# An Interval Estimation Approach to Sample Selection Bias

Matthew Tudball1,∗, Qingyuan Zhao2, Rachael Hughes1, Kate Tilling1, Jack
Bowden1 1 MRC Integrative Epidemiology Unit, University of Bristol
2 Statistical Laboratory, DPMMS, University of Cambridge
∗ Correspondence to<EMAIL_ADDRESS>
https://github.com/matt-tudball/selection-interval

###### Abstract

A widespread and largely unaddressed challenge in statistics is that non-
random participation in study samples can bias the estimation of parameters of
interest. To address this problem, we propose a computationally efficient
interval estimator for a class of population parameters encompassing
population means, ordinary least squares and instrumental variables estimands
which makes minimal assumptions about the selection mechanism. Using results
from stochastic programming, we derive valid confidence intervals and
hypothesis tests based on this estimator. In addition, we demonstrate how to
tighten the intervals by incorporating additional constraints based on
population-level information commonly available to researchers, such as survey
response rates and covariate means. We conduct a comprehensive simulation
study to evaluate the finite sample performance of our estimator and conclude
with a real data study on the causal effect of education on income in the
highly-selected UK Biobank cohort. We are able to demonstrate that our method
can produce informative bounds under relatively few population-level auxiliary
constraints.

##  1 Introduction

Every statistical analysis begins by selecting a study sample from some
population of interest. When the sample is representative of the population
(e.g. a simple random draw), the inference we draw from the study can be
generalized to the whole population. However, if individuals are selected or
drop out of the sample non-randomly, then valid inference for the population
is no longer guaranteed. Non-random selection can systematically distort the
distribution of, and relationships among, variables within the sample. When
this, in turn, distorts the distribution of sample statistics of interest, we
call this selection bias. Because it occurs during data collection, selection
bias cannot be ameliorated with larger samples and it can rarely be detected
using the observed sample alone (Bareinboim et al., 2014).

As a motivating example, UK Biobank is a large population-based cohort widely
analysed by epidemiologists, health economists, clinicians and others. Studies
of this cohort show that participants differ systematically from the rest of
the UK population on measures such as education, health status, age and
geographical location (Fry et al., 2017). This has lead to an intensive
discussion among practitioners on the impact of non-random selection for
studies using UK Biobank data. For example, Naimi et al. (2017) argues that
the protective effects of moderate alcohol consumption on mortality found in
previous studies, several of which used UK Biobank data, may be driven by
higher probabilities of sample selection among both healthier individuals and
moderate drinkers. Inverse probability weighting could be used to correct for
this selection bias (Horvitz and Thompson, 1952; Stuart et al., 2011), but no
individual-level data on the non-selected is available for UK Biobank, so the
weights cannot be estimated.

To address this problem, Aronow and Lee (2013) propose an interval estimator
which provides bounds on population means under unknown probabilities of
sample selection given the assumption that the probabilities are bounded.
However, Aronow and Lee’s estimator has many limitations in practice: it is
restricted to population means; it does not provide a procedure for conducting
statistical inference; the bounds are often implausibly wide for reasonable
choices for the probability bounds; and it assumes no knowledge of the
selection mechanism or population from which the sample is drawn, whereas some
information is typically available to researchers. Miratrix et al. (2018)
builds on Aronow and Lee’s approach by showing that the probability weights
imply a population distribution for the outcome and that imposing shape
constraints, such as symmetry or log-concavity, on this distribution can
tighten the bounds for the population mean. While this is an appealing
approach for tightening the bounds, it relies on imposing structure on the
underlying data distributions, which may become difficult to justify in higher
dimensions. Miratrix et al. (2018) also has no procedure for conducting
statistical inference.

Our paper contributes to the literature on interval estimation for selection
bias by addressing the limitations of existing methods outlined above. Namely,
we extend Aronow and Lee’s estimator to population parameters defined by a
more general class of estimating equations (van der Vaart, 1998). We provide a
formal approach to inference and hypothesis testing based on the asymptotic
properties of stochastic programming problems described in Shapiro (1991). We
demonstrate that the percentile bootstrap confidence intervals of Zhao et al.
(2019) for Rosenbaum-type sensitivity models are also applicable in this
setting. Lastly, we show how domain knowledge or information from
representative samples (e.g. censuses) can be used to tighten the estimated
bounds; this information includes the survey response rate, population means
of covariates and knowledge of the direction in which certain variables
influence sample selection. This work is related to the small but growing
literature on sensitivity analyses for selection bias (see further: Huang and
Lee, 2015; Lash and Fink, 2003; Smith and VanderWeele, 2019).

##  2 Set-Up

We outline key notation and definitions that will be used throughout this
paper. Consider an independent and identically distributed draw of size $N$
from a potentially infinite population. For concreteness, we can think of this
finite draw as the set of individuals who are eligible to enter the sample.
Let $S_{i}\in\\{0,\,1\\}$ be a missingness indicator for whether individual
$i$ enrols in the sample, where $S_{i}=1$ indicates sample participation, and
let the observed sample size be denoted by $n=\sum_{i=1}^{N}S_{i}$. Without
loss of generality, assume $S_{1}=\ldots=S_{n}=1$ and
$S_{n+1}=\ldots=S_{N}=0$.

Denote $T_{i}\in\mathcal{T}\subset\mathbb{R}^{t}$ as some vector-valued random
variable observed within the selected sample. We assume that sample selection
is a deterministic function of some other random variable
$D_{i}\in\mathcal{D}\subseteq\mathbb{R}^{d}$ such that

$\operatorname*{pr}\bigl{(}S_{i}=1\mid
D_{i}\bigr{)}\,\equiv\,1/\lambda_{0}(D_{i})$

where $\lambda_{0}:\mathcal{D}\to[1,\,\infty)$ is individual $i$’s inverse
probability of sample selection.

Denote $E[\,\cdot\,]$ as the population expectation and $E_{S}[\,\cdot\,]$ as
a shorthand for $E[\,\cdot\,|\,S_{i}=1]$ (i.e. expectation in the selected
sample). We assume that our estimand, denoted by $\beta_{0}$, is defined by
the following moment condition:

$\begin{array}[]{l@{}l}0&{}=E_{S}\left[\lambda_{0}(D_{i})\left\\{f(T_{i})-\beta_{0}\,g(T_{i})\right\\}\right]=E\left[f(T_{i})-\beta_{0}\,g(T_{i})\right]\end{array}$
(1)

where $f,\,g:\mathcal{T}\to\mathbb{R}$ are finite-valued. The two expressions
are equivalent because
$E_{S}[\lambda_{0}(D_{i})f(T_{i})]P(S_{i}=1)=E[S_{i}\lambda_{0}(D_{i})f(T_{i})]=E[f(T_{i})]$
by the law of iterated expectations. Elements of $T_{i}$ can overlap or be
correlated with elements of $D_{i}$; indeed, if $T_{i}$ and $D_{i}$ are
independent, then it is easy to see that sample selection will not bias the
estimate. Writing the moment condition (1) in terms of the sample expectation
$E_{S}$ makes explicit that changing the function inside the expectation to
some other function $\lambda:\mathcal{D}\to[1,\,\infty)$ will imply a
different population estimand. Therefore, we will often write $\beta(\lambda)$
as the estimand for a given choice of $\lambda$; in this notation,
$\beta_{0}=\beta(\lambda_{0})$.

$\beta(\lambda)\,\equiv\,\left.E_{S}\bigl{[}\lambda(D_{i})\,f(T_{i})\bigr{]}\middle/E_{S}\bigl{[}\lambda(D_{i})\,g(T_{i})\bigr{]}.\right.$

The class of estimators characterized by (1) embed, among others, population
means and ordinary least squares and instrumental variable estimands.

$\begin{array}[]{r@{}l}&{}\text{1) Population mean:
}T_{i}=Y_{i},\,f(T_{i})=Y_{i},\,g(T_{i})=1\\\\[10.00002pt] &{}\text{2)
Ordinary least squares:
}T_{i}=(X_{i},Y_{i}),\,f(T_{i})=X_{i}\,Y_{i},\,g(T_{i})=X_{i}^{2}\\\\[10.00002pt]
&{}\text{3) Instrumental variables:
}T_{i}=(Z_{i},X_{i},Y_{i}),\,f(T_{i})=Z_{i}\,Y_{i},\,g(T_{i})=Z_{i}\,X_{i}\end{array}$

If $\lambda_{0}(D_{i})$ was known for each individual in the selected sample,
then we could consistently estimate $\beta_{0}$ using the inverse probability-
weighted estimator

$\begin{array}[]{l@{}l}\hat{\beta}_{n}(\lambda_{0})&{}=\displaystyle\left.\left(\sum_{i=1}^{n}\lambda_{0}(D_{i})\,f(T_{i})\right)\middle/\left(\sum_{i=1}^{n}\lambda_{0}(D_{i})\,g(T_{i})\right)\right..\end{array}$
(2)

The term $\lambda_{0}(D_{i})$, however, is not usually observed or even
estimable in an observational study in the absence of individual-level data on
the non-selected observations. In such cases, the parameter of interest
$\beta_{0}$ may not be identifiable. However, it is often reasonable to assume
that each individual’s unobserved sample selection probability is bounded.

######  Assumption 1.

(Bounded selection probabilities) For all $D_{i}\in\mathcal{D}$, with
probability one,

$\begin{array}[]{l@{}l}&{}\,1\,\leq\,1/b\,\leq\,\lambda_{0}(D_{i})\,\leq\,1/a\,<\,\infty.\end{array}$
(3)

where $a$ and $b$ are fixed constants.

Then, defining
$\Lambda(a,\,b)\equiv\left\\{\lambda\,:\,\mathcal{D}\to[1/b,\,1/a]\right\\}$
as the set of all functions mapping the support of $D_{i}$ to the interval
$[1/b,\,1/a]$, we may only identify the set

$I(a,\,b)=[\beta^{L},\,\beta^{H}],~{}\beta^{H}=\displaystyle\sup_{\lambda}\bigl{\\{}\beta(\lambda):\lambda\in\Lambda(a,\,b)\bigr{\\}},~{}\beta^{L}=\displaystyle\inf_{\lambda}\bigl{\\{}\beta(\lambda):\lambda\in\Lambda(a,\,b)\bigr{\\}}$
(4)

The population optimisation problem (4) is infinite-dimensional and the number
of parameters in the corresponding sample optimisation problem grows with the
sample size. In order to apply asymptotic theory for stochastic programs,
which requires a finite number of parameters, we impose a technical assumption
on the support $\mathcal{T}$ requiring it to be discrete.

######  Assumption 2.

(Discrete support)
$T_{i}\in\mathcal{T}=\\{t_{k}\,:\,k=1,2,...,K\\}\subset\mathbb{R}^{t}$.

Let $\hat{p}_{n}(\,\cdot\,)$ denote the empirical mass of $T_{i}$ and let
$p(\,\cdot\,)$ denote the population mass. When Assumption 2 holds, we can
rewrite $\beta(\lambda)$, for some $\lambda\in\Lambda(a,\,b)$, as

$\begin{array}[]{l@{}l}\beta(\lambda)\par&{}=\displaystyle\left.\left(\sum_{k=1}^{K}w_{k}\,f(t_{k})\,p(t_{k})\right)\middle/\left(\sum_{k=1}^{K}w_{k}\,g(t_{k})\,p(t_{k})\right)\right.\par\equiv\,\beta(w)\end{array}$
(5)

where $w_{k}=E_{S}[\lambda_{0}(D_{i})\mid T_{i}=t_{k}]$ is the average of the
inverse probability weights at the value $t_{k}$. We can define the sample
estimator similarly:

$\hat{\beta}_{n}(w)=\displaystyle\left.\left(\sum_{k=1}^{K}w_{k}\,f(t_{k})\,\hat{p}_{n}(t_{k})\right)\middle/\left(\sum_{k=1}^{K}w_{k}\,g(t_{k})\,\hat{p}_{n}(t_{k})\right)\right..$
(6)

Therefore, we need only optimize over $w\in\mathcal{W}$ where
$\mathcal{W}\,\equiv\,[1/b,\,1/a]^{K}$. Under Assumption 2, the variational
optimization problem (4) is equivalent to a finite-dimensional linear
fractional programming problem. Simulations in Section 6 suggest that the
discrete support assumption might not be necessary for our asymptotic theory
to hold.

##  3 Estimation

We estimate $I(a,\,b)$ with natural estimators which replace $\beta(w)$ with
$\hat{\beta}_{n}(w)$ in the optimisation problem.

$\hat{I}_{n}(a,\,b)=[\hat{\beta}_{n}^{L},\,\hat{\beta}_{n}^{H}],~{}\hat{\beta}_{n}^{H}=\displaystyle\sup_{w}\bigl{\\{}\hat{\beta}_{n}(w)\,:\,w\in\mathcal{W}\bigr{\\}},~{}\hat{\beta}_{n}^{L}=\displaystyle\inf_{w}\bigl{\\{}\hat{\beta}_{n}(w)\,:\,w\in\mathcal{W}\bigr{\\}}$
(7)

and $\hat{\beta}_{n}(w)$ is defined as in (6). The optimization problem (7) is
deterministic and can be solved via a simple algorithm described below. For
(4) and (7) to have unique solutions, we need to impose two additional
regularity conditions outlined in the Supplementary Material. The first
regularity condition requires that $f(t_{k})/g(t_{k})\neq\beta^{m}$ for all
$k$ and $m=L,H$. If this is violated for some $k$, then the optimal value is
identical for all values of $w_{k}$, meaning that the solution is not unique.
An example of a function violating this condition is the following:

$\beta(w)\,=\,\displaystyle\frac{w_{1}\,+\,7w_{2}\,+\,10w_{3}}{w_{1}\,+\,w_{2}\,+\,w_{3}},~{}\mathcal{W}\,=\,[1,2]^{3}$

In this example, $w=(1,1,2)$ and $w=(1,2,2)$ are both maximizers over
$\mathcal{W}$. The maximum value is 7 but $f(t_{2})/g(t_{2})=7$ as well, which
violates the condition. The second reguarity condition simply ensures that the
denominators of $\beta(w)$ and $\hat{\beta}_{n}(w)$ are non-zero for all
$w\in\mathcal{W}$ with probability one. This will hold trivially for
estimators with strictly positive denominators (such as population means) but
may not hold, for example, with instrumental variables estimators when the
instrument is weak. We prove that these assumptions imply solution uniqueness
with the following lemma:

######  Lemma 3.1.

For $m=L,H$, under Assumption 1, 2 and some minor regularity conditions, the
set $\mathcal{W}^{m}\equiv\\{w\in\mathcal{W}\,:\,\beta(w)=\beta^{m}\\}$ is a
singleton, $\mathcal{W}^{m}=\\{w^{m}\\}$. Similarly, for $n$ sufficiently
large,
$\hat{\mathcal{W}}^{m}\equiv\\{w\in\mathcal{W}\,:\,\hat{\beta}_{n}(w)=\hat{\beta}_{n}^{m}\\}$
is a singleton, $\hat{\mathcal{W}}^{m}=\\{\hat{w}^{m}\\}$, with probability
one.

It can be seen in the proof of Lemma 3.1 that the maximizing weights under the
empirical (and population) measure will be of the form

$w_{k}^{H}=\begin{cases}1/b&\text{ if
}f(t_{k})/g(t_{k})\,\leq\,\hat{\beta}_{n}^{H}~{}(\beta^{H})\\\ 1/a&\text{ if
}f(t_{k})/g(t_{k})\,>\,\hat{\beta}_{n}^{H}~{}(\beta^{H})\end{cases}$ (8)

and the minimizing weights will be of an equivalent form but with the
inequalities reversed. Both Proposition 1 of Aronow and Lee (2013) and Section
4.4 of Zhao et al. (2019) propose equivalent algorithms for computing the
optimizing weights in the population means setting. Lemma 3.1 shows that we
can generalize this algorithm to our broader class of estimands. In short, we
can order $f(t_{k})/g(t_{k})$ from smallest to largest and evaluate
$\hat{\beta}_{n}(w)$ by enumerating over the weight at which $1/b$ changes to
$1/a$, which has computational complexity $O(n)$. Alternatively, one could use
a general solver for linear fractional programs.

The following theorem establishes consistency of our estimator:

######  Theorem 3.1.

For $m=L,H$, let $\hat{w}^{m}\in\mathcal{W}$ denote the $K$-dimensional weight
vector solving the sample optimization problem (7) and let $w^{m}$ denote the
population optimizing weights. Then, under Assumption 1, 2 and some minor
regularity conditions, $\hat{\beta}_{n}^{m}\,\to\,\beta^{m}$ and
$\hat{w}^{m}\,\to\,w^{m}$ with probability one as $n\to\infty$.

We can also make a statement about the finite sample bias of (7). Theorem 21.3
of Davidson (1994) is a general statement about sample averages and implies
that if $\hat{\beta}_{n}(w)$ is an unbiased estimator for $\beta(w)$ for fixed
$w$, then, on average, $\hat{\beta}_{n}^{L}$ will under-estimate the true
$\beta^{L}$ and $\hat{\beta}_{n}^{H}$ will over-estimate the true $\beta^{H}$.
Since the numerator and denominator of (6) are dependent in sample,
unbiasedness for fixed $w$ is not true in general. The instrumental variables
estimator, for example, is biased in finite sample and falls into the class of
estimators we consider.

##  4 Inference

###  4.1 Inference under asymptotic normality

There is a well-established literature in stochastic optimization theory on
using sample average approximations to solve optimization problems
characterized by functions of expected values. Since optimization of expected
values is often intractable, sample average approximation methods draw a large
number of values from the underlying distribution and optimize the sample
average instead, which is a deterministic function. Value and solution
convergence rates and distributional properties of these approximations are
well known (Shapiro et al., 2009). By framing our estimand as the solution to
a stochastic optimization problem and our estimator as a sample average
approximation, we can utilize many of these known properties to conduct valid
inference. This approach to inference is particularly useful for dealing with
large sample sizes, where bootstrap approaches may be computationally
intensive; for example, our motivating dataset, UK Biobank, has over 500,000
observations.

We begin with an assumption on the convergence properties of our estimator.

######  Assumption 3.

(Convergence in distribution)
a) For all $w\in\mathcal{W}$,
$\sqrt{n}(\hat{\beta}_{n}(w)\,-\,\beta(w))\,\to\,\mathcal{N}\bigl{(}0,\,\sigma^{2}(w)\bigr{)}$
in distribution.
b) Let $C(\mathcal{W})$ denote the space of continuous functions with domain
$\mathcal{W}$. The sequence
$V_{n}(w)=\sqrt{n}(\hat{\beta}_{n}(w)\,-\,\beta(w))$ converges in distribution
to a random element $V(w)\in C(\mathcal{W})$, where $V(w)$ is Gaussian with
mean 0 and variance $\sigma^{2}(w)\in C(\mathcal{W})$.

Assumption 3a means pointwise convergence in distribution of
$\hat{\beta}_{n}(w)$ for any fixed $w$, while Assumption 3b means joint
convergence of the entire stochastic process. Therefore, Assumption 3b is
stronger than Assumption 3a. Assumption 3b is a standard assumption in the
stochastic optimization literature and is typically justified by some uniform
integrability condition. See Shapiro et al. (2009, p.164–165) and Banholzer et
al. (2019, p.10–11) for a discussion.

Theorem 3.2 of Shapiro (1991), which is central to the results in this
section, is restated in the Supplementary Material. It establishes a central
limit theorem for the extremal values of convergent sequences of random
functions. We can apply it to our estimator (7) to obtain the following
theorem:

######  Theorem 4.1.

Under Assumption 1, 2 and 3b and some minor regularity conditions,

$\sqrt{n}\left(\hat{\beta}_{n}^{m}-\beta^{m}\right)\to\mathcal{N}\left(0,\,\sigma^{2}(w^{m})\right),~{}m=L,H$
(9)

in distribution, where $\sigma^{2}(w^{m})$ is the asymptotic variance of
$\hat{\beta}_{n}^{m}$.

Because $\mathcal{W}$ is compact, $\sqrt{n}(\hat{\beta}_{n}\,-\,\beta)$
converges in distribution to a Gaussian with mean zero and variance
$\sigma^{2}$ by Assumption 3b and $\beta(w)$ has a unique minimizer and
maximizer by Lemma 3.1, Theorem 4.1 follows from Theorem 3.2 of Shapiro
(1991). Naturally, in finite sample, $\sigma^{2}(w^{m})$ will need to be
estimated. We establish the conditions needed for a valid estimator with the
following proposition:

######  Proposition 4.1.

Suppose that $\sigma^{2}(w)$ is continuous and there exists some
$\hat{\sigma}^{2}_{n}(w)$ such that
$\sup_{w\in\mathcal{W}}|\hat{\sigma}^{2}_{n}(w)\,-\,\sigma^{2}(w)|\,\to 0$
with probability one. Then

$\hat{\sigma}^{2}_{n}(\hat{w}^{m})\,\to\,\sigma^{2}(w^{m}),~{}m=L,H.$

with probability one.

Some sufficient conditions for almost sure convergence of
$\sup_{w\in\mathcal{W}}|\hat{\sigma}^{2}_{n}(w)\,-\,\sigma^{2}(w)|$ are given
by Theorem 21.10 in Davidson (1994).

Now we can define a two-sided $100(1-\alpha)\%$ confidence interval for
$I(a,\,b)$ by
$[C_{n}^{L}(\hat{w}^{L};\alpha/2),\,C_{n}^{H}(\hat{w}^{H};\alpha/2)]$ such
that

$C_{n}^{L}(\hat{w}^{L};\alpha)=\hat{\beta}_{n}^{L}-Z_{\alpha/2}\,\frac{\hat{\sigma}_{n}(\hat{w}^{L})}{\sqrt{n}},~{}C_{n}^{H}(\hat{w}^{H};\alpha)=\hat{\beta}_{n}^{H}+Z_{\alpha/2}\,\frac{\hat{\sigma}_{n}(\hat{w}^{H})}{\sqrt{n}}$
(10)

where $Z_{\alpha}$ is the upper $\alpha$-quantile of the standard normal
distribution. This is the $100(1-\alpha)\%$ confidence interval for the
identified interval $I(a,\,b)$ and not for the parameter $\beta_{0}$. It
follows from Lemma 1 of Imbens and Manski (2004) that this confidence interval
has at least $1-\alpha$ coverage for $\beta_{0}$ as well, however, it will
likely be too conservative if $\beta_{0}$ is the target of coverage.

We can perform hypothesis testing by inverting these confidence intervals.

$H_{0}:\,\tilde{\beta}_{0}\in[\beta^{L},\,\beta^{H}],~{}H_{1}:\,\tilde{\beta}_{0}\notin[\beta^{L},\,\beta^{H}]$
(11)

where $\tilde{\beta}_{0}$ is a hypothesized value for $\beta_{0}$. The
following proposition gives an asymptotically valid p-value:

######  Proposition 4.2.

A valid p-value for the hypothesis
$\tilde{\beta}_{0}\in[\beta^{L},\,\beta^{H}]$ is

$\displaystyle\Phi\left(\frac{\hat{\beta}_{n}^{H}-\tilde{\beta}_{0}}{\hat{\sigma}_{n}(\hat{w}^{H})/\sqrt{n}}\right)\,-\,\Phi\left(\frac{\hat{\beta}_{n}^{L}-\tilde{\beta}_{0}}{\hat{\sigma}_{n}(\hat{w}^{L})/\sqrt{n}}\right)$
(12)

where $\Phi(\,\cdot\,)$ denotes the standard normal CDF.

We demonstrate that this p-value has good small sample properties via
simulation in Section 6.

###  4.2 Bootstrap confidence intervals

We can also obtain asymptotically valid inference without assuming convergence
of the whole process $\hat{\beta}_{n}(w)$ (Assumption 3b) by using the
percentile bootstrap proposed in Zhao et al. (2019). An additional advantage
of this method over the asymptotic approach in the previous section is that it
does not require Assumption 2 of a discrete support $\mathcal{T}$ for the data
$T_{i}$, provided $\mathcal{T}$ is compact. We refer readers to Zhao et al.
(2019) for a detailed exposition of their approach, which we only briefly
summarize here. Let $T\equiv\\{T_{1},\ldots,T_{n}\\}$ be a sample drawn non-
randomly from the population. For some $\lambda\in\Lambda(a,\,b)$, let
$\hat{\beta}_{n,r}(\lambda)$ be the inverse probability weighted estimate (2)
from the $r$th independent and identically distributed resample from the
empirical distribution of $T$, $r=1,...,R$. Then the percentile bootstrap
confidence interval is written as $[\tilde{C}^{L},\,\tilde{C}^{H}]$ where

$\tilde{C}^{L}_{n}=Q_{\alpha/2}\biggl{(}\biggl{(}\inf_{\lambda\in\Lambda(a,\,b)}\hat{\beta}_{n,r}(\lambda)\biggr{)}_{r=1,\ldots,R}\biggr{)},~{}\tilde{C}^{H}_{n}=Q_{1-\alpha/2}\biggl{(}\biggl{(}\sup_{\lambda\in\Lambda(a,\,b)}\hat{\beta}_{n,r}(\lambda)\biggr{)}_{r=1,...,R}\biggr{)}.$
(13)

where $Q_{\alpha}(\,\cdot\,)$ denotes the $\alpha$ percentile. In practice,
this involves solving $R$ fractional programming problems from the bootstrap
distribution and calculating the desired percentiles.

######  Proposition 4.3.

Given Assumption 1, Assumption 3a, compactness of $\mathcal{T}$ and the
regularity conditions in Theorem 4.1, $[\tilde{C}^{L}_{n},\tilde{C}^{H}_{n}]$
is an asymptotically valid confidence set for $\beta_{0}$ and $I(a,b)$ with
coverage at least $1-\alpha$ under the collection of weight functions
$\Lambda(a,b)$.

Proof of Proposition 4.3 follows directly from Theorem 2 of Zhao et al.
(2019).

##  5 Incorporating population-level information

###  5.1 General framework

In the original set-up in Section 2, we assume no knowledge of the population
or selection mechanism beyond the bounding parameters $a$ and $b$. However, we
often have some additional information about the target population. It seems
natural to use this information to place more constraints on the weights. In
this section, we develop a general framework for incorporating auxiliary
information into the constraint set and conclude with two specific examples:
the survey response rate and covariate means.

Suppose the weights $w$ fall into the constraint set

$\mathcal{W}_{0}\,\equiv\,\biggl{\\{}w\in\mathcal{W}\,:\,h_{j}(w)\,\leq\,0,~{}j=1,...,J\biggr{\\}},~{}h_{j}(w)\,\equiv\,\displaystyle\sum_{k=1}^{K}w_{k}\,H_{j}(t_{k})\,p(t_{k}).$
(14)

We give examples of possible constraint functions in Sections 5.2 and 5.3.
Since $h_{j}(w)$ are functions of the unobserved population mass $p$, we
cannot directly observe $\mathcal{W}_{0}$. The naive approach of simply
replacing $p$ with $\hat{p}_{n}$ and computing the sample expectation is
problematic because there is no guarantee that the resulting set will contain
the population constraint set $\mathcal{W}_{0}$. To solve this problem, we
need to construct a set that contains $\mathcal{W}_{0}$ with high probability
under repeated sampling. The approach we adopt is to relax the sample
constraint set such that solutions are feasible to the population problem with
some level of confidence. This will involve selecting a $J$-dimensional
sequence $\epsilon_{n}(w)\geq 0$ and letting the relaxed sample feasible
region be written as

$\hat{\mathcal{W}}_{\epsilon_{n}}\,\equiv\,\biggl{\\{}w\in\mathcal{W}\,:\,\hat{h}_{j}(w)\,\leq\,\epsilon_{jn}(w),~{}j=1,...,J\biggr{\\}},~{}\hat{h}_{j}(w)\,\equiv\,\displaystyle\sum_{k=1}^{K}w_{k}\,H_{j}(t_{k})\,\hat{p}_{n}(t_{k}).$

Then we can write the sample maximization problem as
$\sup_{w}\\{\hat{\beta}_{n}(w)\,:\,w\in\hat{\mathcal{W}}_{\epsilon_{n}}\\}$.

Setting aside the possible computational complexity of this constraint, it
remains to select a sequence $\epsilon_{n}(w)$ that ensures feasibility with
some level of confidence $0<\alpha_{1}<1$, that is,
$\operatorname*{pr}\bigl{(}\mathcal{W}_{0}\subseteq\hat{\mathcal{W}}_{\epsilon_{n}}\bigr{)}\geq
1-\alpha_{1}$. If $J=1$ then a natural choice is to construct a one-sided
$100(1-\alpha_{1})\%$ confidence interval around $\hat{h}_{1}(w)$. If
$\hat{\sigma}_{h_{1}}(w)$ is the sample standard deviation of $\hat{h}_{1}(w)$
then we can write

$\epsilon_{1n}(w)=Z_{\alpha_{1}}\,\hat{\sigma}_{h_{1}}(w)/\sqrt{n}$ (15)

where $Z_{\alpha_{1}}$ is the upper $\alpha_{1}$-quantile of the standard
normal distribution. The resulting $\hat{\mathcal{W}}_{\epsilon_{n}}$
naturally has the desired coverage as $n$ becomes large. It is straight-
forward to extend this to $J>1$. If we construct confidence intervals around
each $\hat{h}_{j}(w)$ with $1-\alpha_{1j}$ coverage, then, by the intersection
bound, $\hat{\mathcal{W}}_{\epsilon_{n}}$ has coverage of $\mathcal{W}_{0}$ of
at least $1-\sum_{j=1}^{J}\alpha_{1j}$. We denote constraint sets with this
property by

$\begin{array}[]{l@{}l}&{}\hat{\mathcal{W}}_{\alpha}\,\equiv\,\biggl{\\{}w\in\mathcal{W}\,:\,\hat{h}_{j}(w)\,\leq\,Z_{\alpha_{1j}}\,\hat{\sigma}_{h_{j}}(w)n^{-1/2},~{}j=1,...,J\biggr{\\}},~{}\alpha_{1}\,=\,\displaystyle\sum_{k=1}^{K}\alpha_{1j}.\end{array}$
(16)

The remaining challenge is to assess the overall coverage of the confidence
interval (10) computed over the relaxed constraint set (16). To achieve this,
we need to impose some regularity condition on $\hat{\sigma}^{2}_{n}(w)$ of
Proposition 4.1 with the following assumption.

######  Assumption 4.

Suppose $\hat{\sigma}^{2}_{n}(w)$ is a consistent estimator for
$\sigma^{2}(w)$ in the sense of Proposition 4.1. Then, for sufficiently large
$n$, $\hat{\sigma}^{2}_{n}(w)$ is Lipschitz continuous in $w$ with probability
one.

It follows from van Kempen and Vliet (2000) that Assumption 2 implies
Lipschitz continuity in $\hat{\sigma}^{2}_{n}(w)$ up to a first order Taylor
expansion. Therefore, Assumption 4 only imposes some regularity condition on
the remainder term.

The following theorem shows that, for a suitable choice of $\alpha_{1}$,
defined as the significance level of the relaxed sample constraint set (16)
and $\alpha_{2}$, defined as the significance level of the confidence interval
(10), we can achieve nominal coverage for the true identified interval over
$\mathcal{W}_{0}$.

######  Theorem 5.1.

Define $\mathcal{W}_{0}$ as in (14) and define the sequence
$\\{\hat{\mathcal{W}}_{\alpha_{1}}\\}$ as in (16), where $\alpha_{1}$ is some
positive constant such that

$\lim_{n\to\infty}\operatorname*{pr}\bigl{(}\mathcal{W}_{0}\,\subseteq\,\hat{\mathcal{W}}_{\alpha_{1}}\bigr{)}\,\geq\,1-\alpha_{1}$

and let $\alpha_{2}$ denote the significance level of (10). Let

$\hat{w}^{L}_{\alpha_{1}}\,\equiv\,\displaystyle\operatorname*{arg\,min}_{w\in\hat{\mathcal{W}}_{\alpha_{1}}}\hat{\beta}_{n}(w),~{}w^{L}_{0}\,\equiv\,\displaystyle\operatorname*{arg\,min}_{w\in\mathcal{W}_{0}}\beta(w)$

where $\hat{w}^{H}_{\alpha_{1}}$ and $w^{H}_{0}$ are similarly defined for the
$\operatorname*{arg\,max}$. Further, assume $w_{0}^{L}$ and $w_{0}^{H}$ are
unique. Then, under Assumptions 1, 2, 3b and 4,

$\lim_{n\to\infty}\operatorname*{pr}\biggl{(}\bigl{[}\beta\bigl{(}w_{0}^{L}\bigr{)},\,\beta\bigl{(}w_{0}^{H}\bigr{)}\bigr{]}\,\subseteq\,\bigl{[}C_{n}^{L}\bigl{(}\hat{w}_{\alpha_{1}}^{H};\alpha_{2}\bigr{)},\,C_{n}^{H}\bigl{(}\hat{w}_{\alpha_{1}}^{H};\alpha_{2}\bigr{)}\bigr{]}\biggr{)}\,\geq\,1\,-\,\alpha_{1}\,-\,\alpha_{2}.$

To briefly sketch the proof, we begin by showing that, under some regularity
conditions, the interval
$\bigl{[}C_{n}^{L}\bigl{(}\hat{w}_{\alpha_{1}}^{H};\alpha_{2}\bigr{)},\,C_{n}^{H}\bigl{(}\hat{w}_{\alpha_{1}}^{H};\alpha_{2}\bigr{)}\bigr{]}$
becomes arbitrarily close to the interval characterized by the infimum of
$C_{n}^{L}\bigl{(}w)$ and supremum of $C_{n}^{H}\bigl{(}w)$ over
$\hat{\mathcal{W}}_{\alpha_{1}}$ as $n$ becomes large. This is achieved by
applying Theorem 5.5. of Shapiro et al. (2009), which establishes consistency
of sample average approximations solved over estimated constraint sets. From
here, we are able to apply a similar argument to the main lemma of Berger and
Boos (1994), which constructs a valid confidence set for a target parameter in
the presence of nuisance parameters which cannot be estimated but can be
bounded with high probability. In our context, $\beta_{0}$ is the target
parameter and $w_{k}$ are the nuisance parameters. In addition to these,
uniqueness of $w_{0}^{L}$ and $w_{0}^{H}$ is needed to invoke Theorem 3.2 of
Shapiro (1991) over $\mathcal{W}_{0}$. The population maximisation problem
over the set (14) is a linearly-constrained linear fractional programming
problem. Then, since the level sets of $\beta(w)$ are linear, to establish
uniqueness of $w_{0}^{L}$ and $w_{0}^{H}$ it suffices to assume that the
coefficient vectors of $h_{j}$ and the level sets of $\beta(w)$ over
$\mathcal{W}_{0}$ are not parallel. As with the original problem, we also need
all coefficients of the level set for the optimal value to be non-zero on
$\mathcal{W}_{0}$.

In the Supplementary Material we demonstrate via simulation the coverage
frequency of these confidence intervals and the optimal choice of $\alpha_{1}$
and $\alpha_{2}$ for fixed $\alpha_{1}+\alpha_{2}$.

Since the percentile bootstrap confidence intervals (13) require strictly
fewer assumptions, a straight-forward corollary of Theorem 5.1 is they will
also have at least $1-\alpha_{1}-\alpha_{2}$ asymptotic coverage when computed
over the constraint set $\hat{\mathcal{W}}_{\alpha_{1}}$.

###  5.2 Survey response rate

The response rate represents the unconditional probability that a given
individual will enter the sample and is commonly available in survey-based
data. One approach to tighten the bounds is to ensure that the optimizing
weights imply a response rate that is consistent with the true response rate.

Define $r\equiv E\left[S_{i}\right]$ as the population survey response rate
and $\pi_{i}\equiv\operatorname*{pr}\bigl{(}S_{i}=1\bigr{)}$ as individual
$i$’s probability of sample selection, then $E_{S}\left[1/\pi_{i}\right]=1/r$.
This means that the expected value within the sample of the inverse selection
probabilities is equal to the inverse response rate. Since this is a within-
sample expectation, we can use it to construct a constraint set consistent
with the framework in Section 5.1.

Let $c_{n}\equiv Z_{\alpha_{1}/2}/\sqrt{n}$, then we can construct a quadratic
constraint set with $1-\alpha_{1}$ coverage by squaring both sides of the
inequality in (16):

$\begin{array}[]{l@{}l}&{}\hat{\mathcal{W}}_{\alpha_{1}}\,\equiv\,\biggl{\\{}w\in\mathcal{W}\,:\displaystyle\,\bigl{(}1+c_{n}^{2}\bigr{)}\biggl{\\{}\sum_{k=1}^{K}(w_{k}-1/r)\hat{p}_{n}(t_{k})\biggr{\\}}^{2}\,\leq\,c_{n}^{2}\sum_{k=1}^{K}(w_{k}-1/r)^{2}\hat{p}_{n}(t_{k})\biggr{\\}}.\end{array}$
(17)

While this optimisation problem cannot be solved by the algorithm described in
Section 3, it can be solved by other methods (e.g. sequential quadratic
programming).

###  5.3 Covariate means

In many instances, we know population means of elements of $T_{i}$ from domain
knowledge or large representative samples (e.g. national censuses). Suppose we
know $E[Q_{i}]=\bar{q}$, where $Q_{i}\in T_{i}$. Since $\bar{q}$ is known, we
want the optimizing weights to be consistent with this known population value
in the sense that

$E_{S}[\lambda_{0}(D_{i})(Q_{i}-\bar{q})]\,=\,\sum_{k=1}^{K}w_{k}(q_{k}-\bar{q})p(t_{k})\,=\,0.$
(18)

where $q_{k}$ is the value of $Q_{i}$ in the vector $t_{k}$. Let $c_{n}\equiv
Z_{\alpha_{1}/2}/\sqrt{n}$, then we can construct the sample constraint set in
much the same way as (17):

$\begin{array}[]{l@{}l}&{}\hat{\mathcal{W}}_{\alpha_{1}}\,\equiv\,\biggl{\\{}w\in\mathcal{W}\,:\displaystyle\,\bigl{(}1+c_{n}^{2}\bigr{)}\biggl{\\{}\sum_{k=1}^{K}w_{k}(q_{k}-\bar{q})\hat{p}_{n}(t_{k})\biggr{\\}}^{2}\,\leq\,c_{n}^{2}\sum_{k=1}^{K}w_{k}^{2}(q_{k}-\bar{q})^{2}\hat{p}_{n}(t_{k})\biggr{\\}}.\end{array}$
(19)

Conceptually, this constraint is similar to the raking procedure in survey
sampling outlined in Deming and Stephan (1940), which is a finite-population
technique that adjusts estimated survey weights to imply known marginal counts
from the population.

###  5.4 Parametric assumptions

Suppose that $D_{i}$ is observed within the selected sample and
$\mathcal{D}\subseteq\mathbb{R}^{d}$. Until this point, we have made no claims
about whether $D_{i}$ is observable and so this does represent an additional
assumption. If we also suppose that $\lambda_{0}$ falls into some parametric
family, then we can optimize over the parameters of that family, tightening
the bounds over the non-parametric estimators considered thus far. This is
still a selection-on-unobservables model in the classical sense, since $D_{i}$
is only observed within the sample. Let $h:\mathbb{R}\to\mathbb{R}$ be some
continuous and monotonically increasing function. Then we define

$\Theta\,\equiv\,\bigl{\\{}\alpha\,:\,h(\alpha_{0}+\alpha_{1}^{T}D_{i})\in[1/b,\,1/a]~{}\text{for
all}~{}D_{i}\in\mathcal{D}\bigr{\\}}$

as the set of parameters restricting this function to lie in the interval
$[1/b,1/a]$, where $\alpha=(\alpha_{0},\alpha_{1})$. Then we can define a
parametric family for $h(\cdot)$ as

$\mathcal{F}_{h}\,\equiv\,\bigl{\\{}h(\alpha_{0}+\alpha_{1}^{T}D_{i})\,:\,\alpha\in\Theta\bigr{\\}}\,\subseteq\,\Lambda(a,\,b).$
(20)

This embeds a broad class of generalized linear models such as logit and
probit. If $\lambda_{0}\in\mathcal{F}_{h}$, the resulting optimization problem
takes the form

$\begin{array}[]{l@{}l}\hat{\beta}_{n}^{H}&{}=\displaystyle\max_{\alpha}\left\\{\hat{\beta}_{n}(h)\,:\,\alpha\in\Theta\right\\}\\\
&{}=\displaystyle\max_{\alpha}\left\\{\hat{\beta}_{n}(h):h^{-1}(b^{-1})\leq\alpha_{0}+\alpha_{1}^{\prime}D_{i}\leq
h^{-1}(a^{-1}),~{}i=1,\ldots,n\right\\}.\end{array}$ (21)

Because we are optimizing over $d+1$ parameters, Assumption 2, which assumes a
finite support for $T_{i}$, is no longer necessary to prove consistency or
validity of our inference techniques. However, this optimization problem
cannot be solved using fractional programming techniques as before. Since the
constraints are linear, when $d$ is relatively small, this problem can instead
be solved using global optimization methods. As an example, for logit,
$h(\cdot)$ can be written as
$h(\alpha_{0}+\alpha_{1}^{T}D_{i})=1+\exp(\alpha_{0}+\alpha_{1}^{\prime}D_{i})$,
which is continuous and monotonically increasing as desired. Further, if
$\mathcal{D}$ is compact, since the image of $h$ is bounded, $\Theta$ is also
compact. Therefore, consistency of (21) follows from similar arguments to
those in Theorem 3.1. The asymptotic inference established in Section 4.1 will
continue to hold provided the optimizing parameters are unique, however, the
bootstrap confidence intervals of Section 4.2 will be valid under the weaker
condition of pointwise convergence in distribution of $\hat{\beta}_{n}(h)$ for
fixed $h\in\mathcal{F}_{h}$ (Zhao et al., 2019).

Another convenient property of this extension is that external information on
the direction of $D_{i}$’s effects on selection can easily be incorporated.
For example, if years of education is an element of $D_{i}$ and domain
knowledge indicates that better educated individuals are more likely to select
into the sample, then the element of $\alpha_{1}$ associated with years of
education can be constrained to be non-negative. This can be done by a simple
modification of the parameter space. If we believe $\alpha_{11}\geq 0$, for
example, then we can modify the parameter space by adding an additional linear
constraint $\\{\alpha\in\Theta\,:\,\alpha_{11}\geq 0\\}$.

##  6 Simulations

Assumption 2 is restrictive in requiring the support of the data to be
discrete. The aim of these simulations is to assess and visualize the finite
sample and limiting properties of our estimator $\hat{I}(a,\,b)$ when this
assumption does not hold, that is, when the support is continuous and
unbounded. We demonstrate that our results, in particular those of Sections 4,
appear to hold in the absence of this assumption.

We consider estimation of the mean of $Y_{i}$ which follows a standard normal
distribution $\mathcal{N}(0,1)$ within the hypothetical selected sample. For
the purposes of this simulation, we do not need to explicitly model the
population distribution or sample selection process. Since it is difficult to
calculate $I(a,\,b)$ exactly in this setting, we instead compute our
population parameters in a large finite sample of $N=1,000,000$. Sub-samples
of varying size $n$ are drawn randomly without replacement from this sample.
The parameters for this data-generating process are

$a=0.1,\,b=1,\,I(a,\,b)=[\beta^{L},\,\beta^{H}]\approx[-0.902,\,0.902]$

(a) The red line is the average bias of the upper bound
$\hat{\beta}_{n}^{H}-\beta^{H}$ and the blue line is the average bias of the
lower bound $\hat{\beta}_{n}^{L}-\beta^{L}$ averaged over 5,000 samples.

(b) The red line is the coverage frequency of the percentile bootstrap CI (13)
and the blue line is the coverage probability of the asymptotic CI (10)
averaged over 5,000 samples.

(c) The blue line is the power curve for the $p$-value (12) averaged over
1,000 samples. The red lines are the boundaries of the true interval
$[\beta^{L},\,\beta^{H}]$.

(d) The light blue bars are a histogram of $\hat{\beta}_{n}^{H}$ over 5,000
samples. The blue line is the true limiting distribution.

Figure 6.1: Each simulation is estimating the mean of
$Y_{i}\sim\mathcal{N}(0,1)$ with $a=0.1$ and $b=1$. The population identified
interval $[\beta^{L},\beta^{H}]\approx[-0.902,0.902]$ is approximated using a
large sample of size $N=1,000,000$ from which sub-samples are drawn without
replacement to calculate finite sample values.

Figure 6.1 illustrates our main simulation results. Figure 1(a) demonstrates
the finite sample bias of $\hat{\beta}_{n}^{L}$ and $\hat{\beta}_{n}^{H}$. We
see that, on average, the lower bound is underestimated and the upper bound is
overestimated, consistent with the discussion in Section 3. However, this bias
dissipates quickly. Figure 1(b) depicts the coverage frequency of our two
confidence intervals: the asymptotic approxmation (10) in blue and the
percentile bootstrap (13) in red. The two have roughly equivalent coverage
frequencies by $n=500$ and both converge to nominal 95% coverage. Figure 1(c)
shows the power curve for the hypothesis test (11) for $n=100$ using a
$p$-value cut-off of 0.05. The test exhibits some over-rejection around the
boundaries of the true identified interval. This is consistent with the under-
coverage seen at around $n=100$ in Figure 1(b). Lastly, Figure 1(d) shows a
histogram of $\hat{\beta}_{n}^{H}$ for $n=100$ and the true limiting
distribution fitted over it, which appears to be a reasonable approximation.
These simulations suggest that Assumption 2 might not be necessary for this
estimator to have desirable asymptotic properties.

##  7 Applied Example: Effect of Education on Income

We illustrate the utility of this method in an instrumental variable design
looking at the effect of education on income. An in-depth exposition of this
design can be found in Davies et al. (2018). Our instrument is based on a
September 1972 education reform in England which raised the school leaving age
from 15 to 16. Individuals who turned 15 just prior to the implementation of
this reform were allowed to leave school, while individuals who turned 15 just
after were required to remain in school until they were 16. This created a
sharp discontinuity in the policies that the two groups were exposed to, with
age as the forcing variable, which cannot be manipulated. We denote the
instrument by $Z_{i}\in\\{0,\,1\\}$.

Our exposure is whether an individual remained in school at least until age
16, denoted by $X_{i}\in\\{0,\,1\\}$, which associates with the instrument
with $F>100$. Lastly, our outcome is whether an individual earned more than
£31,000 per year at the time of UK Biobank measurement in 2006, denoted by
$Y_{i}\in\\{0,\,1\\}$. We restrict our sample to individuals who turned 15
within 12 months of September 1972 and we control for sex and month-of-birth
indicators. For a given weight vector $w$, we write our estimator as

$\begin{array}[]{l@{}l}\hat{\beta}_{n}(w)=\displaystyle\left.\left(\sum_{i=1}^{n}w_{i}\,Z_{i}\,Y_{i}\right)\middle/\left(\sum_{i=1}^{n}w_{i}\,Z_{i}\,X_{i}\right)\right.\end{array}$

As discussed in the introduction, UK Biobank is known to be non-representative
of the UK population along dimensions such as education, health and sex. Since
data on these characteristics is available within the UK Biobank sample, we
specify a logit model for $\lambda_{0}$ as in Section 5.4, choosing as
selection variables sex, years of education, income and days of physical
activity per week as a proxy for health (Fry et al., 2017; Hughes et al.,
2019).

For our bound constraint, we choose $a$ of 0.01 and $b$ of 1. This choice
allows some individuals to be as much as 100 times more likely to enter the
sample than others. We also impose some additional constraints. Specifically,
for our directionality constraint, we require the coefficients on years of
education, income and days of physical activity per week to be non-negative
and for the male indicator to be non-positive. We also know that the response
rate for UK Biobank is 5.5% and the proportion of males in the UK population
in the relevant age group is 49.5%, so these are included as constraints as in
Sections 5.2 and 5.3 respectively.

The unweighted point estimate is 0.183 with 95% confidence interval of [0.084,
0.282], indicating that an additional year of schooling induced by this reform
increases the likelihood of earning more than £31,000 in later life by 18.3
percentage points. When only the bound constraints are imposed, we estimate an
interval $[-1.509,2.694]$ with a 95% asymptotic confidence interval of
$[-4.022,5.381]$. This is very wide and contains the null. However, when we
impose all available constraints, we obtain an interval $[0.176,0.27]$ with a
95% asymptotic confidence interval of $[0.003,0.456]$. These constraints
substantially reduce the width of the estimated interval and result in
interpretable effect sizes. The unweighted estimate of 0.183 lies near the
lower bound of this interval, however, much of the interval contains values
larger than this. This is what we might expect given our knowledge of the
selection mechanism. Since education and income both positively influence
selection into UK Biobank, this could induce an artificial negative
correlation between the two, thus biasing the effect estimate downward.

##  8 Discussion

In order to directly apply some existing theoretical results from the
stochastic optimization literature, we assume that the data has a discrete
support. This is so that the number of parameters is finite and does not
depend on the sample size. We demonstrate by simulation in Section 6 that
desirable properties such as nominal coverage of confidence intervals appear
to hold for data distributed on a continuous unbounded support. It would be
valuable to provide theoretical guarantees for this by extending the results
in Shapiro (1991) and Shapiro et al. (2009) to allow for variational problems.

## Acknowledgements

This research has been conducted using the UK Biobank Resource. UK Biobank
received ethical approval from the Research Ethics Committee (REC reference
for UK Biobank is 11/NW/0382). This research was approved as part of
application 8786. Tudball would like to acknowledge financial support from the
Wellcome Trust (grant number 220067/Z/20/Z).

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## Supplementary material

### Additional assumptions

######  Assumption 5.

For all $t_{k}\in\mathcal{T}$, $f(t_{k})/g(t_{k})\,\neq\,\beta^{m}$ and, for
sufficiently large $n$, $f(t_{k})/g(t_{k})\,\neq\,\hat{\beta}_{n}^{m}$ almost
surely, $m=L,H$.

######  Assumption 6.

There exists some constant $m>0$ such that, for all $\lambda\in\Lambda(a,b)$
and sufficiently large $n$,

$E\bigl{[}\lambda(D_{i})\,g(T_{i})\bigr{]}\,\geq\,m,~{}\frac{1}{n}\sum_{i=1}^{n}\lambda(D_{i})\,g(T_{i})\,\geq\,m~{}\text{with
probability one.}$

### Constraint simulation

Section 5 introduces a framework for using population-level auxiliary
information, such as means of measured covariates, to tighten the estimated
bounds. The aim of this simulation is to demonstrate that the confidence
interval of Theorem 5.1 has correct coverage. We also show how varying
$\alpha_{1}$ and $\alpha_{2}$ affects the width of this confidence interval.

Consider two binomial-distributed random variables $Q_{i}$ and $E_{i}$ such
that

$Q_{i},E_{i}\,\sim\,B(100,0.5)$

which are scaled to have mean zero and variance one. We use binomial-
distributed random variables in this simulation since we can compute the
constrained population interval exactly. We define our outcome $Y_{i}\equiv
Q_{i}+E_{i}$ and assume the population mean of $Q_{i}$ is $\bar{q}=0.5$. We
can use the constraint set (19) of Section 5.3 to use this mean to tighten the
estimated interval. Figure 1(a) shows the coverage of the 95% confidence
interval of Theorem 5.1 with $\alpha_{1}=0.98$ and $\alpha_{2}=0.97$.
Consistent with our theoretical results, this confidence interval has correct
coverage but is conservative. Figure 1(b) shows how the width of this
confidence interval varies depending on the choice of $\alpha_{1}$ and
$\alpha_{2}$. When $\alpha_{1}$ approaches 0.95, $\alpha_{2}$ approaches 1 and
so the confidence interval becomes arbitrarily wide. When $\alpha_{1}$
approaches 1, $\alpha_{2}$ approaches 0.95 and we arrive at the 95% confidence
interval for the unconstrained problem. The optimal choice of $\alpha_{1}$ in
this simulation is around 0.98. In practice, the optimal choice of
$\alpha_{1}$ and $\alpha_{2}$ can be approximated by generating a number of
$(1-\alpha_{1}-\alpha_{2})$% confidence intervals and selecting the one with
the smallest width.

Figure 8.1: Simulation results

(a) The red line is the coverage frequency averaged over 5,000 samples of the
95% confidence interval of Theorem 5.1 with $\alpha_{1}=0.98$ and
$\alpha_{2}=0.97$.

(b) The red line is the average width over 100 samples of the 95% confidence
interval of Theorem 5.1 for various choices of $\alpha_{1}$ and $\alpha_{2}$.

### Technical details

Theorem 3.2 (Shapiro, 2009). Let $\mathcal{W}$ be a compact subset of
$\mathbb{R}^{k}$, let $C(\mathcal{W})$ denote the space of continuous
functions with domain $\mathcal{W}$ and let

$\begin{array}[]{l@{}l}\bar{\varphi}(\xi)&{}\equiv\,\inf\,\\{\xi(w)\,:\,w\in\mathcal{W}\\}\\\
\mathcal{W}^{*}(\xi)&{}\equiv\,\operatorname*{arg\,min}\,\\{\xi(w)\,:\,w\in\mathcal{W}\\}\end{array}$

where $\xi\in C(\mathcal{W})$. Let $\\{\psi_{n}\\}$ be a sequence of random
elements in $C(\mathcal{W})$, $f\in C(\mathcal{W})$ and
$\varphi_{0}=\bar{\varphi}(f)$, $\hat{\varphi}_{n}=\bar{\varphi}(\psi_{n})$.
Suppose $\sqrt{n}(\psi_{n}\,-\,f)$ converges in distribution to a random
element $Z$ of $C(S)$. Then

$\sqrt{n}(\hat{\varphi}_{n}\,-\,\varphi_{0})\,\to\,\min_{w\in\mathcal{W}^{*}(f)}Z(w)$

in distribution. In particular, if $f(w)$ attains its minimum over
$\mathcal{W}$ at a unique point $w_{0}$, then
$\sqrt{n}(\hat{\varphi}_{n}\,-\,\varphi_{0})$ converges in distribution to
$Z(w_{0})$.

######  Lemma 8.1.

$\beta^{H}=\beta(w^{H})$ is a global maximum over $\mathcal{W}$ if and only
if, for all $k=1,...,K$, $q_{k}\,f(t_{k})\,\geq\,\beta^{H}\,q_{k}\,g(t_{k})$,
where $q_{k}=w^{H}_{k}-(1/a+1/b-w^{H}_{k})$.

###### Proof.

Consider the population problem. We will prove the ‘if’ statement since the
‘only if’ statement follows from some simple algebra. We will begin by noting
that, since $\beta^{H}$ is the solution to a linear fractional programming
problem and since $\mathcal{W}$ is a compact, convex polyhedron, the
maximizing weight vector $w^{H}$ will lie at a vertex. In other words,
$w^{H}\in\\{1/b,\,1/a\\}^{K}$. Take an arbitrary weight vector
$w\in\mathcal{W}$. Suppose there are $1<m\leq n$ elements of $w$ which differ
from $w^{H}$. Without loss of generality, suppose these are the first $m$
elements. Then we can write

$\displaystyle\beta(w)$
$\displaystyle=\displaystyle\frac{\sum_{k=1}^{K}w_{k}\,f(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w_{k}\,g(t_{k})\,p(t_{k})}$
$\displaystyle=\displaystyle\frac{\sum_{k=1}^{K}w^{H}_{k}\,f(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}\,f(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k}\,g(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}\,g(t_{k})\,p(t_{k})}$
$\displaystyle\leq\displaystyle\frac{\beta^{H}\sum_{k=1}^{K}w^{H}_{k}\,g(t_{k})\,p(t_{k})\,-\,\beta^{H}\sum_{k=1}^{m}q_{k}\,g(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k}\,g(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}g(t_{k})p(t_{k})}$
$\displaystyle=\beta^{H}\displaystyle\frac{\sum_{k=1}^{K}w^{H}_{k}\,g(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}\,g(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k}\,g(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}g(t_{k})p(t_{k})}$
$\displaystyle=\beta^{H}$

The same holds almost surely for the sample problem by replacing $p(t_{k})$
with $\hat{p}_{n}(t_{k})$. ∎

### Proof of Theorem 3.1

###### Proof.

We will state this proof for the upper bound since the proof is symmetric for
the lower bound. The proof will follow by an application of the Uniform Strong
Law of Large Numbers (USLLN) for stochastic sequences and Theorem 5.3 of
Shapiro et al. (2009).

The sequence

$\\{G_{n}(w)\\}\,\equiv\,\\{\hat{\beta}_{n}(w)-\beta(w)\\}$

is said to converge uniformly almost surely if

$\sup_{w\in\mathcal{W}}|G_{n}(w)|\,\to\,0~{}\text{almost surely.}$ (22)

This is a rather strong form of convergence needed for many of the results in
stochastic programming to hold. By Theorem 21.8 of Davidson (1994), (22)
occurs if the following are true:

i) $G_{n}(w)\,\to\,0$ almost surely for every $w\in\mathcal{W}$. ii)
$\\{G_{n}(w)\\}$ is strongly stochastically equicontinuous. (23)

Condition i) follows since $\mathcal{T}$ is discrete and so
$\hat{p}_{n}(t_{k})\,\to\,p(t_{k})$ almost surely for all
$t_{k}\in\mathcal{T}$. Condition ii), however, is more difficult to directly
verify. Theorem 21.10 of Davidson (1994) establishes a Lipschitz-type
sufficient condition for strong stochastic equicontinuity. Suppose there
exists some $n\geq 1$ such that

$|\hat{\beta}_{n}(w_{1})\,-\,\hat{\beta}_{n}(w_{2})|\,\leq\,B_{n}\,h(\rho(w_{1},w_{2}))$

holds for all $w_{1},w_{2}\in\mathcal{W}$, where $h(\cdot)$ is non-stochastic,
$\lim_{x\to 0}h(x)=0$, $B_{n}$ is a stochastic sequence not depending on $w$
and $\rho(\cdot,\cdot)$ is a distance metric. Then $\\{G_{n}(w)\\}$ is
strongly stochastically equicontinuous if $\limsup_{n}B_{n}<\infty$. This will
obviously be true if $B_{n}$ is almost surely finite for sufficiently large
$n$. This is implied by our assumptions. For sufficiently large $n$,
Assumption 6 states that the denominator of $\hat{\beta}_{n}(w)$ is almost
surely bounded away from zero. Since $\hat{\beta}_{n}(w)$ is continuous,
differentiable and finite-valued on a compact set, by the mean value theorem
all the partial derivatives are almost surely bounded. Therefore, it is
Lipschitz continuous and there exists some finite Lipschitz constant $B_{n}$,
satisfying condition ii).

With the USLLN established for $\\{G_{n}(w)\\}$, we now want to establish that
sequences $\\{\hat{\beta}_{n}(\hat{w}^{H})-\beta(w^{H})\\}$ and
$\\{\hat{w}^{H}-w^{H}\\}$ converge to 0, where
$\hat{w}^{H}\,\equiv\,\operatorname*{arg\,min}_{w\in\mathcal{W}}\hat{\beta}_{n}(w)$
and $w^{H}\,\equiv\,\operatorname*{arg\,min}_{w\in\mathcal{W}}\beta(w)$

Theorem 5.3 of Shapiro et al. (2009) states that this convergence occurs
almost surely under the following conditions:

i) $\mathcal{W}$ is compact. ii) The set $\mathcal{W}^{H}$ of optimal
solutions to the true problem is non-empty and contained in $\mathcal{W}$.
iii) $\beta(w)$ is finite-valued and continuous on $\mathcal{W}$. iv)
$\hat{\beta}_{n}(w)$ converges to $\beta(w)$ almost surely as $n\to\infty$
uniformly on $\mathcal{W}$. v) With probability 1 and for $n$ sufficiently
large, the set of optimal solutions to the sample problem
$\hat{\mathcal{W}}^{H}$ is non-empty and $\hat{\mathcal{W}}^{H}$ is contained
in $\mathcal{W}$. (24)

Condition i) is true by the Heine-Borel theorem since
$\mathcal{W}=[1/b,\,1/a]^{K}$ is a closed and bounded subset of
$\mathbb{R}^{K}$. Conditions ii) and v) follow from Lemma 3.1 under Assumption
2 and Assumptions 5-6. Condition iii) is satisfied under Assumption 6 and the
fact that $\beta(w)$ is a linear fractional function. Condition iv) has been
verified by the arguments above. ∎

### Proof of Lemma 3.1

###### Proof.

Consider the population problem. Suppose that there are two global maxima,
$\beta^{H}_{1}=\beta(w^{H}_{1})$ and $\beta^{H}_{2}=\beta(w^{H}_{2})$,
$\beta_{1}^{H}=\beta_{2}^{H}=\beta^{H}$ and $w^{H}_{1}\neq w^{H}_{2}$. Since
$\beta^{H}_{1}$ and $\beta^{H}_{2}$ are both global maxima, then by Lemma 8.1,
for all $k=1,...,K$,

$\begin{array}[]{l@{}l}&{}q_{k}\,f(t_{k})\,\geq\,\beta^{H}_{1}\,q_{k}\,g(t_{k})\\\\[5.0pt]
&{}q_{k}\,f(t_{k})\,\geq\,\beta^{H}_{2}\,q_{k}\,g(t_{k})\end{array}$ (25)

Without loss of generality, we assume that $w^{H}_{1}$ and $w^{H}_{2}$ differ
by the first $m$ elements. Then,

$\begin{array}[]{l@{}l}\beta^{H}=\beta^{H}_{1}&{}=\displaystyle\frac{\sum_{k=1}^{K}w^{H}_{k1}\,f(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k1}\,g(t_{k})\,p(t_{k})}\\\\[15.00002pt]
&{}=\displaystyle\frac{\sum_{k=1}^{K}w^{H}_{k2}\,f(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k2}\,g(t_{k})\,p(t_{k})}\\\\[15.00002pt]
&{}=\displaystyle\frac{\sum_{k=1}^{K}w^{H}_{k1}\,f(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}f(t_{k})\,p(t_{k})}{\sum_{k=1}^{K}w^{H}_{k1}\,g(t_{k})\,p(t_{k})\,-\,\sum_{k=1}^{m}q_{k}g(t_{k})\,p(t_{k})}\end{array}$

where $q_{k}=w^{H}_{k1}-(1/a+1/b-w^{H}_{k1})$. Rearranging, we obtain,

$\displaystyle\sum_{k=1}^{m}q_{k}\,f(t_{k})\,p(t_{k})\,=\,\beta^{H}\,\sum_{k=1}^{m}q_{k}\,g(t_{k})\,p(t_{k})$

However, (25) implies that this equality will only hold if, for all
$k=1,...,m$, $f(t_{k})/g(t_{k})=\beta^{H}$, which cannot be true by Assumption
5. Therefore, by contradiction, the set $\mathcal{W}^{H}$ must be a singleton.
The same holds almost surely for the sample problem by replacing $p(t_{k})$
with $\hat{p}_{n}(t_{k})$. ∎

### Proof of Proposition 4.1

###### Proof.

$\begin{array}[]{l@{}l}|\hat{\sigma}^{2}(\hat{w}^{m})-\sigma^{2}(w^{m})|&{}=|\hat{\sigma}^{2}(\hat{w}^{m})-\sigma^{2}(\hat{w}^{m})+\sigma^{2}(\hat{w}^{m})-\sigma^{2}(w^{m})|\\\\[10.00002pt]
&{}\leq|\hat{\sigma}^{2}(\hat{w}^{m})-\sigma^{2}(\hat{w}^{m})|\,+\,|\sigma^{2}(\hat{w}^{m})-\sigma^{2}(w^{m})|\\\\[10.00002pt]
&{}\leq\displaystyle\sup_{w\in\mathcal{W}}|\hat{\sigma}^{2}(w)-\sigma^{2}(w)|\,+\,|\sigma^{2}(\hat{w}^{m})-\sigma^{2}(w^{m})|\\\\[10.00002pt]
&{}\to 0~{}\text{almost surely},\end{array}$

where the last inequality holds by the Uniform Strong Law of Large Numbers and
the second term on the last line goes to zero almost surely since
$\hat{w}^{m}\,\to\,w^{m}$ almost surely by Theorem 3.1 and $\sigma^{2}(w)$ is
continuous. ∎

### Proof of Theorem 5.1

###### Proof.

For simplicity, we prove this theorem for a one-sided confidence interval. The
proof will proceed as follows. If we can show that
$C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{2})$ is asymptotically equivalent
to $C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{2})$ then, for any
$\alpha_{3}>\alpha_{2}$, we can show that, in the limit,
$C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{2})$ has a coverage probability at
least as large as $C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{3})$. Then,
because $C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{3})$ is the infimum of the
confidence interval bounds over $\hat{\mathcal{W}}_{\alpha_{1}}$, it must
surely be less than $C_{n}^{L}(w_{0}^{L};\alpha_{3})$ whenever
$\mathcal{W}_{0}\subseteq\hat{\mathcal{W}}_{\alpha_{1}}$ and therefore have a
higher coverage probability. From here, we can apply a similar argument to the
main lemma of Berger and Boos (1994) to arrive at our main result.

To begin, we require some additional notation. For some $0<\alpha<1$, let

$\hat{v}^{L}_{\alpha_{1}}\,\equiv\,\operatorname*{arg\,min}_{w\in\mathcal{W}_{n}^{\alpha_{1}}}C_{n}^{L}(w;\alpha)$

In addition, let

$\delta_{n}\,\equiv\,\bigl{(}\sigma(w^{L}_{0})-\nu\bigr{)}\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}/\sqrt{n}$

where $\nu$ is some constant such that $0<\nu<\sigma(w_{0})$. Before deriving
our first lower bound, we summarize the relationships among the objects
defined so far in the following figure:

$C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{2})$$C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{2})$$C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{3})$$C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{3})$$\hat{\beta}_{n}(\hat{w}_{\alpha_{1}}^{L})$$\hat{\beta}_{n}(\hat{v}_{\alpha_{1}}^{L})$$\mathbb{R}$

By the relationships above and several applications of the intersection bound,
we can derive

$\displaystyle\operatorname*{pr}\biggl{(}C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{2})\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle=\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{w}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{2}}\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})/\sqrt{n}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle=\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{2}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,+\,\bigl{\\{}(\hat{\beta}_{n}(\hat{w}^{L}_{\alpha_{1}})-\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}}))\bigr{\\}}$
$\displaystyle~{}~{}~{}~{}+\,Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle\geq\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{2}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,+\,Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle\geq\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{2}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,+\,\delta_{n}\,\leq\,\beta^{L}_{0},\,\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}$
$\displaystyle\geq\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{2}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,+\,\delta_{n}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle~{}~{}~{}~{}+\,\operatorname*{pr}\biggl{(}\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}\,-\,1$
$\displaystyle=\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{3}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,+\,\delta_{n}\,-\,\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle~{}~{}~{}~{}+\,\operatorname*{pr}\biggl{(}\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}\,-\,1$
$\displaystyle=\,$
$\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{3}}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,\leq\,\beta^{L}_{0}\biggr{)}\,+\,\operatorname*{pr}\biggl{(}\delta_{n}\,-\,\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})/\sqrt{n}\,\leq\,0\biggr{)}\,-\,1$
$\displaystyle~{}~{}~{}~{}+\,\operatorname*{pr}\biggl{(}\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}\,-\,1$
$\displaystyle=\,$
$\displaystyle\operatorname*{pr}\biggl{(}C_{n}^{L}(\hat{v}_{\alpha_{1}}^{L};\alpha_{3})\,\leq\,\beta^{L}_{0}\biggr{)}\,+\,\operatorname*{pr}\biggl{(}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,\geq\,\sqrt{n}\,\delta_{n}/\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}\biggr{)}\,-\,1$
$\displaystyle~{}~{}~{}~{}+\,\operatorname*{pr}\biggl{(}\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}\,-\,1$

We now want to check what this lower bound converges to as $n$ becomes large.
In particular, we claim

  1. A)

$\operatorname*{pr}\biggl{(}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,\geq\,\sqrt{n}\,\delta_{n}/\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}\biggr{)}\,\to\,1$

  2. B)

$\operatorname*{pr}\biggl{(}\bigl{|}Z_{\alpha_{2}}\bigl{\\{}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,\hat{\sigma}_{n}(\hat{w}^{L}_{\alpha_{1}})\bigr{\\}}/\sqrt{n}\bigr{|}\,\leq\,\delta_{n}\biggr{)}\,\to\,1$

To begin with, we want to show $\hat{w}^{L}_{\alpha_{1}}\,\to\,w^{L}_{0}$ and
$\hat{v}^{L}_{\alpha_{1}}\,\to\,w^{L}_{0}$ almost surely. We will make use of
Theorem 5.5 of Shapiro et al. (2009), which we restate below using our
notation:
Theorem 5.5 (Shapiro et al, 2009). Suppose that, in addition to the
assumptions of Theorem 5.3 of Shapiro et al. (2009) stated in (24), the
following conditions hold:

  1. 1)

If $w_{n}\in\hat{\mathcal{W}}_{\alpha_{1}}$ and $w_{n}$ converges almost
surely to a point $w$, then $w\in\mathcal{W}_{0}$.

  2. 2)

There exists a sequence $w_{n}\in\hat{\mathcal{W}}_{\alpha_{1}}$ such that
$w_{n}\,\to\,w_{0}^{L}$ almost surely.

Then $\hat{\beta}_{n}(\hat{w}_{\alpha_{1}}^{L})\,\to\,\beta_{0}^{L}$ and
$\hat{w}_{\alpha_{1}}^{L}\,\to\,w_{0}^{L}$ almost surely.

To begin, we need to verify conditions 1) and 2). Shapiro et al. (2009),
pp.160-162 discusses when these conditions will hold. Provided
$\hat{h}_{j}(w)$ converges to $h_{j}(w)$ uniformly with probability one for
$j=1,...,J$, condition 1) will be satisfied. Condition 2) is satisfied by
imposing a constraint qualification on the true constraint set
$\mathcal{W}_{0}$. Since $h_{j}(w)$ are simple linear functions in $w$, this
holds trivially.

With the additional conditions satisfied, we must now verify the original
assumptions of Theorem 5.3. We verified these assumptions in the proof of
Theorem 3.1 and so convergence of $\hat{w}^{L}_{\alpha_{1}}$ is given. We need
to do some extra work to establish convergence of $\hat{v}^{L}_{\alpha_{1}}$.
Recall that $\hat{v}^{L}_{\alpha_{1}}$ solves the problem:

$\inf_{w\in\mathcal{W}_{n}^{\alpha_{1}}}C_{n}^{L}(w;\,\alpha)\,=\,\inf_{w\in\mathcal{W}_{n}^{\alpha_{1}}}\bigl{\\{}\hat{\beta}_{n}(w)\,-\,Z_{\alpha}\hat{\sigma}_{n}(w)/\sqrt{n}\bigr{\\}}$

By Assumption 4, $\hat{\sigma}_{n}(w)$ is almost surely Lipschitz continuous,
thus satisfying condition ii) of (23). Then, since
$\hat{\sigma}_{n}(w)/\sqrt{n}\,\to\,0$ almost surely, we have, for any fixed
$w\in\mathcal{W}$, $C_{n}^{L}(w;\alpha)\,\to\,\beta(w)$ almost surely,
satisfying condition i) of (23). Therefore, Theorem 5.5 applies and
$\hat{v}^{L}_{\alpha_{1}}\,\to\,w^{L}_{0}$ almost surely as desired.

Let us begin by tackling claim A) above.

$\displaystyle\operatorname*{pr}\biggl{(}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,\geq\,\sqrt{n}\,\delta_{n}/\bigl{(}Z_{\alpha_{2}}-Z_{\alpha_{3}}\bigr{)}\biggr{)}\,$
$\displaystyle=\,\operatorname*{pr}\biggl{(}\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,\geq\,\sigma(w_{0}^{L})-\nu\biggr{)}$
$\displaystyle=\operatorname*{pr}\biggl{(}\sigma(w_{0}^{L})\,-\,\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,\leq\,\nu\biggr{)}$
$\displaystyle=1\,-\,\operatorname*{pr}\biggl{(}\sigma(w_{0}^{L})\,-\,\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\,>\,\nu\biggr{)}$
$\displaystyle\geq
1\,-\,\operatorname*{pr}\biggl{(}\bigl{|}\sigma(w_{0}^{L})\,-\,\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})\bigr{|}\,>\,\nu\biggr{)}$
$\displaystyle\to\,1~{}\text{as $n\to\infty$}$

by Proposition 4.1. B) follows from a very similar argument to A), using the
fact that $\hat{\sigma}_{n}(w)$ is almost surely continuous on $\mathcal{W}$
and
$\hat{v}^{L}_{\alpha_{1}}\,\to\,w^{L}_{0}\,\leftarrow\,\hat{w}^{L}_{\alpha_{1}}$
almost surely. Therefore, we have that

$\displaystyle\displaystyle\lim_{n\to\infty}\,\operatorname*{pr}\biggl{(}C_{n}^{L}(\hat{w}^{L}_{\alpha_{1}};\alpha_{2})\,\leq\,\beta^{L}_{0}\biggr{)}\,\geq\,\displaystyle\lim_{n\to\infty}\,\operatorname*{pr}\biggl{(}C_{n}^{L}(\hat{v}^{L}_{\alpha_{1}};\alpha_{3})\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle=$
$\displaystyle\displaystyle\lim_{n\to\infty}\,\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{3}}\frac{\hat{\sigma}_{n}(\hat{v}^{L}_{\alpha_{1}})}{\sqrt{n}}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle\geq$
$\displaystyle\displaystyle\lim_{n\to\infty}\,\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{v}^{L}_{\alpha_{1}})\,-\,Z_{\alpha_{3}}\frac{\hat{\sigma}_{n}(\hat{\bm{v}}^{L}_{\alpha_{1}})}{\sqrt{n}}\,\leq\,\beta^{L}_{0},~{}\mathcal{W}^{0}\,\subseteq\,\mathcal{W}_{n}^{\alpha_{1}}\biggr{)}$
$\displaystyle\geq$
$\displaystyle\displaystyle\lim_{n\to\infty}\,\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{w}^{L}_{0})\,-\,Z_{\alpha_{3}}\frac{\hat{\sigma}_{n}(\hat{w}^{L}_{0})}{\sqrt{n}}\,\leq\,\beta^{L}_{0},~{}\mathcal{W}^{0}\,\subseteq\,\mathcal{W}_{n}^{\alpha_{1}}\biggr{)}$
$\displaystyle=$
$\displaystyle\displaystyle\lim_{n\to\infty}\,\biggl{\\{}\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{w}^{L}_{0})\,-\,Z_{\alpha_{3}}\frac{\hat{\sigma}_{n}(\hat{w}^{L}_{0})}{\sqrt{n}}\,\leq\,\beta^{L}_{0}\biggr{)}$
$\displaystyle~{}~{}~{}~{}-\,\displaystyle\operatorname*{pr}\biggl{(}\hat{\beta}_{n}(\hat{w}^{L}_{0})\,-\,Z_{\alpha_{3}}\frac{\hat{\sigma}_{n}(\hat{w}^{L}_{0})}{\sqrt{n}}\,\leq\,\beta^{L}_{0},~{}\mathcal{W}^{0}\,\not\subseteq\,\mathcal{W}_{n}^{\alpha_{1}}\biggr{)}\biggr{\\}}$
$\displaystyle\geq$
$\displaystyle\displaystyle\lim_{n\to\infty}\,\biggl{\\{}\operatorname*{pr}\biggl{(}\sqrt{n}\,\frac{\bigl{(}\hat{\beta}_{n}(\hat{w}^{L}_{0})\,-\,\beta^{L}_{0}\bigr{)}}{\sigma(w^{L}_{0})}\,\frac{\sigma(w^{L}_{0})}{\hat{\sigma}_{n}(\hat{w}^{L}_{0})}\,\leq\,Z_{\alpha_{3}}\biggr{)}\,-\,\operatorname*{pr}\biggl{(}\mathcal{W}^{0}\,\not\subseteq\,\mathcal{W}_{n}^{\alpha_{1}}\biggr{)}\biggr{\\}}$
$\displaystyle\geq$ $\displaystyle\,1\,-\,\alpha_{3}\,-\,\alpha_{1}$

where the last inequality follows by Slutsky’s theorem, Proposition 4.1 and
Theorem 4.1. Since $\alpha_{3}$ is some arbitrary constant such that
$\alpha_{3}>\alpha_{2}$, we can simply take limits of $\alpha_{3}$ to
$\alpha_{2}$ from the right to arrive at our final result. ∎