1}\right|=1$), where $\beta_{0,j}$ denotes the $j$-th
component of $\beta_{0}$. Later, in Section 2.4, we impose an additional
location normalization to fix $\Lambda$. Let
$B\equiv\\{-1,1\\}\times\mathbb{R}^{k-1}$ denote the normalized regression
parameter space. Our first main focus is on the identification of and
inference on the normalized regression parameters $\beta_{0}$ in $B$.

The transformation model contains many kinds of duration models as its special
cases: the accelerated failure time model, Cox’s proportional hazard model,
and the MPH model.444If $\Lambda(Y^{\ast})=\exp(Y^{\ast})$, the transformation
model corresponds to the accelerated failure time model; if
$\Lambda(Y^{\ast})=\exp(\Delta(Y^{\ast}))$, where $\Delta(\cdot)$ is the
integrated baseline hazard function and $U$ has the CDF $F(u)=1-\exp(-e^{u})$,
the transformation model corresponds to Cox’s proportional hazard model; if
$\Lambda(Y^{\ast})=\exp(\Delta(Y^{\ast}))$ and $U=\epsilon+\nu$ where $\nu$ is
unobserved heterogeneity and $\epsilon$ has the CDF
$F(\epsilon)=1-\exp(-e^{\epsilon})$, the transformation model corresponds to
the MPH model. For more details, see Horowitz (2009, Ch. 6). In particular,
the nonparametric MPH model is an important duration model represented by a
nonparametric transformation model. The MPH model extends Cox’s proportional
hazard model by incorporating individual unobserved heterogeneity. Since
introduced in Lancaster (1979), the MPH model has been widely used in various
empirical studies in economics. In the nonparametric MPH model, the normalized
regression parameters $\beta_{0}$ can be interpreted as the logs of the scale-
normalized hazard ratios (see, e.g., Lancaster (1990)).

When data are subject to censoring, the duration outcome $Y^{\ast}$ cannot
always be observed. Instead, for unit $i=1,\ldots,n$, we observe
$W_{i}=\left(Y_{0i},D_{i},X_{i}\right)$ such that
$Y_{0i}=\mbox{min}\left\\{Y_{i}^{\ast},C_{i}\right\\}$ and
$D_{i}=I\left[Y_{i}^{\ast}\leq C_{i}\right]$, where $C_{i}$ is a random
censoring variable and $I\left[\cdot\right]$ denotes the indicator function.
$D_{i}$ is a censoring indicator that takes the value zero if $Y_{i}^{\ast}$
is censored and the value one if $Y_{i}^{\ast}$ is observed. Note that we
consider right censoring in the paper, but all the results presented below are
easily extendable to left and interval censoring. Using $D_{i}$, $Y_{0i}$ can
be expressed as $Y_{0i}=D_{i}Y_{i}^{\ast}+(1-D_{i})C_{i}$. Let $P$ denote the
distribution function of a vector of random variables $\left(X,U,C\right)$ and
$\mathcal{X}\subseteq\mathbb{R}^{k}$ denote the support of $X$.

Throughout this paper, we suppose that the following assumptions hold.

######  Assumption 2.1.

$\Lambda$ is a non-degenerate monotonically increasing function.

######  Assumption 2.2.

The vectors $\left(Y_{i}^{\ast},C_{i},X_{i}\right)$, $i=1,\ldots,n$, are
independent and identically distributed (i.i.d) as $(Y^{\ast},X,C)$ where
$Y^{\ast}$ is distributed according to the latent transformation model (1.1)
and the support of $X$ is $\mathcal{X}$.

######  Assumption 2.3.

$U$ is independent of $X$.

Note that Assumption 2.3 does not restrict the relationship between $U$ and
$C$, allowing for endogenous censoring.555Chiappori et al. (2015) study
identification and estimation of the nonparametric transformation model when
some covariates are endogenous and there is no censoring.

####  2.2 Identification of the Regression Parameters

This section constructs bounds on the regression parameters $\beta_{0}$. The
construction is based on a rank property of the latent outcome $Y^{*}$ and the
regression part $X^{\prime}\beta_{0}$ in the transformation model (1.1). For
explanatory purposes, we first introduce the point identification result of
Han (1987) in the absence of censoring.

Han (1987) supposes that there is no censoring (i.e., $Y^{\ast}$ is always
observed). In this case, under Assumptions 2.1–2.3, a full-support condition
on an element of $X$, a full-rank condition on $X$, and a continuous
distribution of $U$, he shows that $\beta_{0}$ uniquely satisfies the
following rank property,

$x_{i}^{\prime}\beta_{0}\geq x_{j}^{\prime}\beta_{0}\Rightarrow
P(Y_{i}^{\ast}\geq Y_{j}^{\ast}\mid x_{i},x_{j})\geq\frac{1}{2}$ (2.1)

for all $(x_{i},x_{j})\in\mathcal{X}^{2}$, where $P(\cdot\mid x_{i},x_{j})$
denotes the conditional probability given $(X_{i},X_{j})=(x_{i},x_{j})$.666Han
(1987) actually considers a slightly different rank property. Theorem H.1 in
the supplementary material to this paper shows that $\beta_{0}$ uniquely
satisfies (2.1) for all $(x_{i},x_{j})\in\mathcal{X}^{2}$ under the same
assumptions as in his theorem. This rank property means that, for any given
pair of $(x_{i},x_{j})$, the probability that $Y_{i}^{\ast}$ is larger than or
equal to $Y_{j}^{\ast}$ is greater than or equal to 1/2 if and only if
$x_{i}^{\prime}\beta_{0}$ is larger than or equal to
$x_{j}^{\prime}\beta_{0}$. Then $\beta_{0}$ is the unique value in $B$ that
satisfies this rank relation for any pair of
$(x_{i},x_{j})\in\mathcal{X}^{2}$. In other words, for any
$\beta\neq\beta_{0}$, there exists at least one pair
$(x_{i},x_{j})\in\mathcal{X}^{2}$ that violates the rank relation (2.1).

In this paper, we suppose that censoring exists and it may be endogenous.
Hence, we cannot always observe $Y_{i}^{\ast}$ and do not have any information
about the censoring mechanism. The censoring variable $C_{i}$ may be
arbitrarily correlated with the observed covariates $X_{i}$ and unobserved
heterogeneity $U_{i}$.

In this situation, we can still construct bounds on the regression parameters
$\beta_{0}$. Let $Y_{1i}\equiv D_{i}Y_{i}^{\ast}+(1-D_{i})(+\infty)$, which is
an outcome variable that takes an arbitrary large value when the primary
outcome is censored. Recall that $Y_{0i}=D_{i}Y_{i}+(1-D_{i})C_{i}$. Then
because $P(Y_{1i}\geq Y_{0j}\mid x_{i},x_{j})\geq P(Y_{i}^{\ast}\geq
Y_{j}^{\ast}\mid x_{i},x_{j})$ holds for all
$(x_{i},x_{j})\in\mathcal{X}^{2}$, the following rank property holds from
(2.1):

$x_{i}^{\prime}\beta_{0}\geq x_{j}^{\prime}\beta_{0}\Rightarrow P(Y_{1i}\geq
Y_{0j}\mid x_{i},x_{j})\geq\frac{1}{2}$ (2.2)

for all $(x_{i},x_{j})\in\mathcal{X}^{2}$. Therefore, defining

$B_{I}\equiv\\{\beta\in B\mid\ \mbox{$x_{i}^{\prime}\beta\geq
x_{j}^{\prime}\beta\Rightarrow P\left(Y_{1i}\geq Y_{0j}\mid
x_{i},x_{j}\right)\geq\frac{1}{2}$ for all
\mbox{$(x_{i},x_{j})\in\mathcal{X}^{2}$}}\\},$

$\beta_{0}$ is contained in $B_{I}$. This set is derived from a worst-case
analysis where we suppose that censored outcomes may take extreme values, $C$
or $+\infty$, for any given value of $x$. This reflects the fact that
concerning each censored outcome, all we know is that it may take any value at
least larger than its censored time.

The following assumption ensures that $B_{I}$ is a proper subset of
$B(=\\{-1,1\\}\times\mathbb{R}^{k-1})$.

######  Assumption 2.4.

Let $\widetilde{\mathcal{X}}^{2}\equiv\\{(x_{i},x_{j})\in\mathcal{X}^{2}\mid
P(Y_{1i}\geq Y_{0j}\mid x_{i},x_{j})<\frac{1}{2}\\}$. Then
$P((X_{i},X_{j})\in\widetilde{\mathcal{X}}^{2})>0$.

The following theorem summarizes the main result in this section.

######  Theorem 2.1.

Under Assumptions 2.1–2.4, $\beta_{0}\in B_{I}\subset B$ a.s.

A proof of this theorem is given in Appendix A. We make several remarks about
this theorem. First, in this theorem, we do not impose a full-support
condition on an element of $X$, in contrast to many works in the
semiparametric literature, as we no longer focus on point
identification.777Magnac and Maurin (2008), Blevins (2011), and Komarova
(2013) discuss the difficulties of justifying the full-support condition in a
number of cases, and provide partial identification results for different
semiparametric models in the absence of the full-support condition. A full-
rank condition on $X$ is necessary (but not sufficient) for $B_{I}$ to be
bounded. Second, as we will see in the following subsection, $B_{I}$ is not a
sharp identified set. However, this set is easy to compute and to construct a
feasible inference method, as we will see in Section 3. The following
subsection shows how the sharp identified set can be characterized, and its
computational difficulty.

####  2.3 Characterization of the Sharp Identified Set

In this section, we illustrate a way to characterize the sharp identified set
using concepts from random set theory. Subsequently, we compare the sharp
identified set with $B_{I}$. This comparison clarifies why $B_{I}$ is not a
sharp set and in which situations $B_{I}$ approaches the sharp set. For the
definitions and notations for random set theory used in this section, see, for
example, Molchanov (2005) or Beresteanu et al. (2012, Appendix A). Throughout
this section, for any variable $A$, we denote by $\tilde{A}$ an independent
copy of $A$.

Using concepts from random set theory, we can characterize the incomplete
information for the latent outcome variable $Y^{\ast}$. For any random
variable $A$, let $A_{x}$ denote a random variable that has the conditional
distribution of $A$ given $X=x$. Then, for a given $x\in\mathcal{X}$, what we
observe for the latent outcome variable in the presence of endogenous
censoring can be expressed as the random set $\mathcal{Y}_{x}$ defined as

$\displaystyle\mathcal{Y}_{x}$ $\displaystyle=$
$\displaystyle\begin{cases}\begin{array}[]{c}\\{Y_{x}^{\ast}\\}\\\
(C_{x},+\infty)\end{array}&\begin{array}[]{c}\mbox{if}\ D_{x}=1\\\
\mbox{otherwise}\end{array}\end{cases},$

where $Y_{x}^{\ast}$ and $C_{x}$ are, respectively, a latent outcome and a
censoring variable given $X=x$; $D_{x}=I\left[Y_{x}^{\ast}\leq C_{x}\right]$
is a censoring indicator given $X=x$. Hence, all the information for the
latent outcome variable can be expressed by stating that
$Y_{x}^{\ast}\in\mbox{Sel}(\mathcal{Y}_{x})$.888For any random set ${\cal Y}$,
a random variable $Y$ is called a measurable selection of ${\cal Y}$ if
$Y\in{\cal Y}$ a.s., and $\mbox{Sel}\left({\cal Y}\right)$ is defined to be
the set of all measurable selections of ${\cal Y}$. See, for example,
Molchanov (2005, Ch. 1) or Beresteanu et al. (2012, Appendix A). Let $B_{0}$
denote the sharp identified set of $\beta_{0}$. Throughout this section, we
suppose that the full-support condition on one element of $X$, the full-rank
condition on $X$, and the continuity of $F_{U}$ hold to ensure the sharp
identification result.999More specifically, we suppose that the following
three conditions hold: (i) $\mathcal{X}$ is not contained in any proper linear
subspace of $\mathbb{R}^{k}$; (ii) for almost every $x_{-1}=(x_{2}\
\ldots,x_{k})$, the distribution of $X_{(1)}$ conditional on
$(X_{(2)},\ldots,X_{(k)})=x_{-1}$ has an everywhere positive density, where
$X_{(m)}$ denotes the $m$-th element of $X$; (iii) $F_{U}$ is a continuous
distribution function. 101010These conditions might not be needed to derive
the sharp identified set; however, to our knowledge, there is no work that
derives the sharp identified set for the nonparametric transformation model in
the absence of these conditions.

Combining the above random set representation with Han’s (1987) point
identification result that $\beta_{0}$ uniquely satisfies the the rank
property (2.1) for all $(x_{i},x_{j})\in\mathcal{X}^{2}$, the sharp identified
set $B_{0}$ is characterized as the set of $\beta$ such that there exists a
family of pairs of selections
$(Y_{x_{i}},\tilde{Y}_{x_{j}})\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}})$
over $(x_{i},x_{j})\in{\cal{X}}^{2}$ that satisfy the following:

$x_{i}^{\prime}\beta\geq x_{j}^{\prime}\beta\Leftrightarrow
P(Y_{x_{i}}\geq\tilde{Y}_{x_{j}})\geq\frac{1}{2}$ (2.3)

for all $(x_{i},x_{j})\in{\cal X}^{2}$, where $\widetilde{\mathcal{Y}}_{x}$ is
the random set of $\tilde{Y}_{x}$. Therefore, $B_{0}$ is characterized as

$B_{0}=\left\\{\beta\in
B\mid\exists\left\\{(Y_{x_{i}},\tilde{Y}_{x_{j}})\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}})\right\\}_{(x_{i},x_{j})\in{\cal{X}}^{2}},\forall(x_{i},x_{j})\in{\cal
X}^{2},\mbox{(\ref{eq:rank inequality_random set}) holds}\right\\}.$ (2.4)

We next look at how the proposed set $B_{I}$ can be characterized by the
random set. Given $x\in\mathcal{X}$, by definition, $Y_{1x}$ and $Y_{0x}$
satisfy (i) $Y_{1x},Y_{0x}\in\mbox{Sel}(\mathcal{Y}_{x})$ a.s. and (ii)
$Y_{0x}\leq Y_{x}\leq Y_{1x}$ a.s. for all
$Y_{x}\in\mbox{Sel}(\mathcal{Y}_{x})$. Thus for any given pair
$(x_{i},x_{j})\in\mathcal{X}^{2}$, the parameter set

$\\{\beta\in B\mid\mbox{$x_{i}^{\prime}\beta\geq
x_{j}^{\prime}\beta\Rightarrow P\left(Y_{1i}\geq Y_{0j}\mid
x_{i},x_{j}\right)\geq\frac{1}{2}$}\\}$

is equivalent to

$\left\\{\beta\in
B\mid\exists\left(Y_{x_{i}},\tilde{Y}_{x_{j}}\right)\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}}),\mbox{(\ref{eq:rank
inequality_random set}) holds}\right\\},$

which is the set of $\beta$ such that for the fixed $(x_{i},x_{j})$, there
exists a pair of selections
$\left(Y_{x_{i}},\tilde{Y}_{x_{j}}\right)\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}})$
that satisfies the inequality (2.3). Therefore, $B_{I}$ is characterized as
the set of $\beta$ such that for any pair $(x_{i},x_{j})\in{\cal{X}}^{2}$,
there exists a pair of selections
$\left(Y_{x_{i}},\tilde{Y}_{x_{j}}\right)\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}})$
that satisfy the inequality (2.3). Formally, $B_{I}$ is characterized as

$B_{I}=\left\\{\beta\in B\mid\forall(x_{i},x_{j})\in{\cal
X}^{2},\exists\left(Y_{x_{i}},\tilde{Y}_{x_{j}}\right)\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}}),\mbox{(\ref{eq:rank
inequality_random set}) holds}\right\\}.$ (2.5)

The difference between (2.4) and (2.5) (i.e., the different orders of
“$\forall(x_{i},x_{j})\in{\cal X}^{2}$” and
“$\exists\left(Y_{x_{i}},\tilde{Y}_{x_{j}}\right)\in\mbox{Sel}(\mathcal{Y}_{x_{i}})\times\mbox{Sel}(\widetilde{\mathcal{Y}}_{x_{j}})$”
in (2.4) and (2.5)) shows that $B_{0}$ is contained in $B_{I}$, but $B_{0}$
does not necessary contain $B_{I}$. Hence $B_{I}$ is not necessarily a sharp
set. Some intuition for the non-sharpness of $B_{I}$ is as follows. Fix a
triple $\left(x_{i},x_{j},x_{k}\right)\in\mathcal{X}^{3}$ and $\beta\in B$
such that $x_{i}^{\prime}\beta\leq x_{j}^{\prime}\beta\leq
x_{k}^{\prime}\beta$. When we check whether $\beta\in B_{I}$ through the rank
property (2.1), in comparing $x_{j}$ and $x_{k}$, we suppose that the latent
outcome variable $Y_{x_{j}}^{\ast}$ takes its smallest value, $C_{x_{j}}$;
whereas, in comparing $x_{j}$ and $x_{i}$, we suppose that $Y_{x_{j}}^{\ast}$
takes its largest value, $+\infty$. However, when we check whether $\beta\in
B_{0}$ through the characterization of (2.4), we compare fixed selections
$Y_{x}\in\mbox{Sel}(\mathcal{Y}_{x})$ over all $x\in\cal{X}$; that is,
$Y_{x_{j}}$ does not change when comparing different $\tilde{Y}_{x_{i}}$ over
$x_{i}\in\cal{X}$. This difference explains why $B_{I}$ is larger than
$B_{0}$.

######  Remark 2.1.

Although we could characterize the sharp set $B_{0}$ as (2.4), it is hard to
compute. When examining whether a certain $\beta$ is contained in $B_{0}$, one
would have to search for the existence of measurable selections
$Y_{x}\in\mbox{Sel}(\mathcal{Y}_{x})$ for all $x\in\cal{X}$ that satisfy the
rank inequality (2.3) for all pairs $(x_{i},x_{j})\in\mathcal{X}^{{}^{2}}$. In
contrast, $B_{I}$ is much easier to compute. For this reason, we focus on
$B_{I}$ in this paper rather than the sharp set.111111 Beresteanu et al.
(2011, 2012) suggest using the support function and Aumann expectation to make
it easy to compute the sharp identified set. However, when we follow this
approach, we still have to search for the measurable selections
$Y_{x}\in\mbox{Sel}(\mathcal{Y}_{x})$, for all $x\in\cal{X}$, to satisfy a
certain equality. Thus this approach does not greatly ease the computation of
the sharp identified set in our setting.

######  Remark 2.2.

There are some situations when $B_{I}$ is close to $B_{0}$. The first case is
when censoring does not occur frequently. If censoring is unlikely to occur
given any $x\in\mathcal{X}$, measurable selections of $\mathcal{Y}_{x}$
correspond to a single measurable selection $Y_{x}^{\ast}$ with high
probability. The second case is when $Y_{x}^{\ast}$ is not censored at small
values for each $x\in\mathcal{X}$; that is, $C_{x}$ takes a large value when
$Y_{x}^{\ast}$ is censored. In this case, for each $x\in\mathcal{X}$, the
random interval $(C_{x},+\infty)$ in the definition of $\mathcal{Y}_{x}$ is
narrow, and hence the difference between (2.4) and (2.5) does not make much
difference between $B_{I}$ and $B_{0}$. In the empirical example of the heart
transplantation study in Section 5, this case corresponds to the case when
each patient is unlikely to drop out of the study at an early stage.

####  2.4 Identification of the Transformation Function

We next develop a lower bound on the transformation function in the presence
of endogenous censoring. Knowing about the transformation function enables us
to infer the type of duration model. We here suppose that $\Lambda$ is
strictly monotonic.

######  Assumption 2.5.

$\Lambda$ is a strictly increasing function.

Let $T\equiv\Lambda^{-1}$. We call $T$ the transformation function when no
confusion arises. As an additional location normalization to fix $T$, we
suppose $T(\tilde{y})=0$ for some specific outcome value $\tilde{y}<\infty$.
We now construct a bound on the normalized $T(y)$ at a particular value of
$y\in\mathbb{R}$.

Construction of the bound is built on the rank property studied by Chen
(2002). In the case of no censoring, provided that Assumptions 2.2–2.3 and 2.5
hold and that the true regression parameters $\beta_{0}$ are given,121212Under
the supposed conditions with the full-support and full-rank conditions on $X$
and the continuity of $F_{U}$, $\beta_{0}$ can be point identified by, for
example, applying Han’s (1987) maximum rank correlation approach. Chen (2002)
shows that $T\left(y\right)$ satisfies the following rank property:

$x_{i}^{\prime}\beta_{0}-x_{j}^{\prime}\beta_{0}\geq T(y)\Rightarrow
P\left(Y_{i}^{\ast}\geq y\mid x_{i}\right)\geq
P\left(Y_{j}^{\ast}\geq\tilde{y}\mid x_{i}\right)$ (2.6)

for all $(x_{i},x_{j})\in{\cal X}^{2}$, where we recall that $\tilde{y}$ is
such that $T\left(\tilde{y}\right)=0$ for the location normalization.
Moreover, supposing further that the full-support and full-rank conditions on
$X$ and the continuity of $F_{U}$ hold, Chen (2002) shows that
$T\left(y\right)$ can be point identified as the minimum value of $t$ that
satisfies the inequality (2.6) with $t$ in place of $T(y)$ for all
$(x_{i},x_{j})\in\mathcal{X}^{2}$. Chen (2002) also provides an inference
method based on this identification result.

In the presence of endogenous censoring, we can construct a lower bound on
$T\left(y\right)$ using a similar idea to that presented in Section 2.2. If
$\beta_{0}$ were given, because $P(Y_{1i}\geq y\mid x_{i})\geq
P(Y_{i}^{\ast}\geq y\mid x_{i})$ and $P(Y_{j}^{\ast}\geq y\mid x_{j})\geq
P(Y_{0j}\geq y\mid x_{j})$ hold for all $(x_{i},x_{j})\in\mathcal{X}^{2}$, it
follows from (2.6) that $T\left(y\right)$ is contained in the following set:

$\left\\{t\in\mathbb{R}:x_{i}^{\prime}\beta_{0}-x_{j}^{\prime}\beta_{0}\geq
t\Rightarrow P\left(Y_{1i}\geq y\mid x_{i}\right)\geq
P\left(Y_{0j}\geq\tilde{y}\mid x_{i}\right)\ \mbox{for all}\
(x_{i},x_{j})\in{\cal X}^{2}\right\\}.$ (2.7)

In the presence of endogenous censoring, we cannot point identify $\beta_{0}$;
instead, we have $B_{I}$ which contains $\beta_{0}$. Thus, letting

$T_{I,\beta}\left(y\right)\equiv\left\\{t\in\mathbb{R}:x_{i}^{\prime}\beta-
x_{j}^{\prime}\beta\geq t\Rightarrow P\left(Y_{1i}\geq y\mid x_{i}\right)\geq
P\left(Y_{0j}\geq\tilde{y}\mid x_{i}\right)\ \mbox{for all}\
(x_{i},x_{j})\in{\cal X}^{2}\right\\}$ (2.8)

and
$T_{B_{I}}\left(y\right)\equiv\left\\{T_{I,\beta}\left(y\right)\mid\beta\in
B_{I}\right\\}$, we have that $T\left(y\right)\in T_{B_{I}}\left(y\right)$
a.s. Note that since any $t>T(y)$ satisfies all the inequality conditions in
(2.8), $T_{I,\beta}(y)$, as well as $T_{B_{I}}$, does not have a finite upper
bound. Hence we can only obtain a lower bound on $T(y)$. Note also that, due
to a similar reason as that discussed in Section 2.3,
$T_{I,\beta_{0}}\left(y\right)$ is not a sharp identified set of
$T\left(y\right)$ even if $\beta_{0}$ is known. Hence,
$\left\\{T_{I,\beta}\left(y\right)\mid\beta\in B_{0}\right\\}$ is not a sharp
identified set even if we had $B_{0}$.

The following theorem formalizes the main result in this subsection.

######  Theorem 2.2.

Under Assumptions 2.2–2.3 and 2.5, $T\left(y\right)\in
T_{B_{I}}\left(y\right)$ holds a.s. for any $y\in\mathbb{R}$.

The proof is provided in Appendix A. For any fixed $y\in\mathbb{R}$,
$T_{B_{I}}(y)$ has a finite lower bound if there exists at least one pair of
$(x_{i},x_{j})\in\mathcal{X}^{2}$ such that $P\left(Y_{1i}\geq y\mid
x_{i}\right)<P\left(Y_{0j}\geq\tilde{y}\mid x_{i}\right)$ holds and that
$x_{i}^{\prime}\beta-x_{j}^{\prime}\beta>-\infty$ holds for all $\beta\in
B_{I}$. The lower bound approaches the true value $T(y)$ as the likelihood of
censoring diminishes.

######  Remark 2.3.

Once we obtain the bounds on $\beta_{0}$ and $T$, we can derive a bound on the
error distribution $F_{U}$. Let $u\in\mathbb{R}$ be fixed. Because
$U=T(Y^{\ast})-X^{\prime}\beta_{0}$ holds and $U$ is independent of $X$,
$F_{U}(u)$ is equivalent to $P(T(Y^{\ast})-X^{\prime}\beta_{0}\leq u\mid X=x)$
for all $x\in\mathcal{X}$. Due to the censoring, we cannot always observe
$Y^{\ast}$, but we instead have

$\displaystyle P(T(Y_{1})-X^{\prime}\beta_{0}\leq u\mid X=x)\leq F_{U}(u)\leq
P(T(Y_{0})-X^{\prime}\beta_{0}\leq u\mid X=x),$

because $T$ is a non-decreasing function. In the presence of endogenous
censoring, we cannot point identify $\beta_{0}$ and $T$, but we instead have
the sets $B_{I}$ for $\beta_{0}$ and $T_{I,\beta}(\cdot)$ for $T(\cdot)$ with
any $\beta\in B_{I}$. Therefore, $F_{U}(u)$ is contained in the following set:

$\displaystyle\\{t\in[0,1]:\ $ $\displaystyle P(T(Y_{1})-X^{\prime}\beta<u\mid
X=x)\leq t\leq P(T(Y_{0})-X^{\prime}\beta<u\mid X=x)$ $\displaystyle\mbox{
holds for some }(\beta,T)\in B_{I}\times T_{I,\beta}\mbox{ and all
}x\in\mathcal{X}\\},$

where $T_{I,\beta}$ denotes a set of all functions $T$ that satisfy $T(y)\in
T_{I,\beta}(y)$ for all $y$.

###  3 Inference

This section provides a statistical inference approach for the regression
parameters in model (1.1) based on the result presented in Section 2.2. We
suggest a method to construct a confidence set that covers the true parameter
value $\beta_{0}$ with a probability greater than or equal to $1-\alpha$ for
$\alpha\in\left(0,1\right)$. Because $B_{I}$ is characterized by conditional
moment inequalities involving U-statistics, we construct the inference method
by extending the inference approach for conditional moment inequality models
proposed by Andrews and Shi (2013) (hereafter AS) to the U-statistics case.
The approach transforms conditional moment inequalities into an infinite
number of unconditional ones, without information loss, to construct a test
statistic. A confidence set is then constructed by inverting the test
statistic and using critical values obtained via moment selection. The
inference method proposed below is applicable to either continuous or discrete
covariates. The inference method is for U-statistics of order two, but the
approach is extendable to U-statistics of greater order with obvious
modifications. A joint inference procedure for $\beta_{0}$ and $T(\cdot)$ is
presented in Appendix C.

####  3.1 Test Statistic and Critical Value

We first construct a test statistic and then describe the inference procedure.
Let

$\displaystyle m\left(W_{i},W_{j},\beta\right)\equiv-\frac{1}{2}+I[Y_{1i}\geq
Y_{0j}]\cdot I[X_{i}^{\prime}\beta\geq
X_{j}^{\prime}\beta]+I[Y_{1j}>Y_{0i}]\cdot
I[X_{j}^{\prime}\beta>X_{i}^{\prime}\beta],$

where $W_{i}\equiv(Y_{1i},Y_{0i},X_{i})$ for $i=1,\ldots,n$. Then $B_{I}$ is a
set of parameter values $\beta$ that satisfy the following conditional moment
inequalities:

$E_{P}\left[m\left(W_{i},W_{j},\beta\right)\mid x_{i},x_{j}\right]\geq 0\
\mbox{for a.e.}\ (x_{i},x_{j})\in\mathcal{X}^{2},$ (3.1)

where $E_{P}[\cdot\mid x_{i},x_{j}]$ denotes the conditional expectation under
the distribution $P$ given $(X_{i},X_{j})=(x_{i},x_{j})$. To transform all of
the conditional moment inequalities (3.1) into unconditional ones without loss
of information, we adopt AS’s instrumental functions approach. We here suppose
that the first $p$ ($p\leq k$) elements of $X$ are continuous variables or
have infinite support and that the remaining elements of $X$ have finite
support.131313We can also allow all covariates to have finite or infinite
support with modifications. Let $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ denote
the supports of $(X_{i,1},\ldots,X_{i,p})$ and $(X_{i,p+1},\ldots,X_{i,k})$,
respectively, where $X_{i,j}$ denotes the $j$-th element of $X_{i}$. Without
loss of generality, we also suppose that $(X_{i,1},\ldots,X_{i,p})$ is
transformed via a one-to-one mapping so that each of its elements lies in
$[0,1]$ (i.e., $\mathcal{X}_{1}\subseteq\left[0,1\right]^{p}$). 141414Let
$X_{i}^{(1)}\equiv(X_{i,1},\ldots,X_{i,p})$. Following AS, the vector of
transformed covariates in $X_{i}^{(1)}$ may be
$\tilde{X}_{i}^{(1)}\equiv\Phi\left(\hat{\Sigma}_{X^{(1)},n}^{-1/2}\left(X_{i}^{(1)}-\bar{X}_{n}^{(1)}\right)\right)$
where $\bar{X}_{n}^{(1)}\equiv n^{-1}\sum_{i=1}^{n}X_{i}^{(1)}$,
$\hat{\Sigma}_{X^{(1)},n}\equiv
n^{-1}\sum_{i=1}^{n}\left(X_{i}^{(1)}-\bar{X}_{n}^{(1)}\right)\left(X_{i}^{(1)}-\bar{X}_{n}^{(1)}\right)^{\prime}$,
and
$\Phi\left(x^{(1)}\right)\equiv\left(\Phi\left(x_{1}\right),\ldots,\Phi\left(x_{p}\right)\right)^{\prime}$,
where $\Phi\left(\cdot\right)$ denotes the standard normal cumulative
distribution function and $x^{(1)}=\left(x_{1},\ldots,x_{p}\right)^{\prime}$.

The set of instrumental functions that we consider is of the following form:

$\displaystyle\mathcal{G}\equiv\left\\{g(x_{i},x_{j})=I\left[(x_{i},x_{j})\in
J\right]\ \mbox{for}\ J\in\mathcal{J}\right\\},$

where

$\displaystyle\mathcal{J}\equiv$
$\displaystyle\left\\{J_{(a,b),(\tilde{a},\tilde{b}),r}=\left(\vartimes_{u=1}^{p}\left(\frac{a_{u}-1}{2r},\frac{a_{u}}{2r}\right]\times\\{b\\}\right)\times\left(\vartimes_{u=1}^{p}\left(\frac{\tilde{a}_{u}-1}{2r},\frac{\tilde{a}_{u}}{2r}\right]\times\\{\tilde{b}\\}\right):\right.$
$\displaystyle\ \left.a=(a_{1},\ldots,a_{p}),\
\tilde{a}=(\tilde{a}_{1},\ldots,\tilde{a}_{p}),\
(a_{u},\tilde{a}_{u})\in\left\\{1,2,\ldots,2r\right\\}^{2}\right.$
$\displaystyle\left.\mbox{ for }u=1,\ldots,p\mbox{ and }r=1,2,\ldots,\mbox{
and }(b,\tilde{b})\in\mathcal{X}_{2}^{2}\right\\}.$

This set of instrumental functions transforms the conditional moment
inequalities (3.1) into infinitely many unconditional ones without loss of
information. Accordingly, under Assumptions 2.2–2.3, $B_{I}$ is equivalent to

$\left\\{\beta\in B:\ E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\geq 0\
\mbox{for all}\ g\in\mathcal{G}\right\\},$

where $m(W_{i},W_{j},\beta,g)\equiv m\left(W_{i},W_{j},\beta\right)\cdot
g(x_{i},x_{j})$ for $g\in{\cal G}$. We formalize this result as Lemma B.2 in
Appendix B. Other kinds of instrumental functions introduced in AS are
applicable with modifications.

We define the sample moment function and sample variance function of
$m(W_{i},W_{j},\beta,g)$, respectively, as

$\displaystyle\bar{m}_{n}\left(\beta,g\right)\equiv\frac{1}{n(n-1)}\sum_{i\neq
j}m(W_{i},W_{j},\beta,g)$

and

$\displaystyle\hat{\sigma}_{n}^{2}\left(\beta,g\right)\equiv$
$\displaystyle\left\\{\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq k}m(W_{i},W_{j},\beta,g)m\left(W_{i},W_{k},\beta,g\right)\right.$
$\displaystyle\left.-\left(\frac{1}{n(n-1)}\sum_{i\neq
j}m(W_{i},W_{j},\beta,g)\right)^{2}\right\\}.$

Note that $\bar{m}_{n}\left(\beta,g\right)$ and
$\hat{\sigma}_{n}^{2}\left(\beta,g\right)$ are U-statistics of orders two and
three, respectively. Because $\bar{m}_{n}\left(\beta,g\right)$ is a non-
degenerate U-statistic of order two, the asymptotic variance of
$\sqrt{n}\bar{m}_{n}\left(\beta,g\right)$ is
$\mbox{Var}_{P}\left(E_{P}\left[m(W_{i},W_{j},\beta,g)\mid
W_{i}\right]\right)$, which is equivalent to151515For the variance of
U-statistics, see, for example, van der Vaart (1998, Ch. 12).

$E_{P}\left[m(W_{i},W_{j},\beta,g)m\left(W_{i},W_{k},\beta,g\right)\right]-\left(E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\right)^{2}.$

Thus, $\hat{\sigma}_{n}^{2}\left(\beta,g\right)$ is a consistent estimator of
the asymptotic variance of $\sqrt{n}\bar{m}_{n}\left(\beta,g\right)$. However,
in practice, $\hat{\sigma}_{n}^{2}\left(\beta,g\right)$ could be zero for some
$g\in\mathcal{G}$; as such we use the modification proposed by AS for
$\hat{\sigma}_{n}^{2}\left(\beta,g\right)$. The modified version of
$\hat{\sigma}_{n}^{2}\left(\beta,g\right)$ is

$\displaystyle\bar{\sigma}_{n}^{2}\left(\beta,g\right)\equiv\hat{\sigma}_{n}^{2}\left(\beta,g\right)+\epsilon\hat{\sigma}_{n}^{2},$

where $\hat{\sigma}_{n}^{2}=\hat{\sigma}_{n}^{2}\left(\beta,1\right)$, which
is a consistent estimator of

$\displaystyle\sigma_{P}^{2}\left(\beta\right)\equiv
E_{P}\left[m\left(W_{i},W_{j},\beta\right)m\left(W_{i},W_{k},\beta\right)\right]-\left(E_{P}\left[m\left(W_{i},W_{j},\beta\right)\right]\right)^{2},$

and $\epsilon$ is a regularization parameter that takes some fixed positive
value. In the simulation studies in Section 4 and empirical application in
Section 5, we use $\epsilon=0.0001$ and $\epsilon=0.001$.

Then, letting $g_{(a,b),(\tilde{a},\tilde{b}),r}(x_{i},x_{j})\equiv
1\left[(x_{i},x_{j})\in J_{(a,b),(\tilde{a},\tilde{b}),r}\right]$, the test
statistic at $\beta$ takes the form

$\displaystyle
T_{n}(\beta)\equiv\sum_{r=1}^{\infty}\left(r^{2}+100\right)^{-1}\sum_{\left(a,\tilde{a}\right)\in\left\\{1,\cdots,2r\right\\}^{2p}}\sum_{(b,\tilde{b})\in\mathcal{X}_{2}^{2}}\frac{1}{\left((2r)^{p}\cdot|\mathcal{X}_{2}|\right)^{2}}\left[\frac{n^{\frac{1}{2}}\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2},$

where $\left[x\right]_{-}=-x$ if $x<0$ and $\left[x\right]_{-}=0$ if $x\geq
0$. $\left((2r)^{p}\cdot(|\mathcal{X}|)^{k-p}\right)^{2}$ corresponds to the
number of instrumental functions $g_{(a,b),(\tilde{a},\tilde{b}),r}$ for a
fixed $r$. This test statistic is a version of AS’s test statistic that is
extended to U-statistics of order two. Here the inner summation is taken over
two set of indices, $(a,b)$ and $(\tilde{a},\tilde{b})$. In the
implementation, we instead use an approximate test statistic at $\beta$:

$\displaystyle
T_{n,R}(\beta)\equiv\sum_{r=1}^{R}\left(r^{2}+100\right)^{-1}\sum_{\left(a,\tilde{a}\right)\in\left\\{1,\cdots,2r\right\\}^{2p}}\sum_{(b,\tilde{b})\in\mathcal{X}_{2}^{2}}\frac{1}{\left((2r)^{p}\cdot|\mathcal{X}_{2}|\right)^{2}}\left[\frac{n^{\frac{1}{2}}\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2},$

where $R$ is some truncation integer chosen by the researcher.

To compute a critical value for $T_{n,R}(\beta)$, we propose using an
asymptotic approximation version of the critical value. This is a simulated
quantile of

$\displaystyle T_{n,R}^{Asy}(\beta)\equiv$
$\displaystyle\sum_{r=1}^{R}\left(r^{2}+100\right)^{-1}\sum_{(a,\tilde{a})\in\\{1,\cdots,2r\\}^{2p}}\sum_{(b,\tilde{b})\in\mathcal{X}_{2}^{2}}\left\\{\frac{1}{\left((2r)^{p}\cdot|\mathcal{X}_{2}|\right)^{2}}\right.$
$\displaystyle\left.\times\left[\frac{v_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)+\varphi_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2}\right\\},$

where $\left(v_{n}\left(\beta,g\right)\right)_{g\in{\cal G}}$ is a zero mean
Gaussian process with a covariance kernel evaluated by

$\displaystyle\hat{h}_{2}\left(\beta,g,g^{\ast}\right)\equiv$
$\displaystyle\left\\{\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq k}m(W_{i},W_{j},\beta,g)m(W_{i},W_{k},\beta,g^{\ast})\right.$
$\displaystyle\left.-\left[\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}m(W_{i},W_{j},\beta,g)\right]\cdot\left[\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}m\left(W_{i},W_{j},\beta,g^{\ast}\right)\right]\right\\}.$

In the expression of $T_{n,R}^{Asy}(\beta)$,
$\left(v_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)\right)_{(a,b),(\tilde{a},\tilde{b}),r}$
approximates the asymptotic distribution of

$\left(n^{\frac{1}{2}}\left[\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)-E_{P}\left[m\left(W_{i},W_{j},\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)\right]\right]\right)_{(a,b),(\tilde{a},\tilde{b}),r}.$

$\varphi_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)$ is a
generalized moment selection (GMS) function to select binding moment
restrictions and is given by

$\varphi_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)\equiv\hat{\sigma}_{n}^{2}B_{n}I\left[\kappa_{n}^{-1}n^{\frac{1}{2}}\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)/\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)>1\right],$

where $B_{n}$ and $\kappa_{n}$ are two tuning parameters that should satisfy
$B_{n}\rightarrow\infty$, $\kappa_{n}\rightarrow\infty$, and
$\kappa_{n}/n^{1/2}\rightarrow 0$ as $n\rightarrow\infty$ a.s. In Sections 4
and 5, we use
$B_{n}=\left(0.8\ln\left(n\right)/\ln\ln\left(n\right)\right)^{\frac{1}{2}}$
and $\kappa_{n}=\left(\left(1-\hat{p}_{1-D}^{1/3}\right)^{2/5}\times
0.6\ln(n)\right)^{\frac{1}{2}}$, where $\hat{p}_{1-D}\equiv
n^{-1}\sum_{i}^{n}\left(1-D_{i}\right)$ is the sample censoring rate.161616The
values for $\kappa_{n}$ and $B_{n}$ are different from the values recommend by
AS, which are $\kappa_{n}=\left(0.3\ln(n)\right)^{\frac{1}{2}}$ and
$B_{n}=\left(0.4\ln\left(n\right)/\ln\ln\left(n\right)\right)^{\frac{1}{2}}$,
respectively. This value of $\kappa_{n}$ decreases with the sample censoring
rate. The following assumption summarizes the requirements for the tuning
parameters in the GMS function.

######  Assumption 3.1.

The tuning parameters $\left(B_{n},\kappa_{n}\right)$ satisfy
$B_{n}\rightarrow\infty$, $\kappa_{n}\rightarrow\infty$, and
$\kappa_{n}/n^{1/2}\rightarrow 0$ as $n\rightarrow\infty$ a.s.

For a significance level of $\alpha<1/2$, the critical value is set to be the
$1-\alpha+\eta$ simulated quantile of $T_{n,R}^{Asy}(\beta)$, where $\eta$ is
an arbitrarily small positive value (e.g., $10^{-6}$ following AS). Letting
$\hat{c}_{n,\eta,1-\alpha}(\beta)$ be the $1-\alpha+\eta$ quantile of
$T_{n,R}^{Asy}(\beta)$, a nominal level $1-\alpha$ confidence set is computed
as

$\widehat{CS}_{n,\eta,1-\alpha}\equiv\left\\{\beta\in
B:T_{n,R}(\beta)\leq\hat{c}_{n,\eta,1-\alpha}(\beta)\right\\}.$

######  Remark 3.1.

When all the covariates have finite support, the above procedure is applicable
using the test statistic

$\displaystyle
T_{n}(\beta)=\sum_{(b,\tilde{b})\in\mathcal{X}^{2}}\frac{1}{|\mathcal{X}|^{2}}\left[\frac{n^{\frac{1}{2}}\bar{m}_{n}\left(\beta,g_{b,\tilde{b}}\right)}{\bar{\sigma}_{n}\left(\beta,g_{b,\tilde{b}}\right)}\right]_{-}^{2}$

and a simulated quantile of

$\displaystyle
T_{n}^{Asy}(\beta)=\sum_{(b,\tilde{b})\in\mathcal{X}^{2}}\frac{1}{|\mathcal{X}|^{2}}\left[\frac{v_{n}\left(\beta,g_{b,\tilde{b}}\right)+\varphi_{n}\left(\beta,g_{b,\tilde{b}}\right)}{\bar{\sigma}_{n}\left(\beta,g_{b,\tilde{b}}\right)}\right]_{-}^{2},$

where $g_{b,\tilde{b}}(x_{i},x_{j})=I[(x_{i},x_{j})=(b,\tilde{b})]$. In this
case, inference is free from the tuning parameter $R$.

Regardless of whether some covariates have finite support or not, the above
procedure with $\mathcal{J}$ replaced by

$\displaystyle\mathcal{J}^{\prime}\equiv$
$\displaystyle\left\\{J_{a,\tilde{a},r}=\left(\vartimes_{u=1}^{k}\left(\frac{a_{u}-1}{2r},\frac{a_{u}}{2r}\right]\right)\times\left(\vartimes_{u=1}^{k}\left(\frac{\tilde{a}_{u}-1}{2r},\frac{\tilde{a}_{u}}{2r}\right]\right):\right.$
$\displaystyle\ \left.a=(a_{1},\ldots,a_{k}),\
\tilde{a}=(\tilde{a}_{1},\ldots,\tilde{a}_{k}),\
(a_{u},\tilde{a}_{u})\in\left\\{1,2,\ldots,2r\right\\}^{2}\right.$
$\displaystyle\left.\mbox{ for }u=1,\ldots,k\mbox{ and
}r=1,2,\ldots\right\\}.$

is also applicable when $\mathcal{X}\subseteq[0,1]^{k}$. When
$\mathcal{X}_{2}$ does not contain too many points, the original inference
procedure using $\mathcal{J}$ is computationally more efficient. In contrast,
when $\mathcal{X}_{2}$ contains many points, the inference procedure using
$\mathcal{J}^{\prime}$ with a moderate number of $R$ incurs a lower
computational cost.

Note again that the inference approach presented above is for U-statistics of
order two, but can be extended to U-statistics of a greater order with some
modification. Namely, the class of instrumental functions, the moment
function, and the sample variance function need to be modified for higher
order; the inner double summations in the test statistics need to be replaced
by more summations.

####  3.2 Asymptotic Size and Power Properties

This subsection provides the uniform asymptotic size and power properties of
the inference method. Let ${\cal Q}$ be the collection of all pairs of the
regression parameters in $B$ and distributions, $(\beta,P)$, such that the
conditional moment inequality condition (3.1) holds, $\\{W_{i}:i\geq 1\\}$ are
i.i.d. under $P$, and the support of $X$ is contained in
$\mathcal{X}(=\mathcal{X}_{1}\times\mathcal{X}_{2})$. Define

$\displaystyle h_{2,P}(\beta,g,g^{\ast})\equiv$ $\displaystyle
E_{P}\left[E_{P}\left[m(W_{i},W_{j},\beta,g)m(W_{i},W_{k},\beta,g^{\ast})\mid
W_{i}\right]\right]$ $\displaystyle-
E_{P}\left[m(W_{i},W_{j},\beta,g)\right]E_{P}\left[m\left(W_{i},W_{j},\beta,g^{\ast}\right)\right],$
(3.2)

which is the covariance kernel between $m(W_{i},W_{j},\beta,g)$ and
$m(W_{i},W_{j},\beta,g^{\ast})$ under the distribution $P$. Let ${\cal
H}_{2}\equiv\\{h_{2,P}(\beta,\cdot,\cdot):(\beta,P)\in\mathcal{Q}\\}$ be the
collection of all possible covariance kernel functions on ${\cal G}\times{\cal
G}$. For any $\beta\in B$ and the true distribution $P$, let

$\displaystyle\mathcal{X}_{P}^{2}(\beta)\equiv\\{(x_{i},x_{j})\in\mathcal{X}^{2}:x_{i}^{\prime}\beta\geq
x_{j}^{\prime}\beta\Rightarrow P(Y_{1i}\geq Y_{0j})<1/2\\}.$

The following theorem gives the uniform size and power properties of the
proposed inference method.

######  Theorem 3.1.

Suppose that $R=\infty$ and $\alpha<1/2$.
(a) For every compact subset ${\cal H}_{2,cpt}$ of ${\cal H}_{2}$, the
confidence set $\widehat{CS}_{n,\eta,1-\alpha}$ satisfies

$\underset{\eta\rightarrow
0}{\lim}\liminf_{n\rightarrow\infty}\inf_{\left\\{\left(\beta,P\right)\in{\cal
Q}:h_{2,P}(\beta,\cdot,\cdot)\in{\cal
H}_{2,cpt}\right\\}}P\left(\beta\in\widehat{CS}_{n,\eta,1-\alpha}\right)=1-\alpha.$

(b) Suppose further that Assumptions 2.1–2.3 and 3.1 hold for the true
distribution $P$, and let $\tilde{\beta}\in B$ be a vector of regression
parameters such that
$P((X_{i},X_{j})\in\mathcal{X}_{P}^{2}(\tilde{\beta}))>0$. Then
$\lim_{n\rightarrow\infty}P(\tilde{\beta}\in\widehat{CS}_{n,\eta,1-\alpha})=0$.

A proof of this theorem is provided in Appendix B. Theorem 3.1 (a) states that
the proposed confidence set is asymptotically conservative, which corresponds
to Theorem 2(b) of AS. The uniformity in the statement enables the asymptotic
result to provide a good finite sample approximation, which is well discussed
in AS. Theorem 3.1 (b) states that the test is consistent against a fixed
alternative.

###  4 Simulation Studies

This section presents numerical examples and Monte Carlo simulation results.
The numerical examples show how $B_{I}$ and $T_{B_{I}}$ vary with the degree
of censoring and the support of covariates. The Monte Carlo simulations show
the finite sample performance of the proposed inference method for the
regression parameters and demonstrate how it varies with the choice of the
tuning parameters. Some additional numerical example and Monte Carlo
simulation studies are presented in the supplementary material to this paper.

####  4.1 Numerical Examples

This subsection illustrates how $B_{I}$ and $T_{B_{I}}$ vary depending on the
degree of censoring and the support of covariates. We consider three MPH
models (Models 1–3) with endogenous censoring that have the following form:

$\displaystyle\log Y^{\ast}$ $\displaystyle=$
$\displaystyle\beta_{1}X_{1}+\beta_{2}X_{2}+\log U+\log V,$ $\displaystyle\log
C$ $\displaystyle=$
$\displaystyle\alpha_{0}+\left(\gamma_{0}+\gamma_{1}X_{1}+\gamma_{2}X_{2}\right)\times\log
U+\log W.$

In all the models, we set
$\left(\beta_{1},\beta_{2}\right)=\left(0.5,1.5\right)$ and
$\left(\gamma_{0},\gamma_{1},\gamma_{2}\right)=\left(-0.5,0.5,-1\right)$. In
Models 1–3, we set $\alpha_{0}=+\infty$, $3$, and $1.6$, respectively. In
Model 1, there is no censoring; in Models 2 and 3, there is censoring
correlated with the covariates and unobserved heterogeneity. The outcome is
more likely to be censored in Model 3 than in Model 2. In all the models, $U$,
$V$, and $W$ have independent unit exponential distributions, and $X_{2}$
takes values in $\left\\{0,1\right\\}$. Regarding $X_{1}$, we consider three
cases; $X_{1}$ takes values in (i) $\left\\{-2.5,-2.0,\ldots,2.5\right\\}$,
(ii) $\left\\{-5,-2.5,\ldots,5\right\\}$, or (iii)
$\left\\{-5,-4.8,\ldots,5\right\\}$. In these data generating processes
(DGPs), the censoring is endogenous because $Y^{\ast}$ is correlated with $C$
even conditional on $X_{1}$ and $X_{2}$, which occurs due to the presence of
$U$ in both the equations for $\log Y^{\ast}$ and $\log C$.

In the computation of $B_{I}$ and $T_{B_{I}}$, we set $|\beta_{1}|=1$ for the
scale normalization and $T(\tilde{y})=0$ for the location normalization, where
we set $\tilde{y}=0.77$, which is the median of $Y^{\ast}$ when $X_{1}$ is
independently distributed as $N(0,2)$ and $X_{2}$ is independently and
uniformly distributed over $\\{0,1\\}$. This joint distribution will be used
in the Monte Carlo simulation in the following subsection. After the scale and
location normalizations, the true normalized values of $\beta_{2}$ and $T(y)$
are $3$ and $T_{0}(y):=2(\log y-\log 0.77)$, respectively.

Given the DGPs described above, $B_{I}$ is the set of regression parameter
values that satisfy the conditional moment inequality (3.1) for all pair of
values of $\left(X_{1},X_{2}\right)$, and $T_{B_{I}}(y)$ for a fixed
$y\in\mathbb{R}$ is the set of values $t$ that satisfy the inequality in (2.8)
for some $\beta\in B_{I}$ and all pair of values of $(X_{1},X_{2})$. We
numerically obtain $B_{I}$ and $T_{B_{I}}(y)$ by simulating the distributions
of $\log Y^{\ast}$ and $\log C$ for each pair of values of
$\left(X_{1},X_{2}\right)$ using 20,000 random draws from each DGP.

Table 1 presents the numerical results for $B_{I}$. Each cell in the table
shows the interval of $\beta_{2}$ obtained by the projection of the computed
$B_{I}$ for each model and each support of $X_{1}$. As expected, the interval
shrinks as the censoring rate decreases or the support becomes wider or finer.
In each model, expanding the support from (i) to (ii) does not shrink the
lower or upper bounds on $\beta_{2}$.

Table 1: Computed $B_{I}$ for Models 1–3 and Supports (i)–(iii)

Model / Support of $X_{1}$ | Support (i) | Support (ii) | Support (iii)
---|---|---|---
Model 1 | $[2.51,3.49]$ | $[2.51,3.49]$ | $[2.81,3.19]$
Model 2 | $[2.00,4.00]$ | $[2.00,3.49]$ | $[2.21,3.49]$
Model 3 | $[1.50,5.00]$ | $[1.50,3.99]$ | $[1.80,3.79]$

  * Note: Each cell shows the interval of $\beta_{2}$ obtained by the projection of the computed $B_{I}$.

Figure 1 illustrates the computed $T_{B_{I}}$ for each model and each support
of $X_{1}$, along with the true normalized transformation function $T_{0}$. As
mentioned in Section 2.4, $T_{B_{I}}$ is not bounded from above. The result
shows that as the censoring rate decreases or the support of $X_{1}$ becomes
wider or finer, the lower bound on $T_{B_{I}}$ becomes closer to the true
transformation function. The lower bound also becomes smoother as the support
of $X_{1}$ becomes finer.

The supplementary material to this paper gives two additional numerical
examples for $B_{I}$. In the first example, we compute $B_{I}$ in models with
three covariates, where the results in Table 1 are extended to two-dimensional
figures. In the second example, using slightly different models and supposing
that the transformation model is parametrically specified (i.e., the
functional forms of $\Lambda$ and $F_{U}$ are known), we compute the sharp
identified set of $\beta_{0}$, based on the result in Szydłowski (2019), and
compare it with $B_{I}$.

Figure 1: Computed $T_{B_{I}}$ for Models 1-3 and Supports (i)-(iii)

(a-1) Model 1 & Support (i) (a-2) Model 1 & Support (ii) (a-3) Model 1 &
Support (iii)

---
(b-1) Model 2 & Support (i) (b-2) Model 2 & Support (ii) (b-3) Model 3 &
Support (iii)

(c-1) Model 3 & Support (i) (c-2) Model 3 & Support (ii) (c-3) Model 3 &
Support (iii)

  * Notes: The shaded area in each figure is the computed $T_{B_{I}}(y)$ for $0.1\leq y\leq 10$. The solid line in each figure is the graph of the true normalized transformation function $T_{0}(y)=2(\log y-\log 0.77)$.

####  4.2 Monte Carlo Experiments

This subsection presents Monte Carlo simulation results to evaluate the finite
sample size and power properties of the inference method for the regression
parameters presented in Section 3. We here use two DGPs (DGP1 and DGP2). In
DGP1 and DGP2, the data are derived from Models 2 and 3, respectively, where
$X_{1}$ is distributed as $N(0,2)$; $X_{2}$ is uniformly distributed over
$\left\\{0,1\right\\}$; $U$, $V$, and $W$ have independent unit exponential
distributions. The censoring rates in DGP1 and DGP2 are approximately $16\%$
and $30\%$, respectively.

For the Monte Carlo experiments, 500 samples are randomly drawn with sample
sizes $n=100$, $250$, and $500$. Based on the inference method, we conduct a
test of $H_{0}:$ (3.1) holds against $H_{1}:$ (3.1) is violated at each value
of
$\left(\beta_{1},\beta_{2}\right)\in\left\\{1\right\\}\times\left\\{-1,-0.5,\ldots,9\right\\}$.
The true value of the normalized ${\beta}_{2}$ is 3. Critical values are
simulated using 1,000 repetitions for the significance level $\alpha=0.05$. As
a baseline case, we set the tuning parameters in the test statistics and GMS
function to $R=5$, $\epsilon=0.0001$,
$B_{n}=B_{n}^{bc}\equiv\left(0.8\ln\left(n\right)/\ln\ln\left(n\right)\right)^{\frac{1}{2}}$,
and
$\kappa_{n}=\kappa_{n}^{bc}\equiv\left(\left(1-\hat{p}_{1-D}^{1/3}\right)^{2/5}\times
0.6\ln(n)\right)^{\frac{1}{2}}$. We set $\eta=10^{-6}$ throughout this and the
following sections. We do not assume that the researcher knows the exact
distribution of $X_{1}$; hence, we transform $X_{1}$ into $\tilde{X}_{1}$ as
described in Section 3.1 and use $\tilde{X}_{1}$ instead of $X_{1}$. We assume
that the researcher knows the support of $X_{2}$. Thus we apply the inference
method presented in Section 3 with
$\mathcal{X}=\mathcal{X}_{1}\times\mathcal{X}_{2}=[0,1]\times\\{0,1\\}$.

Figure 2 shows the graphs of rejection frequencies for DGP1 and DGP2 given the
baseline case of the tuning parameters with $\epsilon=0.0001$ or
$\epsilon=0.001$. All the rejection frequencies at the true value of
normalized $\beta_{2}$ are close to or lower than the nominal size
$\alpha=0.05$ (though the rejection frequency substantially exceeds 0.05 in
DGP1 with $n=100$ and $\epsilon=0.0001$). The rejection frequencies are also
close to or lower than 0.05 in the corresponding intervals computed in Table 1
with Support (iii). When $\epsilon=0.001$, the test has less power but is more
likely to be conservative even in small sample. The power of the test is
higher at small values of $\beta_{2}$ than at large values.

Table 2 shows the rejection frequencies at the true parameter value and at
$\left(\beta_{1},\beta_{2}\right)=\left(1,0\right)$ for several choices of the
tuning parameters in DGP1 and DGP2. The point
$\left(\beta_{1},\beta_{2}\right)=\left(1,0\right)$ is not contained in
$B_{I}$, as seen from the results of the numerical examples. Table 2 shows the
degree of sensitivity of the test to variation in the sample size $n$, the
choice of the truncation integer $R$ in the approximate test statistic, the
value of $\epsilon$ for the modified variance estimator
$\bar{\sigma}_{n}^{2}\left(\beta,g\right)$, and the choice of $B_{n}$ and
$\kappa_{n}$ in the GMS function. In the case of the last row of Table 2,
$\kappa_{n}$ does not depend on the sample censoring rate $\hat{p}_{1-D}$. The
results show that there is some sensitivity to the sample size, the choices of
$\kappa_{n}$ and $R$, and the value of $\epsilon$. In particular, the
sensitivity to the choice of $\kappa_{n}$ is high. A small value of $\epsilon$
leads to the test having high power, but can lead to the test having incorrect
size in small samples.

The supplementary material contains two additional Monte Carlo simulation
results. One examines inference on the regression parameters in models with
three covariates. The other evaluates the performance of a joint inference
method for the regression parameters and transformation function introduced in
Appendix C.

Figure 2: Rejection Frequencies in DGP1 and DGP2

(a) DGP1 & $\epsilon=0.0001$ (b) DGP1 & $\epsilon=0.001$

---
(c) DGP2 & $\epsilon=0.0001$ (d) DGP2 & $\epsilon=0.001$

  * Notes: The dashed, dotted, dash-dotted lines in each figure indicate rejection frequencies for sample sizes of $100$, $250$, and $500$, respectively. The solid horizontal line in each figure indicates the significance level of 0.05.

Table 2: Rejection Frequencies for the Inference Method: Variation in Sample
Size and Choice of the Tuning Parameters

| DGP1 |  | DGP2
---|---|---|---
Case | $\left(\beta_{1},\beta_{2}\right)=\left(1,3\right)$ | $\left(\beta_{1},\beta_{2}\right)=\left(1,0\right)$ |  | $\left(\beta_{1},\beta_{2}\right)=\left(1,3\right)$ | $\left(\beta_{1},\beta_{2}\right)=\left(1,0\right)$
Baseline Case | 0.054 | 0.786 |  | 0.022 | 0.297
$((n,R,B_{n},\kappa_{n},\epsilon)=(250,5,B_{n}^{bc},\kappa_{n}^{bc},0.0001))$ |
$\epsilon=0.001$ | 0.042 | 0.692 |  | 0.018 | 0.244
$\epsilon=0.00001$ | 0.061 | 0.805 |  | 0.022 | 0.046
$R=3$ | 0.052 | 0.594 |  | 0.046 | 0.192
$R=7$ | 0.033 | 0.824 |  | 0.019 | 0.322
$n=100,\epsilon=0.001$ | 0.064 | 0.345 |  | 0.038 | 0.150
$n=100,\epsilon=0.0001$ | 0.082 | 0.456 |  | 0.056 | 0.205
$n=100,\epsilon=0.00001$ | 0.089 | 0.489 |  | 0.058 | 0.227
$n=500,\epsilon=0.001$ | 0.030 | 0.948 |  | 0.012 | 0.380
$n=500,\epsilon=0.0001$ | 0.030 | 0.971 |  | 0.016 | 0.471
$n=500,\epsilon=0.00001$ | 0.032 | 0.970 |  | 0.018 | 0.492
$n=1000,\epsilon=0.001$ | 0.021 | 1.000 |  | 0.004 | 0.543
$n=1000,\epsilon=0.0001$ | 0.031 | 1.000 |  | 0.010 | 0.675
$n=1000,\epsilon=0.00001$ | 0.037 | 1.000 |  | 0.012 | 0.694
$B_{n}=B_{n}^{bc}/2$ | 0.053 | 0.786 |  | 0.021 | 0.297
$B_{n}=2B_{n}^{bc}$ | 0.054 | 0.786 |  | 0.022 | 0.297
$\kappa_{n}=\kappa_{n}^{bc}/2$ | 0.293 | 0.944 |  | 0.148 | 0.552
$\kappa_{n}=2\kappa_{n}^{bc}$ | 0.000 | 0.243 |  | 0.000 | 0.045
$\left(B_{n},\kappa_{n}\right)=1/2\left(B_{n}^{bc},\kappa_{n}^{bc}\right)$ | 0.288 | 0.944 |  | 0.140 | 0.550
$\left(B_{n},\kappa_{n}\right)=2\left(B_{n}^{bc},\kappa_{n}^{bc}\right)$ | 0.000 | 0.243 |  | 0.000 | 0.045
$\kappa_{n}=\left(0.6\ln(n)\right)^{\frac{1}{2}}$ | 0.026 | 0.683 |  | 0.010 | 0.217

###  5 Empirical Illustration

We apply the proposed inference method to evaluate the effect of heart
transplants on patients’ survival duration using the Stanford Heart Transplant
Data taken from Kalbfleisch and Prentice (1980). The data set consists of
survival times (in days) of 103 patients; an indicator of censoring, which
takes the value one if the patient is dead (uncensored) or zero if the patient
is censored; an indicator of receiving a heart transplant, which takes the
value one if the patient receives a heart transplant or zero otherwise; and
the age (in years) of patients at the time of acceptance into the program.
Among the 103 patients, $27\%$ (28 patients) are censored due to attrition or
administrative censoring. The censoring rates for the treated (receive a
transplant) and untreated (do not receive a transplant) groups are $35\%$ and
$22\%$, respectively.

We consider the following censored transformation model,

$\displaystyle
T\left(Y_{0i}\right)=\min\left\\{\beta_{cons}+X_{i,age}\beta_{age}+X_{i,treat}\beta_{treat}+U_{i},T\left(C_{i}\right)\right\\},$

where, for each patient $i$, $Y_{0i}$ is the observed survival time,
$X_{i,age}$ is the age, $X_{i,treat}$ is the transplant indicator, $U_{i}$ is
unobserved heterogeneity, $C_{i}$ is the censoring time, and $T$ is a strictly
increasing function. Applying the proposed method, we allow the censoring to
be arbitrarily correlated with the patient’s age and unobserved heterogeneity.
Furthermore, we do not specify the transformation function or the distribution
function of the patient’s unobserved heterogeneity. For scale normalization,
we set $\left|\beta_{age}\right|=1$; for location normalization, we set
$\beta_{cons}=0$ and $T(\tilde{y})=0$ with $\tilde{y}=90$ being the median of
$Y_{0}$ in the sample. Our interest is in the normalized regression parameter
$\beta_{treat}$ and the normalized transformation function $T$. We compare the
proposed method with the partial rank estimator (PRE) proposed by Khan and
Tamer (2007). This estimator is robust up to covariate-dependent censoring and
consistently estimates the normalized regression parameters in the
nonparametric transformation model.

Table 3 shows the inference results for $\beta_{treat}$. It presents the point
estimate obtained from the PRE and 95% confidence intervals obtained from the
PRE and the proposed method. The confidence interval obtained from the PRE is
computed based on 1,000 bootstrap pseudo samples from the data. For the
proposed method, we set the tuning parameters to $R=5$,
$B_{n}=\left(0.8\ln\left(n\right)/\ln\ln\left(n\right)\right)^{\frac{1}{2}}$,
$\kappa_{n}=\left(\left(1-\hat{p}_{1-D}^{1/3}\right)^{2/5}\times
0.6\ln(n)\right)^{\frac{1}{2}}$, and $\eta=10^{-6}$ as in the baseline case in
the Monte Carlo simulation in the previous section. For $\epsilon$, we use
both $\epsilon=0.0001$ and $\epsilon=0.001$. Setting $\epsilon=0.001$ is more
conservative in a small sample like the Stanford Heart Transplant data set
according to the Monte Carlo simulation results in the previous section. The
confidence intervals obtained from the proposed method do not have finite
upper bounds. This would be because age does not have sufficiently large
support to derive a finite upper bound on $\beta_{treat}$ in $B_{I}$. The
estimate obtained from the PRE is positive and is significantly different from
zero. The 95% confidence interval obtained from the proposed method is also
entirely positive regardless of the choice of $\epsilon$. With the
conservative choice of $\epsilon=0.001$, the $95\%$ confidence interval
obtained from the proposed method covers the confidence interval obtained from
the PRE. The inference results relating to the proposed method show that even
if censoring is arbitrarily correlated with a patient’s age or unobserved
heterogeneity, a heart transplant has a positive effect on the patient’s
survival time.

Table 3: Empirical Results for $\beta_{treat}$ | PRE | Proposed Method
---|---|---
| $\epsilon=0.001$ | $\epsilon=0.0001$
Estimate | 42.6 | - | -
95% Confidence Interval | [17.2, 57.3] | [10.4, +$\infty$] | [31.3, +$\infty$]

We next apply the joint inference procedure for the regression parameters and
transformation function presented in Appendix C. Figure 3 shows the marginal
$95\%$-confidence set of $T(y)$ at each $y\in[0,900]$, which is the projection
of the three-dimensional confidence interval of
$(\beta_{age},\beta_{treat},T(y))$ on the one-dimension. The same tuning
parameters as those used in Table 3 are used. Since $T_{I,\beta}(y)$ does not
have a finite upper bound, neither do the confidence intervals. The estimated
confidence intervals also do not have finite lower bounds for small values of
$y$. The $95\%$-confidence lower bound on $T(y)$ increases rapidly with $y$ in
the approximate interval $[100,250]$, but it changes slightly when $y$ is
large.

Figure 3: Empirical Results for the Transformation Function

(a) $\epsilon=0.001$ (b) $\epsilon=0.0001$
---

  * Notes: The shaded are in each figure is the computed confidence interval of $T(y)$ at each $y\in[0,900]$. The confidence intervals are computed on the range $[-10,3]$.

###  6 Concluding Remarks

In this paper, we propose a partial identification and inference approach for
a nonparametric transformation model in the presence of endogenous censoring.
We develop bounds on the regression parameters and the transformation
function, each of which is characterized by conditional moment inequalities
involving U-statistics. We also characterize the sharp identified set of the
regression parameters, using concepts from random set theory, though this set
is hard to compute. A comparison of the proposed set and the sharp identified
set characterization makes it clear when the proposed set approaches the sharp
set. Based on the identification result, we propose an inference method for
the regression parameters by extending the inference approach for conditional
moment inequality models, proposed by Andrews and Shi (2013), to the
U-statistics case. We also derive the asymptotic properties of this approach.
A joint inference procedure for the regression parameters and transformation
function is presented in Appendix C. Numerical examples illustrate the
characteristics of the proposed sets for the regression parameters and
transformation function, and the results of Monte Carlo experiments
demonstrate the size and power properties of the proposed inference methods.
As an empirical application, we apply the inference methods to evaluate the
effect of heart transplants on patients’ survival duration using data from the
Stanford Heart Transplant Study, for which we find that heart transplants have
a positive effect on patients’ survival duration regardless of the censoring
mechanism.

## Appendix

In this appendix, Section A provides proofs of Theorems 2.1 and 2.2. Section B
provides a proof of Theorem 3.1 with some auxiliary lemmas. Section C presents
a joint inference procedure for the regression parameters and the
transformation function based on the identification results presented in
Sections 2.2 and 2.4. Some additional numerical studies and Monte Carlo
simulation studies are presented in the supplementary material to this paper.

###  Appendix A Proofs of Theorems 2.1 and 2.2

This section provides proofs of Theorems 2.1 and 2.2.

###### Proof of Theorem 2.1.

From the definitions of $Y_{1i}$ and $Y_{0i}$, the following holds for all
$(x_{i},x_{j})\in\mathcal{X}^{2}$,

$\displaystyle P(Y_{1i}\geq Y_{0j}\mid x_{i},x_{j})\geq P(Y_{i}^{\ast}\geq
Y_{j}^{\ast}\mid x_{i},x_{j}).$

For the conditional multinomial distribution $P(Y_{i}^{\ast},Y_{j}^{\ast}\mid
x_{i},x_{j})$, it follows that

$\displaystyle P(Y_{i}^{\ast}\geq Y_{j}^{\ast}\mid x_{i},x_{j})$
$\displaystyle=P(\Lambda(x_{i}^{\prime}\beta_{0}+U_{i})\geq\Lambda(x_{j}^{\prime}\beta_{0}+U_{j}))$
$\displaystyle\geq P(x_{i}^{\prime}\beta_{0}+U_{i}\geq
x_{j}^{\prime}\beta_{0}+U_{j})$
$\displaystyle=P(U_{i}-U_{j}\geq-(x_{i}-x_{j})^{\prime}\beta_{0}).$

The first line follows from Assumptions 2.2 and 2.3. The second line follows
because $\Lambda$ is a monotonically increasing function by Assumption 2.1.
When $(x_{i}-x_{j})^{\prime}\beta_{0}\geq 0$,

$\displaystyle
P(U_{i}-U_{j}\geq-(x_{i}-x_{j})^{\prime}\beta_{0})\geq\frac{1}{2}$

holds because $U_{i}-U_{j}$ is symmetrically distributed about zero under
Assumption 2.2. Therefore it follows that

$\displaystyle x_{i}^{\prime}\beta_{0}\geq x_{j}^{\prime}\beta_{0}$
$\displaystyle\Rightarrow P\left(Y_{i}^{\ast}\geq Y_{j}^{\ast}\mid
x_{i},x_{j}\right)\geq\frac{1}{2}$ $\displaystyle\Rightarrow P(Y_{1i}\geq
Y_{0j}\mid x_{i},x_{j})\geq\frac{1}{2}$

for all $(x_{i},x_{j})\in\mathcal{X}^{2}$. This implies that $\beta_{0}\in
B_{I}$ a.s..

We next show that $B_{I}$ is a proper subset of $B$. Let
$(\tilde{x}_{i},\tilde{x}_{j})\in\widetilde{\mathcal{X}}^{2}$. Then there
exist $\beta\in B$ such that

$\displaystyle\tilde{x}_{i}^{\prime}\beta\geq\tilde{x}_{j}^{\prime}\beta\Rightarrow
P(Y_{1i}\geq Y_{0j}\mid\tilde{x}_{i},\tilde{x}_{j})<\frac{1}{2}.$

Such a $\beta$ is not a member of $B_{I}$. Hence, because
$P((X_{i},X_{j})\in\widetilde{\mathcal{X}}^{2})>0$ by Assumption 2.4,
$B_{I}\subset B$ holds a.s. ∎

###### Proof of Theorem 2.2.

Under Assumptions 2.2, 2.3, and 2.5, $\beta_{0}\in B_{I}$ a.s. from Theorem
2.1. Then it suffices to show that $T\left(y\right)\in
T_{I,\beta_{0}}\left(y\right)$ a.s. for any $y\in\mathbb{R}$.

Let $y\in\mathbb{R}$ be fixed. Note that from the definitions of $Y_{1i}$ and
$Y_{0i}$,

$\displaystyle P(Y_{1i}\geq y\mid x_{i})-P(Y_{0j}\geq\tilde{y}\mid x_{j})\geq
P(Y_{i}^{\ast}\geq y\mid x_{i})-P(Y_{j}^{\ast}\geq\tilde{y}\mid x_{j}).$

holds for all $(x_{i},x_{j})\in\mathcal{X}^{2}$.

For the conditional multinomial distribution $P(Y_{i}^{\ast},Y_{j}^{\ast}\mid
x_{i},x_{j})$, it follows that

$\displaystyle P(Y_{i}^{\ast}\geq y\mid x_{i})-P(Y_{j}^{\ast}\geq\tilde{y}\mid
x_{j})$ $\displaystyle=$ $\displaystyle\
P\left(x_{i}^{\prime}\beta_{0}+U_{i}\geq T\left(y\right)\mid
x_{i}\right)-P\left(x_{j}^{\prime}\beta_{0}+U_{j}\geq 0\mid x_{j}\right)$
$\displaystyle=$ $\displaystyle\
F_{U}\left(-x_{j}^{\prime}\beta_{0}\right)-F_{U}\left(T\left(y\right)-x_{i}^{\prime}\beta_{0}\right).$

The first equality follows from Assumptions 2.2 and 2.5 and the fact that
$T\left(\tilde{y}\right)=0$ for the location normalization. The second
equality follows from Assumptions 2.2 and 2.3. Since

$\displaystyle x_{i}^{\prime}\beta_{0}-x_{j}^{\prime}\beta_{0}\geq
T\left(y\right)\Rightarrow F_{U}\left(-x_{j}^{\prime}\beta_{0}\right)\geq
F_{U}\left(T\left(y\right)-x_{i}^{\prime}\beta_{0}\right),$

we have

$\displaystyle x_{i}^{\prime}\beta_{0}-x_{j}^{\prime}\beta_{0}\geq
T\left(y\right)$ $\displaystyle\Rightarrow P(Y_{i}^{\ast}\geq y\mid
x_{i})-P(Y_{j}^{\ast}\geq\tilde{y}\mid x_{j})\geq 0$ $\displaystyle\Rightarrow
P(Y_{1i}\geq y\mid x_{i})-P(Y_{0j}\geq\tilde{y}\mid x_{j})\geq 0$

for all $(x_{i},x_{j})\in\mathcal{X}^{2}$. This implies that
$T\left(y\right)\in T_{I,\beta_{0}}\left(y\right)$ a.s. ∎

###  Appendix B Proof of Theorem 3.1

This section provides a proof of the uniform asymptotic probability results
for the inference method presented in Section 3. The outline of the proof is
same as that of the proofs of Theorems 2(b) and 3 in AS, but we modify them
for the case of U-statistics. Let $\rightsquigarrow$ denote weak convergence
of a stochastic process in the sense of Pollard (1990). The following
notations are similar to the notations introduced in AS,

$\displaystyle v_{n,P}\left(\beta,g\right)\equiv
n^{\frac{1}{2}}\left(\bar{m}_{n}\left(\beta,g\right)-E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\right)/\sigma_{p}\left(\beta\right)$

and

$\displaystyle\hat{h}_{2,n,P}\left(\beta,g,g^{\ast}\right)\equiv$
$\displaystyle\left\\{\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq k}m(W_{i},W_{j},\beta,g)m(W_{i},W_{k},\beta,g^{\ast})\right.$
$\displaystyle\left.-E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\cdot
E_{P}\left[m\left(W_{i},W_{j},\beta,g^{\ast}\right)\right]\right\\}/\sigma_{P}^{2}\left(\beta\right).$

Let $\left\\{v_{h_{2}}\left(g\right):g\in{\cal G}\right\\}$ be a mean zero
Gaussian process with some covariance kernel $h_{2}\left(\cdot,\cdot\right)$
on ${\cal G}\times{\cal G}$. Let $\\{a_{n}:n\geq 1\\}$ denote a subsequence of
$\\{n\\}$.

To prove Theorem 3.1, we first prove that the following two lemmas hold. Lemma
B.1 implies that Assumption EP in AS holds. Lemmas B.2 implies that a version
of Assumption CI in AS, which is modified for the case of U-statistics, holds.

######  Lemma B.1.

For any subsequence $\left\\{\left(\beta_{a_{n}},P_{a_{n}}\right)\in{\cal
Q}:n\geq 1\right\\}$ such that

$\displaystyle\underset{n\rightarrow\infty}{\lim}\underset{g,g^{\ast}\in{\cal
G}}{\sup}\left\|h_{2,P_{a_{n}}}\left(\beta_{a_{n}},g,g^{\ast}\right)-h_{2}\left(g,g^{\ast}\right)\right\|=0$

for some covariance kernel $h_{2}(\cdot,\cdot)$ on ${\cal G}\times{\cal G}$,
we have
(a)
$\sqrt{a_{n}}v_{a_{n},P_{a_{n}}}\left(\beta_{a_{n}},\cdot\right)\rightsquigarrow
v_{h_{2}}\left(\cdot\right)$ as $n\rightarrow\infty$, and
(b) $\underset{\left(g,g^{\ast}\right)\in{\cal
G}\times\mathcal{G}}{\sup}\left\|\hat{h}_{2,a_{n},P_{a_{n}}}\left(\beta_{a_{n}},g,g^{\ast}\right)-h_{2}\left(g,g^{\ast}\right)\right\|\underset{p}{\rightarrow}0$
as $n\rightarrow\infty$.

######  Lemma B.2.

For any $\beta\in B$ and any distribution function $P$ that satisfies
Assumptions 2.1–2.3, let

$\displaystyle{\cal X}_{P}\left(\beta\right)=\left\\{(x_{i},x_{j})\in{\cal
X}^{2}:E_{P}\left[m\left(W_{i},W_{j},\beta\right)\mid
x_{i},x_{j}\right]<0\right\\}.$

Then, for any $\beta\in B$ and $P$ that satisfies Assumptions 2.1–2.3 and
$P\left((x_{i},x_{j})\in{\cal X}_{P}\left(\beta\right)\right)>0,$ there exists
some $g\in{\cal G}$ such that

$\displaystyle E_{P}\left[m(W_{i},W_{j},\beta,g)\right]<0.$

The next lemma with a proof is auxiliary to Lemma B.1.

######  Lemma B.3.

Let $(\beta,P)\in\mathcal{{Q}}$. Define classes of functions ${\cal
F}_{1}=\left\\{f_{1}(\cdot,\cdot,\beta,g):g\in{\cal G}\right\\}$ and ${\cal
F}_{2}=\left\\{f_{2}(\cdot,\cdot,\cdot,\beta,g,g^{\ast}):\left(g,g^{\ast}\right)\in{\cal
G}\times{\cal G}\right\\}$, where

$\displaystyle
f_{1}(w_{i},w_{j},\beta,g)=m(w_{i},w_{j},\beta,g)-E_{P}\left[m(W_{i},W_{j},\beta,g)\right]$

and

$\displaystyle f_{2}(w_{i},w_{j},w_{k},\beta,g,g^{\ast})=$ $\displaystyle\
m(w_{i},w_{j},\beta,g)\cdot m(w_{i},w_{k},\beta,g^{\ast})$ $\displaystyle-
E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\cdot
E_{P}\left[m\left(W_{i},W_{j},\beta,g^{\ast}\right)\right].$

Then $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are Euclidean classes of
functions for constant envelopes $1$ and $1/2$, respectively, in the sense of
Nolan and Pollard (1987).

###### Proof of Lemma B.3.

We consider the class of function $\mathcal{G}$ defined in Section 3.1. For
any $x\in\mathbb{R}^{k}$, let $x=(x^{(1)\prime},x^{(2)\prime})$ where
$x^{(1)}$ is a vector of the first $p$ elements of $x$ and $x^{(2)}$ is a
vector of the remaining $k-p$ elements of $x$. $\mathcal{G}$ is represented as

$\displaystyle\mathcal{G}=$
$\displaystyle\left\\{I\left[\frac{a-\mathbf{1}_{p}}{2r}<x_{i}^{(1)}\leq\frac{a}{2r},x_{i}^{(2)}=b\right]\cdot
I\left[\frac{\tilde{a}-\mathbf{1}_{p}}{2r}<x_{j}^{(1)}\leq\frac{\tilde{a}}{2r},x_{j}^{(2)}=\tilde{b}\right]:\right.$
$\displaystyle a=(a_{1},\ldots,a_{p})^{\prime},\
\tilde{a}=\left(\tilde{a}_{1},\ldots,\tilde{a}_{p}\right)^{\prime},\
\left(a_{u},\tilde{a}_{u}\right)\in\left\\{1,2,\ldots,2r\right\\}^{2}$
$\displaystyle\left.\mbox{for }u=1,\ldots,p\ \mbox{and }r=1,2,\ldots,\mbox{
and }(b,\tilde{b})\in\mathcal{X}_{2}^{2}\right\\},$

where $\mathbf{1}_{p}$ is a $p$-dimensional vector of ones. Because the
collection of cells in $\mathbb{R}^{2k}$ is a Vapnik-Chervonenkis (VC) class
of sets (see van der Vaart and Wellner (1996, Example 2.6.1)), the collection
of all subgraphs,
$\left\\{\left(x_{i},x_{j},t\right):t<g(x_{i},x_{j})\right\\}$, of the
function in $\mathcal{G}$ forms a VC class of sets in
$\mathcal{X}^{2}\times\mathbb{R}$. Hence, $\mathcal{G}$ is a VC class of
functions. Combining this result with Lemma 2.6.18 in van der Vaart and
Wellner (1996), $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are VC-classes of
functions. Thus, from Corollary 19 in Nolan and Pollard (1987),
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are Euclidean classes of functions.
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ obviously have the constant envelopes
$1$ and $1/2$, respectively, from their definitions. ∎

We provide proofs of Lemmas B.1 and B.2 and Theorem 3.1 below.

###### Proof of Lemma B.1.(a).

While Lemma B.1 is stated in terms of a subsequence $\left\\{a_{n}\right\\}$,
for notational simplicity, we prove it for the sequence $\left\\{n\right\\}$.
All of the arguments in this and the next proofs proceed with
$\left\\{a_{n}\right\\}$ instead of $\left\\{n\right\\}$.

We use Theorem 5 in Nolan and Pollard (1988) to show that the weak convergence
result in Lemma B.1.(a) holds. Let $(\beta,P)$ be the limit of
$(\beta_{n},P_{n})$ and $h_{2}(\cdot,\cdot)=h_{2,P}(\beta,\cdot,\cdot)$. Let
$N_{p}\left(\epsilon,Q,{\cal F},F\right)$ denote the
$L_{p}\left(Q\right)$-covering number of radius $\epsilon$ for a functional
space ${\cal F}$ with a envelope function $F$ where $Q$ is some probability
measure on $\mathcal{X}$. For any distribution $Q$ on $\mathcal{X}$, let
$Q{\cal F}_{1}\equiv\\{Qf_{1}(x,\cdot,\beta,g):g\in\mathcal{G}\\}$ be the
class of functions $Qf_{1}\left(x,\cdot,\beta,g\right)$ on ${\cal X}$, where
$f_{1}$ and $\mathcal{F}_{1}$ are defined in Lemma B.3 and
$Qf_{1}(x,\cdot,\beta,g)\equiv\int_{\mathcal{X}}f_{1}(x,x_{2},\beta,g)dQ_{X}(x_{2})$.

To apply Theorem 5 in Nolan and Pollard (1988), it suffices to show that the
following conditions hold:

(i)

$\sup_{Q}\intop_{0}^{1}\log N_{2}\left(\epsilon,Q\times Q,{\cal
F}_{1},F\right)d\epsilon<\infty$, $\sup_{Q}\left(\intop_{0}^{1}\log
N_{2}\left(\epsilon,Q\times
Q,\mathcal{F}_{1},F\right)d\epsilon\right)^{2}<\infty$, and
$\sup_{Q}\left(\intop_{0}^{1}\log
N_{2}\left(\epsilon,Q,P\mathcal{F}_{1},PF\right)d\epsilon\right)^{2}<\infty$;

(ii)

as $\gamma\searrow 0$,

$\displaystyle\sup_{Q}\intop_{0}^{\gamma}\log N_{2}\left(\epsilon,Q,P{\cal
F}_{1},PF\right)\rightarrow 0,$

where $Q$ is any probability measure on $\mathcal{X}$. We show below that
these two conditions are satisfied.

We first consider condition (i). From Lemma B.3, the class of functions
$\mathcal{F}_{1}$ is Euclidean with the constant envelope $F=1$. Then, from
Corollary 21 in Nolan and Pollard (1987), the class of functions $P{\cal
F}_{1}$ is also a Euclidean class with the constant envelope $1$. From page
789 in Nolan and Pollard (1987), if a Euclidean class has a constant envelope
function, then the upper bound on the $L_{p}(Q)$-covering number of radius
$\epsilon$ for it is uniform in any probability measure $Q$ on $\mathcal{X}$.
Therefore, since $\mathcal{F}_{1}$ and $P{\cal F}_{1}$ are Euclidean classes
with constant envelopes, for any $0<\epsilon\leq 1$, there exist some
constants $K_{2}$, $K_{2}^{\ast}$, $V_{2}$, and $V_{2}^{\ast}$ such that
$N_{2}\left(\epsilon,Q\times Q,{\cal F}_{1},F\right)\leq
K_{2}\epsilon^{-2V_{2}}$ and $N_{2}\left(\epsilon,Q,P{\cal
F}_{1},PF\right)\leq K_{2}^{\ast}\epsilon^{-2V_{2}^{\ast}}$, for any
probability measure $Q$ on $\mathcal{X}$. Then it follows that

$\displaystyle\sup_{Q}\intop_{0}^{1}\log N_{2}\left(\epsilon,Q\times Q,{\cal
F}_{1},F\right)d\epsilon$ $\displaystyle\leq\intop_{0}^{1}\left(\log
K_{2}\epsilon^{-2V_{2}}\right)d\epsilon<\infty,$
$\displaystyle\sup_{Q}\left(\intop_{0}^{1}\log N_{2}\left(\epsilon,Q\times
Q,{\cal F}_{1},F\right)d\epsilon\right)^{2}$
$\displaystyle\leq\left(\intop_{0}^{1}\left(\log
K_{2}\epsilon^{-2V_{2}}\right)d\epsilon\right)^{2}<\infty,$

and

$\displaystyle\sup_{Q}\left(\intop_{0}^{1}\log N_{2}\left(\epsilon,Q,P{\cal
F}_{1},PF\right)d\epsilon\right)^{2}\leq\left(\intop_{0}^{1}\left(\log
K_{2}^{\ast}\epsilon^{-2V_{2}^{\ast}}\right)d\epsilon\right)^{2}<\infty.$

These imply that condition (i) is satisfied.

Next, as $\gamma\searrow 0$,

$\displaystyle\sup_{Q}\intop_{0}^{\gamma}\log N_{2}\left(\epsilon,Q,P{\cal
F}_{1},PF\right)d\epsilon$
$\displaystyle\leq\intop_{0}^{\gamma}\log(K_{2}^{\ast}\epsilon^{-2V_{2}^{\ast}})d\epsilon$
$\displaystyle=\gamma\log K_{2}^{\ast}-2V_{2}^{\ast}\gamma(\log\gamma-1)$
$\displaystyle\rightarrow 0.$

This implies that condition (ii) is satisfied. Therefore, from Theorem 5 in
Nolan and Pollard (1988) and the fact that $\beta_{n}\rightarrow\beta$ and
$\sigma_{P_{n}}\left(\beta_{n}\right)\underset{p}{\rightarrow}\sigma_{P}\left(\beta\right)$,
Lemma B.1.(a) holds by the extended continuous mapping theorem (Theorem 1.11.1
in van der Vaart and Wellner (1996)). ∎

###### Proof of Lemma B.1.(b).

Let $(\beta,P)$ be the limit of $(\beta_{n},P_{n})$. Since $\mathcal{F}_{1}$
and $\mathcal{F}_{2}$ are Euclidean classes with constant envelopes from Lemma
B.3, by applying Corollary 7 in Sherman (1994), it follows that

$\displaystyle\sup_{g\in{\cal G}}\left\|\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}f_{1}(W_{i},W_{j},\beta,g)\right\|\underset{p}{\rightarrow}0$

and

$\displaystyle\sup_{(g,g^{*})\in\mathcal{G}\times\mathcal{G}}\left\|\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq
k}f_{2}(W_{i},W_{j},W_{k},\beta,g,g^{\ast})\right\|\underset{p}{\rightarrow}0.$

Therefore, letting
$h_{2}\left(\beta,g,g^{\ast}\right)=h_{2,P}\left(\beta,g,g^{\ast}\right)$ be
given by (3.2) and further dividing by $\sigma_{P}^{2}\left(\beta\right)$, as
$\beta_{n}\rightarrow\beta$ and
$\sigma_{P_{n}}\left(\beta_{n}\right)\underset{p}{\rightarrow}\sigma_{P}\left(\beta\right)$,
we have

$\displaystyle\
\underset{(g,g^{\ast})\in\mathcal{G}\times\mathcal{G}}{\sup}\left\|\hat{h}_{2,a_{n},P_{n}}\left(\beta_{n},g,g^{\ast}\right)-h_{2}\left(\beta,g,g^{\ast}\right)\right\|$
$\displaystyle\leq$ $\displaystyle\
\sup_{(g,g^{\ast})\in\mathcal{G}\times\mathcal{G}}\left\|\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq
k}f_{2}(W_{i},W_{j},W_{k},\beta,g,g^{\ast})/\sigma_{P}\left(\beta\right)\right\|$
$\displaystyle\ +\left\\{\sup_{g\in{\cal
G}}\left\|\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j}f_{1}(W_{i},W_{j},\beta,g)/\sigma_{P}\left(\beta\right)\right\|\right\\}^{2}+o_{p}\left(1\right)$
$\displaystyle\underset{p}{\rightarrow}$ $\displaystyle\ 0.$

∎

###### Proof of Lemma B.2.

It suffices to show that

$\displaystyle E_{P}\left[m(W_{i},W_{j},\beta,g)\right]\geq 0\ \ \forall
g\in{\cal G}$ $\displaystyle\Rightarrow\ $ $\displaystyle
E_{P}\left[m\left(W_{i},W_{j},\beta\right)\mid x_{i},x_{j}\right]\geq 0\ \
\mbox{for almost every $(x_{i},x_{j})\in\mathcal{X}^{2}$.}$ (B.1)

Let $X=(X^{(1)\prime},X^{(2)\prime})^{\prime}$ where $X^{(1)}$ and $X^{(2)}$
are distributed on $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$, respectively. Let
$(b,\tilde{b})\in\mathcal{X}_{2}^{2}$ be fixed, and define a class of
instrumental functions

$\displaystyle\widetilde{\mathcal{G}}=\left\\{g\left(x_{i}^{(1)},x_{j}^{(1)}\right)=I\left[\left(x_{i}^{(1)},x_{j}^{(1)}\right)\in
J_{1}\right]\ \mbox{for}\ J_{1}\in\mathcal{J}_{1}\right\\},$

where

$\displaystyle\mathcal{J}_{1}\equiv$
$\displaystyle\left\\{J_{(a,\tilde{a},r)}=\left(\vartimes_{u=1}^{p}\left(\frac{a_{u}-1}{2r},\frac{a_{u}}{2r}\right]\right)\times\left(\vartimes_{u=1}^{p}\left(\frac{\tilde{a}_{u}-1}{2r},\frac{\tilde{a}_{u}}{2r}\right]\right):\right.$
$\displaystyle\ \left.a=(a_{1},\ldots,a_{p}),\
\tilde{a}=(\tilde{a}_{1},\ldots,\tilde{a}_{p}),\
a_{u}\in\left\\{1,2,\ldots,2r\right\\},\
\tilde{a}_{u}\in\left\\{1,2,\ldots,2r\right\\}\right.$ $\displaystyle\
\left.\mbox{for}\ u=1,\ldots,p\ \mbox{and}\ r=1,2,\ldots\right\\}.$

Then, to prove (B.1), it suffices to show that

$\displaystyle
E_{P}\left[m(W_{i},W_{j},\beta,g)\mid(X_{i}^{(2)},X_{j}^{(2)})=(b,\tilde{b})\right]\geq
0\ \ \forall g\in\widetilde{\mathcal{G}}$ $\displaystyle\Rightarrow\ $
$\displaystyle
E_{P}\left[m\left(W_{i},W_{j},\beta\right)\mid(X_{i}^{(1)},X_{j}^{(1)})=(x_{i}^{(1)},x_{j}^{(1)}),(X_{i}^{(2)},X_{j}^{(2)})=(b,\tilde{b})\right]\geq
0$ $\displaystyle\mbox{ for almost every
$(x_{i}^{(1)},x_{j}^{(1)})\in\mathcal{X}_{1}^{2}$}.$ (B.2)

We invoke Lemma C1 in AS. Let ${\cal R}$ be a semiring of subsets of
$\mathbb{R}^{2p}$ and

$\displaystyle\mu(J_{1})\equiv E_{P}\left[m\left(W_{i},W_{j},\beta\right)\cdot
I\left[\left(X_{i}^{(1)},X_{j}^{(1)}\right)\in
J_{1}\right]\mid(X_{i}^{(2)},X_{j}^{(2)})=(b,\tilde{b})\right]$

for $J_{1}\in\sigma\left(\mathcal{J}_{1}\right)={\cal
B}\left(\mathbb{R}^{2p}\right)$, where $\sigma\left(\mathcal{J}_{1}\right)$
denotes the $\sigma$-field generated by $\mathcal{J}_{1}$ and ${\cal
B}\left(\mathbb{R}^{2p}\right)$ is the Borel $\sigma$-field on
$\mathbb{R}^{2p}$. $\sigma\left(\mathcal{J}_{1}\right)={\cal
B}\left(\mathbb{R}^{2p}\right)$ is a well known result. We show that all
conditions of Lemma C1 in AS are satisfied. Then condition (B.2) holds from
Lemma C1 in AS.

First, $\mathcal{J}_{1}$ is a semiring of subsets of $\mathbb{R}^{2p}$. Since
$m\left(W_{i},W_{j},\beta\right)$ and
$I\left[\left(X_{i}^{(1)},X_{j}^{(1)}\right)\in J_{1}\right]$ are bounded
functions, $\mu(\cdot)$ satisfies the boundedness condition of Lemma C1 in AS.
The other conditions of Lemma C1 in AS also hold by the same argument as the
proof of Lemma 3 in AS. Thus, by applying Lemma C1 in AS, the fact that
$\mu(J_{1})\geq 0$ for all $J_{1}\in\mathcal{J}_{1}$ implies that
$E_{P}\left[m\left(W_{i},W_{j},\beta\right)\cdot
I\left[\left(X_{i}^{(1)},X_{j}^{(1)}\right)\in
J_{1}\mid(X_{i}^{(2)},X_{j}^{(2)})=(b,\tilde{b})\right]\right]\geq 0$ for all
$J_{1}\in\sigma(\mathcal{J}_{1})$, which is equivalent to ${\cal
B}\left(\mathbb{R}^{2p}\right)$. This implies that the result of Lemma B.2
holds. ∎

Let $Q$ be a measure on $\mathcal{G}$ such that, for
$g_{(a,b),(\tilde{a},\tilde{b}),r}\in\mathcal{G}$,

$\displaystyle
Q(g_{(a,b),(\tilde{a},\tilde{b}),r})=(r^{2}+100)^{-1}\cdot((2r)^{p}\cdot|\mathcal{X}_{2}|)^{-2}.$

Then $T_{n}(\beta)$ is expressed as

$\displaystyle
T_{n}(\beta)=\int_{g\in\mathcal{G}}S(n^{\frac{1}{2}}\bar{m}_{n}(\beta,g),\bar{\sigma}_{n}(\beta,g))dQ(g),$
(B.3)

where $S(m,\sigma)\equiv[m/\sigma]_{-}^{2}$.

The following lemma shows that $Q$ satisfies Assumption Q in AS.

######  Lemma B.4.

Suppose that Assumption 2.2 holds. Define the pseud-metric $\rho_{X}$ on
$\mathcal{G}$ by

$\displaystyle\rho_{X}(g,g^{\ast})\equiv(E_{P_{X}}[|g(X_{i},X_{j})-g^{\ast}(X_{i},X_{j})|])^{1/2}\mbox{
for }(g,g^{*})\in\mathcal{G}\times\mathcal{G},$

where $E_{P_{X}}[\cdot]$ denotes the expectation under $X\sim P_{X}$. Let
$\mathcal{B}_{\rho_{X}}(g,\delta)$ denote an open $\rho_{X}$-ball in
$\mathcal{G}$ centered at $g$ with radius $\delta$. Then the support of $Q$
under $\rho_{X}$ is $\mathcal{G}$. That is, for all $\delta>0$,
$Q(\mathcal{B}_{\rho_{X}}(g,\delta))>0$ for all $g\in\mathcal{G}$.

###### Proof.

We prove the statement in the case of $p=k$. Then the result for the other
case $p<k$ immediately follows. Let
$AR\equiv\\{(a,\tilde{a},r):(a,\tilde{a})\in\\{1,\ldots,2r\\}^{2k}\mbox{ and
}r=1,2,\ldots\\}$. Then there is a one-to-one mapping
$\Pi:\mathcal{G}\rightarrow AR$. The weight function $Q$ can be expressed as
$Q=\Pi^{-1}Q_{AR}$, where $Q_{AR}$ is a measure on $2^{AR}$ such that
$Q((a,\tilde{a},r))=(r^{2}+100)^{-1}\cdot(2r)^{-2k}$ for any
$(a,\tilde{a},r)\in AR$. Then the statement follows by applying Lemma 4 in AS
to $Q=\Pi^{-1}Q_{AR}$. ∎

###### Proof of Theorem 3.1.

To show that Theorem 3.1 holds, it suffices to show that all required
conditions in Theorems 2(b) and 3 in AS are satisfied. We first check that the
functions $S$ and $Q$ in (B.3) satisfy the required conditions. Since the
function $S$ corresponds to the modified method moments function in AS
(Equation (3.8) in AS), Lemma 1 in AS guarantees that the function $S$ in
(B.3) satisfies Assumptions S1–4 in AS. Lemma B.4 shows that the weight
function $Q$ in (B.3) satisfies Assumption Q in AS. Lemma B.1 implies that
Assumption EP in AS holds. Then Lemma B3 in AS guarantees that Assumption
GMS2(a) in AS holds. Assumption 3.1 implies that the tuning parameters in the
GMS function satisfy Assumptions GMS1, GMS2(b), and GMS2(c) in AS. Therefore,
Theorem 3.1(a) follows from Theorem 2(b) in AS.

For result (b), Lemma B.2 in this appendix has the same role as Assumption CI
in AS. The assumption
$P((X_{i},X_{j})\in\mathcal{X}_{P}^{2}(\tilde{\beta}))>0$ in Theorem 3.1(b)
corresponds to Assumption FA(a) in AS. Note that Lemma B.1 applies for
$(\beta_{a_{n}},P_{a_{n}})=(\tilde{\beta},P)$ by the same argument as in the
proof of Lemma B.1. Therefore, by the same arguments as in Section 14.2 in AS,
where we replace Lemma A1 (in AS) for
$(\beta_{a_{n}},P_{a_{n}})=(\tilde{\beta},P)$ by Lemma B.1 for
$(\beta_{a_{n}},P_{a_{n}})=(\tilde{\beta},P)$, Theorem 3.1(b) holds. ∎

###  Appendix C Inference for the Transformation Function

This section provides a joint inference procedure for $\beta_{0}$ and
$\\{T\left(y\right)\\}_{y\in\mathbf{y}}$, where $\mathbf{y}$ is a finite set
of $y$. Let $q$ be the dimension of $\mathbf{y}$ and denote
$\mathbf{y}=(y_{1},\ldots,y_{q})^{\prime}$. We first note that with fixed
$y\in\mathbb{R}$ and $\beta$, $T_{I,\beta}(y)$ is equivalent to the set of all
values $t\in\mathbb{R}$ that satisfy the following conditional moment
inequality condition:

$\displaystyle E_{P}\left[m^{\dagger}(W_{i},W_{j},\beta,y,t)\mid
x_{i},x_{j}\right]\geq 0\mbox{ for all $(x_{i},x_{j})\in\mathcal{X}^{2}$},$
(C.1)

where $m^{\dagger}(W_{i},W_{j},\beta,y,t)$ is a moment function defined as

$\displaystyle m^{\dagger}\left(W_{i},W_{j},\beta,y,t\right)$
$\displaystyle\equiv\left(I\left[Y_{1i}\geq
y\right]-I\left[Y_{0j}\geq\tilde{y}\right]\right)\cdot I[X_{i}^{\prime}\beta-
X_{j}^{\prime}\beta\geq t]$ $\displaystyle+\left(I\left[Y_{1j}\geq
y\right]-I\left[Y_{0i}\geq\tilde{y}\right]\right)\cdot I[X_{j}^{\prime}\beta-
X_{i}^{\prime}\beta\geq t].$

This is obvious from the definition of $T_{I,\beta}(y)$. The joint inference
procedure for $\beta_{0}$ and $\\{T\left(y\right)\\}_{y\in\mathbf{y}}$ is then
constructed by applying the conditional moment inequality inference approach
presented in Section 3 to the conditional moment inequalities (3.1) and (C.1).

Fix $y\in\mathbf{y}$. Let
$m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)\equiv
m^{\dagger}\left(W_{i},W_{j},\beta,y,t\right)\cdot g(x_{i},x_{j})$ for any
$g\in\mathcal{G}$, where $\mathcal{G}$ is the set of instrumental functions
defined in Section 3.1. Then $T_{I,\beta}\left(y\right)$ is equivalent to

$\left\\{t\in\mathbb{R}:\
E_{P}\left[m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)\right]\geq 0\
\mbox{for all}\ g\in\mathcal{G}\right\\}.$

Define the sample moment function and sample variance function, respectively,
by

$\displaystyle\bar{m}_{n}^{\dagger}\left(\beta,y,t,g\right)\equiv\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)$

and

$\displaystyle\hat{\sigma}_{n}^{\dagger 2}\left(\beta,y,t,g\right)\equiv$
$\displaystyle\left\\{\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq
k}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)m^{\dagger}\left(W_{i},W_{k},\beta,y,t,g\right)\right.$
$\displaystyle\left.-\left(\frac{1}{n(n-1)}\sum_{i\neq
j}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)\right)^{2}\right\\}.$

Since the function $\bar{m}_{n}^{\dagger}\left(\beta,y,t,g\right)$ is a
U-statistic of order two, the estimator of its asymptotic variance,
$\hat{\sigma}_{n}^{\dagger 2}\left(\beta,y,t,g\right)$, is constructed with a
similar form to $\hat{\sigma}_{n}^{2}\left(\beta,g\right)$ in Section 3.1. In
practice, we use the modified sample variance function:

$\displaystyle\bar{\sigma}_{n}^{\dagger
2}\left(\beta,y,t,g\right)\equiv\hat{\sigma}_{n}^{\dagger
2}\left(\beta,y,t,g\right)+\epsilon\hat{\sigma}_{n}^{\dagger 2}(y),$

where $\hat{\sigma}_{n}^{\dagger 2}(y)\equiv\hat{\sigma}_{n}^{\dagger
2}\left(\beta,y,t,1\right)$ and $\epsilon$ is the regularization parameter
(e.g., $\epsilon=0.0001$).

Then, for any $\beta\in B$ and
$\mathbf{t}=(t_{1},\ldots,t_{q})^{\prime}\in\mathbb{R}^{q}$, an approximate
test statistic for a null hypothesis:

$\displaystyle H_{0}:$ $\displaystyle\ E_{P}\left[m(W_{i},W_{j},\beta)\mid
x_{i},x_{j}\right]\geq 0\mbox{ and
}E_{P}\left[m^{\dagger}(W_{i},W_{j},\beta,y_{\ell},t_{\ell})\mid
x_{i},x_{j}\right]\geq 0$ $\displaystyle\mbox{for all }\ell=1,\ldots,q\mbox{
and }(x_{i},x_{j})\in\mathcal{X}^{2}$

is constructed as

$\displaystyle T_{\mathbf{y},n,R}^{\dagger}(\beta,\mathbf{t})\equiv$
$\displaystyle\
\sum_{r=1}^{R}\left(r^{2}+100\right)^{-1}\sum_{\left(a,\tilde{a}\right)\in\left\\{1,\cdots,2r\right\\}^{2p}}\sum_{(b,\tilde{b})\in\mathcal{X}_{2}^{2}}\frac{1}{(2r)^{2p}\cdot|\mathcal{X}_{2}|^{2}}\left(\left[\frac{n^{\frac{1}{2}}\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2}\right.$
$\displaystyle\left.+\sum_{j=1}^{q}\left[\frac{n^{\frac{1}{2}}\bar{m}_{n}^{\dagger}\left(\beta,y_{j},t_{j},g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}^{\dagger}\left(\beta,y_{j},t_{j},g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2}\right),$

where $R$ is a truncation integer chosen by the researcher. Note that this
test statistic comprises a normalized sample moment function for $\beta$,
$\bar{m}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)/\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)$,
and $q$ normalized sample moment functions for $T(y_{j})$,
$\bar{m}_{n}^{\dagger}\left(\beta,y_{j},t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)/\bar{\sigma}_{n}^{\dagger}\left(\beta,y_{j},t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)$,
with $j=1,\ldots,q$.

We can compute the critical value for
$T_{\mathbf{y},n,R}^{\dagger}(\beta,\mathbf{t})$ as a simulated quantile of

$\displaystyle T_{\mathbf{y},n,R}^{\dagger,Asy}(\beta,\mathbf{t})\equiv$
$\displaystyle\
\sum_{r=1}^{R}\left(r^{2}+100\right)^{-1}\sum_{\left(a,\tilde{a}\right)\in\left\\{1,\cdots,2r\right\\}^{2p}}\sum_{(b,\tilde{b})\in\mathcal{X}_{2}^{2}}\frac{1}{(2r)^{2p}\cdot|\mathcal{X}_{2}|^{2}}$
$\displaystyle\times\left(\left[\frac{v_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)+\varphi_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}\left(\beta,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2}\right.$
$\displaystyle\left.+\sum_{j=1}^{q}\left[\frac{v_{n}^{\dagger}\left(\beta,y_{j},t_{j},g_{(a,b),(\tilde{a},\tilde{b}),r}\right)+\varphi_{n}^{\dagger}\left(\beta,y_{j},t_{j},g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}{\bar{\sigma}_{n}^{\dagger}\left(\beta,y_{j},t_{j},g_{(a,b),(\tilde{a},\tilde{b}),r}\right)}\right]_{-}^{2}\right),$

where $\left(v_{n}^{\dagger}\left(\beta,y,t,g\right)\right)_{g\in{\cal G}}$ is
a zero mean Gaussian process with a covariance kernel evaluated by

$\displaystyle\hat{h}_{2}^{\dagger}\left(\beta,y,t,g,g^{\ast}\right)$
$\displaystyle\equiv$
$\displaystyle\left\\{\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq
j\neq
k}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)m^{\dagger}\left(W_{i},W_{k},\beta,y,t,g^{\ast}\right)\right.$
$\displaystyle\left.-\left[\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g\right)\right]\cdot\left[\frac{1}{n\left(n-1\right)}\sum_{i\neq
j}m^{\dagger}\left(W_{i},W_{j},\beta,y,t,g^{\ast}\right)\right]\right\\},$

and
$\varphi_{n}^{\dagger}\left(\beta,y,t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)$
is a GMS function given by

$\varphi_{n}^{\dagger}\left(\beta,y,t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)\equiv\hat{\sigma}_{n}^{\dagger
2}(y)B_{n}I\left[\kappa_{n}^{-1}n^{\frac{1}{2}}\bar{m}_{n}^{\dagger}\left(\beta,y,t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)/\bar{\sigma}_{n}^{\dagger}\left(\beta,y,t,g_{(a,b),(\tilde{a},\tilde{b}),r}\right)>1\right].$

Here $B_{n}$ and $\kappa_{n}$ are two tuning parameters that should satisfy
Assumption 3.1. (e.g.,
$\kappa_{n}=\left(\left(1-\hat{p}_{1-D}^{1/3}\right)^{2/5}\times
0.6\ln(n)\right)^{\frac{1}{2}}$ and
$B_{n}=\left(0.8\ln\left(n\right)/\ln\ln\left(n\right)\right)^{\frac{1}{2}}$).
For a significance level of $\alpha<1/2$, let
$\hat{c}_{\mathbf{y},\eta,1-\alpha}\left(\beta,\mathbf{t}\right)$ be the
$1-\alpha+\eta$ sample quantile of
$T_{\mathbf{y},n,R}^{Asy}(\beta,\mathbf{t})$ given an arbitrarily small value
$\eta$ (e.g., $\eta=10^{-6}$). Then the $\left(1-\alpha\right)$-level
confidence set for
$\left(\beta_{0},T\left(y_{1}\right),\ldots,T\left(y_{q}\right)\right)$ is
computed as

$\left\\{\left(\beta,\mathbf{t}\right)\in
B\times\mathbb{R}^{q}:T_{\mathbf{y},n,R}(\beta,\mathbf{t})\leq\hat{c}_{\mathbf{y},\eta,1-\alpha}\left(\beta,\mathbf{t}\right)\right\\}.$

The size and power properties stated in Theorem 3.1 should apply to this
confidence set. Monte Carlo simulation in Section D in the supplementary
material examines the finite sample performance of this joint inference
procedure.

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