#  Empirical Likelihood for Right Censored Lifetime Data

Shuyuan He1∗, Wei Liang2∗, Junshan Shen3∗ and Grace Yang 4 111Research of
authors 1, 2 and 3 were Supported by National Natural Science Foundation of
China(11171230, 10801003). Research of author 4 was supported by the US
National Science Foundation, while working at the Foundation.
Capital Normal University1, Xiamen University2,
Peking University3, University of Maryland4

Abstract

This paper considers the empirical likelihood (EL) construction of confidence
intervals for a linear functional ${\theta}$ based on right censored lifetime
data. Many of the results in literature show that $-2\log$(empirical
likelihood ratio) has a limiting scaled-$\chi^{2}_{1}$ distribution, where the
scale parameter is a function of the unknown asymptotic variance. The scale
parameter has to be estimated for the construction. Additional estimation
would reduce the coverage accuracy for ${\theta}$. This diminishes a main
advantage of the EL method for censored data. By utilizing certain influence
functions in an estimating equation, it is shown that under very general
conditions, $-2\log$(EL ratio) converges weakly to a standard $\chi^{2}_{1}$
distribution and thereby eliminates the need for estimating the scale
parameter. Moreover, a special way of employing influence functions eases the
otherwise very demanding computations of the EL method. Our approach yields
smaller asymptotic variance of the influence function than those comparable
ones considered by Wang & Jing (2001) and Qin & Zhao (2007). Thus it is not
surprising that confidence intervals using influence functions give a better
coverage accuracy as demonstrated by simulations.

Key words and phrases. Empirical likelihood, Right censored lifetimes,
Influence function, Parameter estimation, Confidence intervals

AMS 1991 subject classifications. Primary 62H99; Secondary 62H05.

##  1 Introduction

Let $Y$ be a non-negative random variable with distribution function
$F(y)=P[Y\leq y]$. Let $C$ be a nonnegative random variable with distribution
function $G(x)=P[C\leq x]$ and $C$ is independent of $Y$. Instead of $Y$, we
observe $Z=\min(Y,C)$ and the indicator $\delta=I[Y\leq C]$ of the event
$[Y\leq C].$ Both $F$ and $G$ are assumed continuous but unknown. In this
paper we use the empirical likelihood method with right-censored data to study
the problem of constructing confidence intervals for ${\theta}$, a functional
of the distribution $F$ defined by $\text{E}g(Y,{\theta})=0$. For example, if
$g(x,{\theta})=\xi(x)-{\theta}$, for some function $\xi$ we get

${\theta}=\text{E}\,\xi(Y)=\int_{0}^{\infty}\xi(x)\,\text{d}F(x).$ (1. 1)

The problem of estimating ${\theta}$ in (1.1) with a sample of $n$ i.i.d.
observations of $(Z,\delta)$ has been studied by many authors. If
$\xi(Y)=I[Y\leq y]$ for a fixed $y$, then ${\theta}=F(y)$. A well-known
nonparametric maximum likelihood and asymptotically optimal estimator of
$F(y)$ is the Kaplan-Meier (KM) estimator $F_{n}$ as defined in (2.3). A
natural estimator of ${\theta}$ is

${\theta}_{n}=\int_{0}^{\infty}\xi(s)\,\text{d}F_{n}(s).$ (1. 2)

For an arbitrary function $\xi$, several authors, e.g. Yang (1994) have shown
that under the condition of finite second moment,

$\int_{0}^{\infty}\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)<\infty,$
(1. 3)

the asymptotic distribution of $\sqrt{n}({\theta}_{n}-{\theta})$, as $n$ goes
to infinity, is normal $N(0,\,\sigma^{2})$, where

$\sigma^{2}=\int_{0}^{\infty}\frac{(\overline{F}(s)\xi(s)-\psi(s))^{2}}{\overline{F}^{2}(s)\overline{G}(s)}\,\text{d}F(s),\
\psi(s)=\int_{s}^{\infty}\xi(x)\,\text{d}F(x),s\geq 0,$ (1. 4)

and $\overline{G}=1-G,\overline{F}=1-F$.

Confidence intervals for ${\theta}$ can be constructed using the asymptotic
normal distribution $N(0,\,\sigma^{2})$. Alternatively, the EL method can be
used as to be investigated in this paper. Employing either method, one needs
to deal with a rather complicated form of the asymptotic variance
$\sigma^{2}$. Among other things, it is computationally demanding.

To use the normal distribution $N(0,\,\sigma^{2})$, it is necessary to
estimate the unknown variance $\sigma^{2}$. Stute (1996) proposed a jackknife
estimator to replace $\sigma^{2}$ in the calculation. Although any consistent
estimator $\sigma_{n}^{2}$ of $\sigma^{2}$ can be used, the convergence rate
of $\sigma_{n}^{2}$ is generally unknown. Substitution by the estimate
$\sigma_{n}^{2}$ tends to reduce the coverage accuracy for ${\theta}$ as
compare to the case of known $\sigma^{2}$.

The usefulness of the EL method for constructing confidence interval/regions
has been well established in a wide variety of situations, see e.g., DiCiccio
et al. (1991) and Chen (1994), and an extensive literature review in Owen
(2001) to that day. Let $R({\theta})$ denote the EL ratio function of a one-
dimensional parameter ${\theta}$ for $n$ i.i.d “complete” observations. Owen
(1988) proved that under certain regularity conditions, $-2\log R({\theta})$
converges to a chi-squared distribution with one degree of freedom. The EL
method gives confidence intervals for ${\theta}$ as $\\{{\theta}:\,-2\log
R({\theta})\leq c_{1-\alpha}\\}$, where $c_{1-\alpha}$ is the $(1-\alpha)$th
quantile of the $\chi_{1}^{2}$ distribution. Here the construction of
confidence intervals does not require estimation of asymptotic variance. In
view of a complicated variance formula in (1.4), this would have provided a
welcome method for censored data. However, as far as we know, for censored
data, the asymptotic standard chi-squared distribution holds only in some
special cases see, e.g. Owen (Chapter 6, 2001). More recent literature shows
that most of the asymptotic distributions involve weights which are functions
of unknown variances or covariance matrices. This is the case, for example, in
the following papers. Li and Wang (2003) studied right-censored regression
models, Ren (2008) used weighted EL under a variety of censoring models, Wang
& Jing (2001) and Qin & Zhao (2007) estimated functionals ${\theta}$, and
Hjort& McKeague & van Keilegom, in their extension of the scope of the EL
method (2009), obtained an asymptotic distribution (Theorem 2.1) which is a
sum of weighted chi-squared distributions with unknown weights. Therefore,
using these results to construct confidence intervals for ${\theta}$ still
require an additional estimation of the unknown $\sigma^{2}$. This diminishes
a main advantage of the EL method for censored data.

The EL ratio $R({\theta})$ is obtained by utilizing auxiliary information on
${\theta}$ through a set of estimating equations. In this paper, we show that
by using certain influence functions with a special construction of estimating
equations in the EL ratio, the asymptotic distribution of -2$\log R({\theta})$
of the functional ${\theta}$ in (1.1) is a standard $\chi_{1}^{2}$ without
involving any unknown scale parameter. Our approach transfers the problem of
estimating $\sigma^{2}$ to the influence functions. As a result, it also
significantly simplifies the often intensive computations of the EL method for
censored data.

Our work is motivated by the work of Wang & Jing (2001) and Qin & Zhao (2007).
Wang & Jing (2001) obtained an EL ratio by first finding an estimating
equation for a certain complete sample and then modifying the estimating
equation for the right censored sample. The resulting estimating equation is a
sum instead of a product (inherent of the product limit estimator). With this
approach, Wang & Jing (2001) use the estimating function
$M_{1}(Z,\delta,{\theta})$ for ${\theta}$ (see (3.7) ) and Qin & Zhao (2007)
use $M_{2}(Z,\delta,{\theta})$ (see (3.8)) for estimating the mean residual
life $\text{E}(Y-t_{0}|Y\geq t_{0})$ at age $t_{0}.$ However, both of these
papers obtain an asymptotic scaled $\chi^{2}_{1}$ distribution for $-2\log
R({\theta}).$

Instead of $M_{1}$ and $M_{2}$, we use influence functions. We compute the
influence functions $W(Z,\delta,\theta)$ of

$\mu_{n}=\int_{0}^{\infty}g(x,{\theta})\,\text{d}F_{n}(x),$

as defined by (3.6), where $F_{n}$ is the Kaplan-Meier estimator. The
influence functions $W^{\prime}s$ are to be utilized to construct an
estimation function for the EL method. Numerous examples of the function $g$
are given in Section 2.

The paper is organized as follows. Preliminaries assumptions and examples of
${\theta}$ and $g$ are given in Section 2. The influence function
$W(Z,\delta,{\theta})$ are given in Section 3. It is shown in Theorem 3.1 that
asymptotically $\sqrt{n}(\mu_{n}-\mu)$ is a partial sum of $n$ independent
influence functions $W(Z_{j},\delta_{j},{\theta})$ (an IID representation), or
is asymptotically linear. Here $\mu$ denotes $E\,g(Y,{\theta})$ and the the
condition $\text{E}\,g(X,{\theta})=0$ is not imposed in Section 3. These
results are general for any $\xi(x)$ having finite second moment (1.3) and no
restrictions are placed on the upper boundaries of $X$ and $C$.

An IID representation of the Kaplan-Meier estimator has been obtained by many
authors e.g., Lo & Singh (1986), Stute & Wang (1993) and Chen & Lo (1997)
using different approaches, under different conditions and in different forms.
See Yang (1997) and references therein. Here, we use an influence function
$W(Z,\delta,{\theta})$ of ${\theta}_{n}$ in (1.2) obtained in He &
Huang(2003). We show that the variance of the influence function is smaller
than that of $M_{1}$ and $M_{2}$ (see Remark 3.1 in Section 3), which results
in an improvement of the asymptotic coverage accuracies of ${\theta}$. In
Section 4, estimation of the influence functions is carried out in Theorem
4.1. The weak convergence of $-2\log R({\theta})$ to the standard
$\chi_{1}^{2}$ distribution without any scale parameter is proved in Theorem
4.2 which justifies the EL construction of confidence intervals for censored
data. In Section 5, simulation comparison of the new method with that of the
scaled $\chi_{1}^{2}$ distribution is presented. The amount of improvement
depends on the form of ${\theta}$. For survival function or mean, the coverage
ratios computing from the traditional normal approximation and the EL method
are about the same. The EL method performs better for more complicated
${\theta}$. Most of the proofs are relegated to the Appendix.

##  2 Preliminaries, Assumptions and Examples

For any right continuous monotone function $h({x})$, let $h({x-})$ or
$h_{-}(x)$ denote the left continuous version of $h({x})$ and the curly
brackets $h\\{{x}\\}$ denote the difference $h({x})-h({x-})$. Then
$h\\{x\\}=\text{d}h(x)$. For any cumulative distribution function $F$, let
$\overline{F}=1-F$. • Assume that

$F(x)=P(Y\leq x),\quad G(x)=P(C\leq x)\ \ \text{and}\ \ H(x)=P(Z\leq x)$ (2.
1)

are continuous cumulative distributions of $Y$, $C$ and $Z=\min(Y,C)$,
respectively. Let $[0,b_{H}]$ be the range of $H$, where

$b_{H}=\sup\\{x:H(x)<1\\}.$

$b_{F}$ and $b_{G}$ are similarly defined for $F$ and $G$. Then
$b_{H}=\min(b_{F},b_{G})$.

Given a sample of $n$ i.i.d. random vectors $(Z_{i},\delta_{i})$,
$i=1,2,\cdots,n$, of $(Z,\delta)$, their empirical distribution functions are
given by:

$\displaystyle H_{n}^{1}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x,\delta_{j}=1],$ $\displaystyle
H_{n}^{0}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x,\delta_{j}=0],$ (2. 2) $\displaystyle
H_{n}(x)=H_{n}^{0}(x)+H_{n}^{1}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x].$

Asymptotic optimal nonparametric estimators of $F(x)$ and $G(x)$ are the well-
known Kaplan-Meier estimators given by

$F_{n}(x)=1-\prod_{s\leq
x}\left[1-\frac{H^{1}_{n}\\{s\\}}{{\overline{H}}_{n}(s-)}\right]\hbox{ and }\
\ G_{n}(x)=1-\prod_{s\leq
x}\left[1-\frac{H^{0}_{n}\\{s\\}}{{\overline{H}}_{n}(s-)}\right],$ (2. 3)

respectively, where an empty product is set equal to one. It can be checked
that for all $x$,

${\overline{H}}_{n}(x)={\overline{F}}_{n}(x){\overline{G}}_{n}(x).$ (2. 4)

Applying (2.3) and (2.4) we get

$\displaystyle\text{d}F_{n}(x)=F_{n}(x)-F_{n}(x-)=\overline{F}_{n}(x-)\frac{\text{d}H_{n}^{1}(x)}{{\overline{F}}_{n}(x-){\overline{G}}_{n}(x-)}.$
(2. 5)

It follows that

$\text{d}H_{n}^{1}(x)={\overline{G}}_{n}(x-)\,\text{d}F_{n}(x),\quad\text{d}H_{n}^{0}(x)={\overline{F}}_{n}(x-)\,\text{d}G_{n}(x).$
(2. 6)

Put

$\displaystyle H^{0}(x)=P(Z\leq x,\delta=0),\quad H^{1}(x)=P(Z\leq
x,\delta=1),\mbox{and}$ (2. 7) $\displaystyle\overline{H}(x)=P(Z>x).$

Then

$\displaystyle
H^{0}(x)=\text{E}\,H_{n}^{0}(x)=\int_{0}^{x}\overline{F}(s)\,\text{d}G(s),$
$\displaystyle
H^{1}(x)=\text{E}\,H_{n}^{1}(x)=\int_{0}^{x}\overline{G}(s)\,\text{d}F(s),$
(2. 8)
$\displaystyle\overline{H}(x)=\text{E}\,\overline{H}_{n}(x)=\overline{F}(x)\overline{G}(x).$

Here and after, the integral sign $\int^{b}_{a}$ stands for $\int_{(a,b]}$ and
$\int$ stands for $\int_{0}^{\infty}$.

Examples of ${\theta}$ and $g(x,{\theta})$

1\. $g(x,{\theta})=I[x>y]-{\theta}$ with $y$ fixed. Then

$\text{E}\,g(Y,{\theta})=\int(I[x>y]-{\theta})\,\text{d}F(x)=0.$

Solving this equation yields ${\theta}=\overline{F}(y)$, the survival function
of $Y$.

2\. $g(x,{\theta})=x^{k}-{\theta}$. Then ${\theta}=\text{E}\,Y^{k}$, the $k$th
moments of $Y$.

3\. $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$ with $t_{0}$ fixed. Then

${\theta}=\text{E}(Y-t_{0}|Y\geq t_{0})=\frac{\text{E}(Y-t_{0})I[Y\geq
t_{0}]}{P(Y\geq t_{0})},$

the mean residual life of $Y$.

4\. $g(x,{\theta})=x(I[x>y]-{\theta})$. Then

${\theta}=\frac{1}{\text{E}\,Y}\int_{y}^{\infty}s\,\text{d}F(s),$

the length biased survival function of $Y$. See Vardi (1982), for example.

5\. $g(x,{\theta})=x^{2}-{\theta}x$. Then

${\theta}=\frac{1}{\text{E}\,Y}\int_{0}^{\infty}x^{2}\,\text{d}F(x),$

the mean of the length-biased lifetime.

6\. $g(x,{\theta})=x^{2}-2{\theta}x$. Then

${\theta}=\frac{1}{2\text{E}\,Y}\int_{0}^{\infty}x^{2}\,\text{d}F(x),$

the mean of the length biased residual lifetime.

7\. $g(x,{\theta})=I[x\leq{\theta}]-p$ with $p\in(0,1)$. Then
${\theta}=F^{-1}(p)$, the $p$th quantile of $Y$.

Examples (4)-(6) often appear in renewal processes and their applications.

##  3 Influence function of $\mu_{n}$

Throughout Section 3, ${\theta}$ is a fixed value. Then it is convenient to
suppress ${\theta}$ in the exposition, by setting $\xi(x)=g(x,{\theta})$,
$\mu=\text{E}\,g(X,{\theta})$, and

$\displaystyle\mu_{n}=\int\xi(x)\,\text{d}F_{n}(x)=\int\frac{\xi(x)}{\overline{G}_{n}(x-)}\,\text{d}H_{n}^{1}(x).$
(3. 1)

Likewise, set $W=W(Z,\delta)=W(Z,\delta,{\theta}).$

In Theorem 3.1, we prove that the estimator $\mu_{n}$ for $\mu$ is asymptotic
linear. That is, there is a function $W=W(Z,\delta)$, such that
$\text{E}\,W=0$, $\text{Var}(W)<\infty$ and

$\sqrt{n}(\mu_{n}-\mu)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W(Z_{i},\delta_{i})+o_{p}(1).$

The function $W(Z,\delta)$ is defined with respect to the true distributions
$(F,G)$. Following literature, we call $W(Z_{i},\delta_{i})$ the $i$-th
influence function of $\mu_{n}$. See, for e.g. van der Vaart (1998) or Tsiatis
(2006).

Theorem 3.1 will be proved by first establishing a similar result for the
truncated $W(Z_{i},\delta_{i})$ as defined in (3.3). Let
$\xi_{b}(x)=\xi(x)I[x\leq b]$ be the restriction of $\xi(x)$ on $(-\infty,b]$,
where $b<b_{H}$ is an arbitrarily chosen constant. By similar truncation, put

$\mu_{b}=\int\xi_{b}(x)\,\text{d}F(x),\ \ \psi_{b}(s)=\int_{x\geq
s}\xi_{b}(x)\,\text{d}F(x),\ \
{\overline{\delta}}_{i}=I[Y_{i}>C_{i}]=1-\delta_{i}.$ (3. 2)

We consider the i.i.d. random variables,

$W_{i}(b)=\frac{\xi_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\mu_{b}+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi_{b}(Z_{i})-\int\psi_{b}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}\,\text{d}H^{0}(s),$ (3. 3)

for $i=1,\cdots,n.$

Under finite variance condition (1.3), it can be calculated that,

$\displaystyle\text{E}\,\frac{\xi_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}$
$\displaystyle=$ $\displaystyle\mu_{b},\ \ \text{E}\,W_{i}(b)=0,$
$\displaystyle\text{Var}(W_{i}(b))$ $\displaystyle=$
$\displaystyle\int\frac{\xi_{b}^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu_{b}^{2}-\int\frac{\psi_{b}^{2}(s)}{\overline{F}(s)\overline{G}^{2}(s)}\,\text{d}G(s).$
(3. 4)

As $b$ approaches the upper bound $b_{H}$, we have

$\mu_{b}\to\mu,\quad\psi(s)=\lim_{b\to b_{H}}\psi_{b}(s)=\int_{x\geq
s}\xi(x)\,\text{d}F(x),\quad\mbox{and}$ (3. 5)

$W_{i}=\lim_{b\to
b_{H}}W_{i}(b)=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\mu+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i})-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}\,\text{d}H^{0}(s),$ (3. 6)

for $i=1,\cdots,n$. The $W_{i}^{\prime}s$, for $i=1,\cdots,n,$ are i.i.d.
random variables.

Wang & Jing (2001) use the estimating function based on

$M_{1}(Z,\delta,{\theta})=\frac{\xi(Z)\delta}{\overline{G}(Z)}-{\theta}$ (3.
7)

to estimate ${\theta}$ in (1.1). Qin & Zhao (2007) used the estimating
function based on

$M_{2}(Z,\delta,{\theta})=\frac{g(Z,{\theta})\delta}{\overline{G}(Z)},$ (3. 8)

where $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$, to estimate the mean
residual life ${\theta}=\text{E}(Y-t_{0}|Y\geq t_{0})$ at a specified age
$t_{0}.$ This case is covered in our formulation, see example (3) in Section
2\. Comparing with $M_{1}$ and $M_{2}$, our $W_{i}^{\prime}s$ contain two
additional terms. Note that $W_{i}^{\prime}s$ are not observable random
variables and whose estimation will be addressed in Section 4.

Under finite variance condition (1.3), applying the dominated convergence
theorem and the Lebesgue-Stieltjes integration by parts, we obtain

$\displaystyle\text{E}\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}$
$\displaystyle=$ $\displaystyle\mu,\ \ \text{E}\,W_{i}=0,$ (3. 9)
$\displaystyle\text{Var}(W_{i})$ $\displaystyle=$
$\displaystyle\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2}-\int\frac{\psi^{2}(s)}{\overline{F}(s)\overline{G}^{2}(s)}\,\text{d}G(s)$
(3. 10) $\displaystyle=$
$\displaystyle\int\frac{(\overline{F}(s)\xi(s)-\psi(s))^{2}}{\overline{F}^{2}(s)\overline{G}(s)}\,\text{d}F(s).$
(3. 11)

Remark 3.1 Formulas (3.6) are (3.10) are obtained in He $\&$ Huang (2003), and
(3.11) is given in Yang (1994). Under condition (1.3), it can be shown that
(3.10) and (3.11) are equal. The variance of $W_{i}$ is smaller than that of
$M_{1}$ and $M_{2}$ defined in (3.7) and (3.8). The variance of the latter two
equals

$\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2}$

with the corresponding choices of $\xi(z)=g(z,{\theta})$, in $M_{1}$ and
$M_{2}$.

We proceed to prove Theorem 3.1. For the restricted $W_{i}(b)^{\prime}s$, the
following lemma is taken from (3.11) of He & Huang (2003).

Lemma 3.1 Let $F$ and $G$ be continuous. For each ${\theta}$ fixed, set
$\xi(x)=g(x,{\theta})$. Assume that $\int\xi^{2}(x)\,\text{d}F(x)<\infty$ and
$b<b_{H}$. Let $\xi_{b}(x)$ be the restriction of $\xi$ on $(0,b]$ for
$b<b_{H}$. Then, as $n\to\infty$,

$\sqrt{n}\int\xi_{b}(x)\,\text{d}(F_{n}(x)-F(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}(b)+o_{p}(1).$

The following result will be used repeatedly and for easy reference, it is
stated in Lemma 3.2. Its proof is given the appendix.

Lemma 3.2 For $b<b_{H}$, let $\\{h_{n}(b)\\}$ be a random sequence such that
$h_{n}(b)\to h(b)$ in distribution as $n\to\infty$, and $h(b)=o_{p}(1)$ as
$b\to b_{H}$. As $n\to\infty$, if $V_{n}=O_{p}(1)$ and the random sequence
$\\{S_{n}\\}$ can be written as $S_{n}=o_{p}(1)+V_{n}h_{n}(b)$ for any
$b<b_{H}$, then $S_{n}=o_{p}(1)$.

Remark 3.2 In what follows, $h_{n}(b)$ is used as a generic notation to denote
any random sequence $\\{h_{n}(b)\\}$ that satisfies the assumptions of Lemma
3.2. This simplifies many of the statements later. For example, under
condition (1.3) and $b<b_{H}$, put

$\displaystyle
h_{n}(b)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s).$
(3. 12)

Then, by the SLLN and (2.8)

$\displaystyle\lim_{n\to\infty}h_{n}(b)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H^{1}(s)$
$\displaystyle=$
$\displaystyle\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\overline{G}(s)\,\text{d}F(s)=h(b)\to
0,\ \mbox{as $b\to b_{H}.$}$

######  Theorem 3.1.

Let $W_{i}$ be given by (3.6). Suppose $F$ and $G$ are continuous, and for
each fixed ${\theta}$ set $\xi(x)=g(x,{\theta})$. Then under condition (1.3)
as $n\to\infty$,

$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$

###### Proof.

For $b<b_{H}$, put $\overline{\xi}_{b}=\xi(x)-\xi_{b}(x)=\xi(x)I[x>b]$.
Decompose the following difference as

$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}\equiv
J_{1}(b)+J_{2}(b)-J_{3}(b),$ (3. 13)

where

$\displaystyle
J_{1}(b)=\sqrt{n}\int\xi_{b}(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}(b),$
$\displaystyle
J_{2}(b)=\sqrt{n}\int\overline{\xi}_{b}(x)\,\text{d}(F_{n}(x)-F(x)),$
$\displaystyle J_{3}(b)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(W_{i}-W_{i}(b)).$

Let $\overline{\psi}_{b}(x)=\int_{s\geq
x}\overline{\xi}_{b}(s)\,\text{d}F(s)$. It follows that

$\overline{\psi}_{b}^{2}(x)\leq\overline{F}(x)^{2}\int_{s\geq
x}\overline{\xi}_{b}^{2}(s)\,\text{d}F(s).$ (3. 14)

By Lemma 3.1, $J_{1}(b)=o_{p}(1)$. We shall show that $J_{2}(b)=h_{n}(b)$ and
$J_{3}(b)=h_{n}(b)$ as $n\to\infty$. Applying Corollary 1 of Yang (1994),
$J_{2}(b)$ converges weakly to $N(0,\,\overline{\sigma}_{b}^{2})$, where
$\overline{\sigma}_{b}^{2}$ is similarly defined as in (3.12) with $\xi$ and
$\psi$ replacing by their restrictions $\overline{\xi}_{b}$ and
$\overline{\psi}_{b}$ respectively.

Now

$\displaystyle\overline{\sigma}_{b}^{2}\leq\int\frac{\overline{\xi}_{b}^{2}(x)}{\overline{G}(x)}\,\text{d}F(x)\to
0,\ \text{as}\ b\to b_{H}.$ (3. 15)

Therefore $J_{2}(b)$ converges to $h(b)=Z_{0}\overline{\sigma}_{b}$ in
distribution, and $h(b)=o_{p}(1)$ as $b\to b_{H}$, where $Z_{0}$ is a
$N(0,\,1)$ random variable. It follows that $J_{2}(b)=h_{n}(b)$.

To prove $J_{3}(b)=h_{n}(b)$, note that the difference $W_{i}-W_{i}(b)$, as
given in (3.6) and (3.3) equals to

$\frac{\overline{\xi}_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\int\overline{\xi}_{b}(x)\,\text{d}F(x)+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\overline{\psi}_{b}(Z_{i})-\int\overline{\psi}_{b}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s).$ (3. 16)

Therefore $W_{i}-W_{i}(b)$ are i.i.d. random variables with mean zero and
variance $\overline{\sigma}_{b}^{2}$. Hence, $J_{3}(b)=h_{n}(b)$ follows for
the same reason as that of $J_{2}(b)$.

We conclude that the following holds for (3.13),

$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}=o_{p}(1)+h_{n}(b)+h_{n}(b).$

Theorem 3.1 follows from Lemma 3.2. ∎

Remark 3.3 If $b_{F}<b_{G}$, then (1.3) is equivalent to $\xi$ having finite
second moment. If $\xi$ is bounded and away from zero, then (1.3) is
equivalent to $\int_{0}^{\infty}\,\text{d}F(s)/{\overline{G}(s)}.$

##  4 Empirical Likelihood Ratios and Confidence Intervals for ${\theta}$

To develop an EL inference procedure, we consider a specific $g(x,{\theta})$.
For each fixed ${\theta}$, as before, set $\xi(x)=g(x,{\theta})$. We shall
utilize the i.i.d. random variables

$\displaystyle
W_{i}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i})-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s).$ (4. 1)

to obtain an estimating equation for the EL ratio. Recall that
$\mu=\text{E}\,W_{i}=\int_{0}^{\infty}\xi(x)\,\text{d}F(x)$ and
$\text{Var}(W_{i})$ are given by (3.10). Note that setting
$\xi(x)=g(x,{\theta})$ above has nothing to do with defining ${\theta}$ from
the equation $\text{E}\,g(X,{\theta})=0$ as given in (1.5). If, however, the
true parameter ${\theta}_{0}$ is the solution of the equation

$\text{E}\,g(Y,{\theta})=\int g(x,{\theta})\,\text{d}F(x)=0,$ (4. 2)

then $\xi(x)=g(x,{\theta}_{0})$ is such that

$\mu=\text{E}\,W_{i}=\int_{0}^{\infty}\xi(x)\,\text{d}F(x)=0.$

Regarding $W_{i}$ for $i=1,\cdots,n$ as a “complete” random sample, one could
formulate an EL likelihood ratio $R({\theta}_{0})$ with multinomial
probability $p_{i}$ assigned to $W_{i}$ and the constraint
$\sum_{i=1}^{n}W_{i}p_{i}=0$. However, $W_{i}^{\prime}s$ are not observable
because of the unknown distributions $G$, $F$ $H$ and $H^{0}$. We shall
replace them by the KM estimates, $F_{n}$, $G_{n}$ given by (2.3) and an
estimate of $\psi$,

$\psi_{n}(x)=\int_{s\geq x}\xi(s)\,\text{d}F_{n}(s).$ (4. 3)

Replacing $\overline{G},\overline{H},H^{0}$ in (4.1) by their corresponding
empirical distributions in (2.2) gives an approximation of $W_{i}$ in (4.1) by

$W_{ni}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}\,\text{d}H_{n}^{0}(s).$ (4. 4)

The price to pay for the estimation is that ${W_{ni}}^{\prime}s$ are not
stochastically independent which complicates the ensuing analysis.

The following theorem indicates the possibility of using $W_{ni}$ to construct
empirical likelihood ratio and to obtain asymptotically a standard $\chi^{2}$
distribution.

######  Theorem 4.1.

Let $W_{ni}$ be given by (4.4) and $\text{E}\,\xi(Y)=0$. Then under condition
(1.3), as $n\to\infty$, we have

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$
(4. 5)

###### Proof.

By (2.6), we have

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}W_{ni}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)\Big{)}$ $\displaystyle=$
$\displaystyle\int\frac{\xi(s)}{\overline{G}_{n}(s-)}\,\text{d}H^{1}_{n}(s)+\int\frac{\psi_{n}(s)}{{\overline{H}_{n}}(s-)}\,\text{d}H_{n}^{0}(s)-\int\psi_{n}(s)\frac{\overline{H}_{n}(s-)}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)$
$\displaystyle=$
$\displaystyle\int\frac{\xi(s)}{\overline{G}_{n}(s-)}\,\text{d}H^{1}_{n}(s)=\int\xi(s)\,\text{d}F_{n}(s).$

Applying $\int_{0}^{\infty}\xi(s)\,\text{d}F(s)=\text{E}\xi(Y)=0$ and Theorem
3.1, we arrive at

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\sqrt{n}\int\xi(s)\,\text{d}(F_{n}(s)-F(s))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$
(4. 6)

∎

Following Owen (2001), define the EL ratio of ${\theta}$ by a multinomial
likelihood subject to constraints as

$R({\theta})=\sup_{\\{p_{i}\\}}\left\\{\ \prod_{i=1}^{n}np_{i}\Big{|}\
\sum_{i=1}^{n}p_{i}=1,\,\sum_{i=1}^{n}p_{i}W_{ni}=0,\,p_{i}\geq
0,\,i=1,2,\cdots,n\right\\}.$ (4. 7)

To determine $R({\theta})$, we solve, as usual, for the Lagrange multipliers
$\mu$ and $\lambda$ in

$A=\sum_{i=1}^{n}\log(np_{i})-n\lambda(\sum_{i=1}^{n}p_{i}W_{ni})-\mu(1-\sum_{i=1}^{n}p_{i}).$

Then $\mu=-n$ and $p_{i}=\dfrac{1}{n}(1+\lambda W_{ni})^{-1},\
i=1,2,\cdots,n,$ where $\lambda$ is the solution of

$\dfrac{1}{n}\sum\limits_{i=1}^{n}\dfrac{W_{ni}}{1+\lambda W_{ni}}=0.$

The uniqueness of $\lambda$ will be addressed in the proof of Theorem 4.2. The
EL ratio of ${\theta}$ can be written as

$R({\theta})=\prod_{i=1}^{n}(np_{i})=\prod_{i=1}^{n}(1+\lambda W_{ni})^{-1}.$
(4. 8)

######  Theorem 4.2.

Suppose that ${\theta}_{0}$ is the unique solution of (4.2) and finite second
moment (1.3) holds. Set $\xi(x)=g(x,{\theta}_{0})$. Then
$l({\theta}_{0})=-2\log R({\theta}_{0})$ converges in distribution to a
$\chi_{1}^{2}$ random variable with one degree of freedom, as $n\to\infty$.

Applying Theorem 4.2, confidence intervals for ${\theta}$ can be constructed
as

$I_{1}=\\{{\theta}:l({\theta})\leq c_{1-\alpha}\\},$ (4. 9)

where $c_{1-\alpha}$ is the $(1-\alpha)$th quantile of the $\chi_{1}^{2}$
distribution. $I_{1}$ has asymptotic coverage probability of $1-\alpha$, as
$n\to\infty.$

To prove Theorem 4.2, we shall make use of the following Taylor’s expansion of
$-2\log R({\theta}_{0})$,

$\displaystyle-2\log R({\theta}_{0})$ $\displaystyle=$ $\displaystyle
2\sum_{i=1}^{n}\ln(1+\lambda_{n}W_{ni})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}2\Big{(}\lambda_{n}W_{ni}-\frac{1}{2}\lambda_{n}^{2}W_{ni}^{2}+\eta_{i}\Big{)}\qquad(\
\text{here}\ |\eta_{i}|\leq|X_{i}|^{3})$ $\displaystyle=$ $\displaystyle
2\lambda_{n}n\overline{W}_{n}-\lambda_{n}^{2}nS_{n}^{2}+2\sum_{i=1}^{n}\eta_{i}$
$\displaystyle=$ $\displaystyle
2n\frac{\overline{W}_{n}^{2}}{\sigma^{2}}-\frac{1}{\sigma^{4}}n\overline{W}_{n}^{2}(\sigma^{2}+o_{p}(1))+o_{p}(1)$
$\displaystyle=$
$\displaystyle\frac{n\overline{W}_{n}^{2}}{\sigma^{2}}+o_{p}(1)\to\chi_{1}^{2},\qquad\text{in
distribution},$

where

$\displaystyle\overline{W}_{n}=\sum_{i=1}^{n}W_{ni},\quad\mbox{and \;
$S_{n}^{2}=\sum_{i=1}^{n}W_{ni}^{2}$.}$ (4. 10)

Asymptotic analysis of $\overline{W}_{n}=\sum_{i=1}^{n}W_{ni}$ and
$S_{n}^{2}=\sum_{i=1}^{n}W_{ni}^{2}$ are needed for establishing the last two
equalities in the expansion. It will be proven in Lemma 4.3 and Theorem 4.2
that both of these averages are related to the asymptotic variance (1.4) or
(3.10). The following lemmas are needed for proving Theorem 4.2.

Lemma 4.1 Let $f_{n}(x)$ and $f(x)$ be monotone functions defined on the range
of $Z$. If $f(x)$ is continuous and for $x\in[0,b_{H}]$, $f_{n}(x)\to f(x)$,
as $n\to\infty$, then $f_{n}(x)$ converges to $f(x)$ uniformly on $[0,b_{H}]$.
The proof is omitted.

Lemma 4.2 Let $V_{ni}=W_{ni}{\overline{H}}_{n}(Z_{i}-){\overline{H}}(Z_{i})$
and $V_{i}=W_{i}{\overline{H}}^{2}(Z_{i})$. Under the conditions of Theorem
4.2 and ${\theta}={\theta}_{0}$, as $n\to\infty$,

(1) $\dfrac{1}{n}\sum\limits_{i=1}^{n}(W_{ni}-W_{i})^{2}=o_{p}(1)$,

(2) $\dfrac{1}{n}\sum\limits_{i=1}^{n}(V_{ni}-V_{i})^{2}\to 0,\ \text{a.s.}$.

The proof is relegated to the Appendix. Lemma 4.2 is needed for showing that
with probability 1, for large $n$ the set $\\{W_{ni}\\}$ contains a positive
and a negative value. To facilitate the proof, $V_{ni}^{\prime}s$, a
modification of $W_{ni}^{\prime}s$, are introduced to deal with the problem at
the boundary $b_{H}.$ It follows that for large $n$ there exists a unique
$\lambda_{n}$ for $R({\theta})$ in (4.8).

Lemma 4.3 Let $W_{ni}$ and $W_{i}$ be given by (4.4) and (4.1), respectively.
Under the conditions of Theorem 4.2 and ${\theta}={\theta}_{0}$, as
$n\to\infty$,

$\max\limits_{1\leq i\leq n}|W_{ni}|=o_{p}(\sqrt{n}),\ \ \
\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\sigma^{2}+o_{p}(1),$

and

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}\to N(0,\,\sigma^{2}),\ \ \text{in
dist.},$

where $\sigma^{2}=\text{Var}(W_{i})$ is given by (3.10).

The proof is relegated to the Appendix. We now prove Theorem 4.2.

###### Proof of Theorem 4.2.

The Lagrange multiplier $\lambda$ in (4.6) appears in the equation

$\displaystyle h(\lambda)=\frac{1}{n}\sum_{i=1}^{n}\frac{W_{ni}}{1+\lambda
W_{ni}}=0.$ (4. 11)

We shall show that for large $n$, $h(\lambda)=0$ has a unique solution
$\lambda_{n}$ such that $\lambda_{n}W_{ni}>-1$ for all $i$. Put

$U_{i}=\begin{cases}-W_{ni}^{-1},&W_{ni}\neq 0,\\\
\infty,&W_{ni}=0.\end{cases}$

Let $U_{(1)}\leq U_{(2)}\leq\cdots\leq U_{(n)}$ be the ordered statistics of
$U_{1},U_{2},\cdots,U_{n}$. Then

$h(\lambda)=\frac{1}{n}\sum_{i=1}^{n}\frac{W_{ni}}{1+\lambda
W_{ni}}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda-U_{(i)}}$

is monotone and differentiable in $\lambda$ on each nonempty interval
$(U_{(i)},U_{(i+1)})$. We claim that for large $n$, there exists an $i$ such
that $U_{(i)}<0<U_{(i+1)}$. To see this, we note that for every
$\varepsilon>0$, by Lemma 4.2(2)

$\frac{1}{n}\sum_{i=1}^{n}I[\,|V_{ni}-V_{i}|\geq\varepsilon\,]\leq\frac{1}{n}\sum_{i=1}^{n}(V_{ni}-V_{i})^{2}/\varepsilon^{2}=o(1),\
\text{a.s.}.$

Using the fact that $I[V_{i}\geq\varepsilon]\leq
I[V_{ni}\geq\varepsilon/2]+I[{|V_{ni}-V_{i}|\geq\varepsilon/2}]$, we get

$\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\geq\varepsilon/2]\geq\frac{1}{n}\sum_{i=1}^{n}I[V_{i}\geq\varepsilon]+o(1),\
\text{a.s.}.$

Using the fact that $P(V_{1}>0)=P(W_{1}>0)>0$, it is seen that for some
$\varepsilon>0$,

$\displaystyle\liminf_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\geq\varepsilon/2]\geq
P(V_{1}\geq\varepsilon)>0,\ \text{a.s.}.$

Similarly, we have

$\liminf_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\leq-\varepsilon/2]\geq
P(V_{1}\leq-\varepsilon)>0,\ \text{a.s.}.$

Since $W_{ni}$ and $V_{ni}$ have the same sign, hence the claim is true. It
follows that there is a unique
$\lambda_{n}\in(U_{(i)},U_{(i+1)})=(-1/\max\\{W_{ni}\\},-1/\min\\{W_{ni}\\})$
such that $h(\lambda_{n})=0$ and $\lambda_{n}\max\\{W_{ni}\\}>-1$ and
$\lambda_{n}\min\\{W_{ni}\\}>-1$.

The rest of the proof is similar to that of Owen (2001). In fact, setting

$X_{i}=\lambda_{n}W_{ni},\ \
\overline{W}_{n}=\frac{1}{n}\sum_{i=1}^{n}W_{ni},\ \
S_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}W_{ni}^{2},$

we have $S_{n}^{2}=\sigma^{2}+o_{p}(1)$,

$\frac{1}{n}\sum_{i=1}^{n}\frac{X_{i}^{2}}{1+X_{i}}=\frac{1}{n}\sum_{i=1}^{n}\frac{(X_{i}+1-1)X_{i}}{1+X_{i}}=\overline{X}_{n}-\lambda_{n}h(\lambda_{n})=\lambda_{n}\overline{W}_{n}$

and

$\lambda_{n}^{2}S_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}\leq\frac{1}{n}\sum_{i=1}^{n}\frac{X_{i}^{2}}{1+X_{i}}\big{(}1+\max_{1\leq
j\leq n}|X_{j}|\big{)}=\lambda_{n}\overline{W}_{n}+\lambda_{n}^{2}o_{p}(1).$
(4. 12)

It follows that

$\lambda_{n}=\frac{\overline{W}_{n}}{\sigma^{2}+o_{p}(1)}=O_{p}(n^{-1/2})$

and

$\overline{W}_{n}=\lambda_{n}\sigma^{2}+o_{p}(n^{-1/2}).$ (4. 13)

Applying Lemma 4.3, we have

$\sum_{i=1}^{n}|X_{i}|^{3}\leq\lambda_{n}^{3}\sum_{i=1}^{n}|W_{ni}|^{2}\max_{1\leq
j\leq n}|W_{ni}|=O_{p}(n^{-3/2})O_{p}(n)o_{p}(n^{1/2})=o_{p}(1).$

Therefore the Taylor expansion (above (4.10)) is valid from which the theorem
follows.

∎

Remark 4.1 We are able to obtain the standard asymptotic $\chi^{2}_{1}$
distribution for $-2\log R({\theta}_{0})$ is because the asymptotic variance
of

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1)$

(which is $\sigma^{2}=\text{Var}(W_{i})$) equals the limit of (see Lemma 4.3)

$\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\frac{1}{n}\sum_{i=1}^{n}W^{2}_{i}+o_{p}(1).$

If $-2\log(EL\,ratio)$ is based on the estimating function
$M_{2}=M_{2}(Z,\delta,{\theta})$ in (3.8) ( or in (3.7)), then

${V}_{ni}=\frac{g(Z_{i},{\theta})\delta_{i}}{1-G_{n}(Z_{i})}$

will be used to construct $-2\log(EL\,ratio)$. Now, the asymptotic variance of

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}V_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1)$

is $\sigma^{2}$ (see(3.10)), but the limit of
$\frac{1}{n}\sum_{i=1}^{n}V^{2}_{ni}$ or

$\frac{1}{n}\sum_{i=1}^{n}\Big{(}V_{ni}-\frac{1}{n}\sum_{j=1}^{n}V_{nj}\Big{)}^{2}$

is (see Remark 3.1)

$\sigma_{1}^{2}=\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2},\
\ \mu=0.$

Therefore, a scaled parameter $r=\sigma_{1}^{2}/\sigma^{2}$ must be introduced
in order to obtain the asymptotic distribution for $-2\log(EL\,ratio)$ as in
Wang $\&$ Jing (2001).

##  5 Simulation

Simulations are carried out to study and compare finite sample performance of
confidence intervals $I_{1}$ in (4.9) derived from Theorem 4.2 and $I_{2}$
from the scaled $\chi_{1}^{2}$ distribution given by Wang & Jing (2001) and
Qin & Zhao (2007).

To calculate $I_{1}$, $W_{ni}$ in (4.4) is used, where

$W_{ni}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\frac{1}{n}\sum_{j=1}^{n}\psi_{n}(Z_{j})\frac{I[Z_{i}\geq
Z_{j}]{\overline{\delta}}_{j}}{\overline{H}_{n}^{2}(Z_{j}-)},$

and $\psi_{n}(x)$ is given by (4.3).

Confidence intervals $I_{2}$ are calculated as follows. Let $F_{n},$ and
$G_{n}$ be the Kaplan-Meier estimators defined by (2.3). Suppose
$\hat{\theta}$ is the unique solution of $\int
g(s,\,\theta)\,\mathrm{d}F_{n}(s)=0$. Set

$\xi_{i}=g(Z_{i},\,\theta),\qquad\hat{\xi}_{i}=g(Z_{i},\,\hat{\theta}),$
$V_{ni}=\frac{\xi_{i}\,\delta_{i}}{1-G_{n}(Z_{i})},\qquad\hat{V}_{ni}=\frac{\hat{\xi}_{i}\,\delta_{i}}{1-G_{n}(Z_{i})},$
(5. 1)
$\sigma_{1}^{2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{V}_{ni}-\overline{V}_{n})^{2},\qquad\overline{V}_{n}=\frac{1}{n}\sum_{i=1}^{n}\hat{V}_{ni},$
$\hat{r}=\frac{\sigma_{1}^{2}}{n\,\widehat{\mathrm{Var}}^{*}(jack)},$

where $n\,\widehat{\mathrm{Var}}^{*}(jack)$ is the modified jackknife
estimator of the asymptotic variance of $\hat{\xi}$ given in Stute(1996).
Then, the EL-based confidence interval for ${\theta}$ is

$I_{2}=\left\\{\theta:\,\,2\,\hat{r}\sum_{i=1}^{n}\log(1+\lambda V_{ni})\leq
c_{1-\alpha}\right\\},$ (5. 2)

where $\lambda$ is the solution of $\sum_{i=1}^{n}V_{ni}/(1+\lambda
V_{ni})=0$.

Simulations were performed in two scenarios. In scenario I, the parameter of
interest is ${\theta}_{0}=\text{E}\,Y$ and in scenario II, the mean residual
lifetime. Scenario I: The parameter of interest is ${\theta}_{0}=\text{E}\,Y$
and $\xi(x)=g(x,{\theta})=x-{\theta}$ is used for calculating $I_{1}$. Two
cases were simulated:

(i) The lifetime $Y$ is uniformly distributed on $(0,1)$ and the censoring
time $C$ is uniformly distributed on (0, c). We selected $c=2.5$ and $c=1.3$
which corresponds respectively to 20% and 30% censoring proportions.

(ii) $Y$ has a Weibull(1, 10) distribution and $C$ has an Exp($\lambda$)
distribution. Then for $\lambda=4.3$ and $\lambda=2.7$, the corresponding
censoring proportions are 20% and 30%. The simulated observations are $n$
i.i.d. copies of $Z=\min(Y,\,C),\delta=I[Y\leq C]$. Based on the simulated
observations, confidence intervals $I_{1}$ derived from Theorem 4.2 and
$I_{2}$ from (5.2) were calculated. The process was repeated for $N=2\times
10^{4}$ times and the coverage proportions and the average width of the
confidence intervals were calculated using the $N$ data sets. The results are
summarized in Table 1 and Table 2.

Table 1: The coverage proportions for the true ${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion |  |  |  |  |
---|---|---|---|---|---
Nominal Value | Sample Size | Uniform(0, 1) |  | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ |  | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.876 | 0.881 |  | 0.871 | 0.871
| 40 | 0.895 | 0.897 |  | 0.889 | 0.890
| 60 | 0.897 | 0.897 |  | 0.893 | 0.893
| 80 | 0.897 | 0.898 |  | 0.896 | 0.896
0.95 | 20 | 0.928 | 0.935 |  | 0.922 | 0.924
| 40 | 0.946 | 0.949 |  | 0.939 | 0.941
| 60 | 0.947 | 0.948 |  | 0.945 | 0.946
| 80 | 0.947 | 0.947 |  | 0.947 | 0.948
30% censoring proportion |  |  |  |  |
Nominal Value | Sample Size | Uniform(0, 1) |  | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ |  | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.841 | 0.861 |  | 0.867 | 0.869
| 40 | 0.885 | 0.890 |  | 0.890 | 0.891
| 60 | 0.888 | 0.892 |  | 0.890 | 0.891
| 80 | 0.897 | 0.900 |  | 0.893 | 0.894
0.95 | 20 | 0.897 | 0.916 |  | 0.916 | 0.924
| 40 | 0.934 | 0.941 |  | 0.939 | 0.943
| 60 | 0.941 | 0.946 |  | 0.944 | 0.946
| 80 | 0.945 | 0.947 |  | 0.945 | 0.947
Table 2: The average width of confidence intervals for ${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion | width |  | width
---|---|---|---
Nominal Value | Sample Size | Uniform(0, 1) |  | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ |  | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.217 | 0.218 |  | 0.092 | 0.091
| 40 | 0.157 | 0.157 |  | 0.066 | 0.065
| 60 | 0.129 | 0.129 |  | 0.054 | 0.053
| 80 | 0.112 | 0.112 |  | 0.046 | 0.046
0.95 | 20 | 0.258 | 0.259 |  | 0.110 | 0.109
| 40 | 0.187 | 0.187 |  | 0.079 | 0.078
| 60 | 0.154 | 0.154 |  | 0.064 | 0.064
| 80 | 0.133 | 0.133 |  | 0.056 | 0.055
30% censoring proportion | width |  | width
Nominal Value | Sample Size | Uniform(0, 1) |  | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ |  | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.220 | 0.227 |  | 0.097 | 0.096
| 40 | 0.162 | 0.164 |  | 0.069 | 0.069
| 60 | 0.134 | 0.134 |  | 0.057 | 0.057
| 80 | 0.116 | 0.116 |  | 0.049 | 0.049
0.95 | 20 | 0.260 | 0.270 |  | 0.116 | 0.116
| 40 | 0.192 | 0.196 |  | 0.083 | 0.083
| 60 | 0.159 | 0.160 |  | 0.068 | 0.068
| 80 | 0.138 | 0.139 |  | 0.059 | 0.059

The following are noted.

(1) As the sample size $n$ increases, all of the coverage proportions converge
to the nominal level $1-\alpha$.

(2) For Uniform(0, 1) distribution, $I_{1}$ has better coverage proportions.
In 8/16 of the cases, the average width of $I_{2}$ is slightly shorter than
that of $I_{1}$. In 8/16 of the cases, $I_{2}$ and $I_{1}$ have the same
average width.

(3) For Weibull(1, 10) distribution, $I_{1}$ has better coverage proportion
and width.

In the $j$th simulation, $\\{W_{ni}\\}$ and $\\{V_{ni}\\}$ were calculated
according to (4.4) and (5.1) respectively. Then the sample means of
$\\{W_{ni}\\}$ and $\\{V_{ni}\\}$ are the same (see the proof of Theorem 4.1).
But the sample variance $s_{W}^{2}(j)$ of $\\{W_{ni}\\}$ and the sample
variance $s_{V}^{2}(j)$ of $\\{V_{ni}\\}$ are different. Let

$s_{W}^{2}=\frac{1}{N}\sum_{j=1}^{N}s_{W}^{2}(j),\ \ \
s_{V}^{2}=\frac{1}{N}\sum_{j=1}^{N}s_{V}^{2}(j).$

Table 3 shows that the sample variance of $\\{W_{ni}\\}$ is smaller than that
of $\\{V_{ni}\\}$. This is proved in Remark 3.1 for the population variances.

Table 3: The sample variances of $\\{W_{ni}\\}$ and $\\{V_{ni}\\}$,
${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion
---
| Uniform(0, 1) |  | Weibull(1, 10)
Sample Size $n$ | $s_{W}^{2}$ | $s_{V}^{2}$ |  | $s_{W}^{2}$ | $s_{V}^{2}$
20 | 0.0935 | 0.1121 |  | 0.0157 | 0.0163
40 | 0.0938 | 0.1115 |  | 0.0157 | 0.0162
60 | 0.0937 | 0.1107 |  | 0.0157 | 0.0161
80 | 0.0934 | 0.1100 |  | 0.0158 | 0.0162
30% censoring proportion
| Uniform(0,1) |  | Weibull(1, 10)
Sample Size $n$ | $s_{W}^{2}$ | $s_{V}^{2}$ |  | $s_{W}^{2}$ | $s_{V}^{2}$
20 | 0.1005 | 0.1386 |  | 0.0175 | 0.0185
40 | 0.1013 | 0.1401 |  | 0.0176 | 0.0184
60 | 0.1016 | 0.1402 |  | 0.0176 | 0.0183
80 | 0.1012 | 0.1393 |  | 0.0176 | 0.0183

Scenario II: Let $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$. Then by
solving the equation $\text{E}\,g(Y,{\theta})=0$, we obtain the mean residual
life of $Y$,

${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})=\frac{\text{E}(Y-t_{0})I[Y\geq
t_{0}]}{P(Y\geq t_{0})},$ (5. 3)

as studied in Qin & Zhao (2007). Let $Y$ have a Weibull (1, 10) distribution
and $C$ have an Exp($\lambda$) distribution. By setting $\lambda=4.3$ and
$\lambda=2.7$, we achieved 20% and 30% censoring proportions respectively. As
in Scenario I, each simulation was repeated $N=2\times 10^{4}$ times. The
coverage proportion of the $N$ data sets and their average width were
calculated. The results are summarized in Table 4 and Table 5, respectively.

Table 4: The coverage proportion and average width of confidence intervals for ${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})$ under the assumptions of $Y\sim$ Weibull(1, 10), 20% censoring proportion, and $1-\alpha=0.90$.  Sample Size $n$ | Method | Coverage Ratio |  | Average Width
---|---|---|---|---
|  | $P(Y\geq t_{0})$ |  | $P(Y\geq t_{0})$
|  | $0.90$ | $0.70$ | $0.50$ | $0.30$ |  | $0.90$ | $0.70$ | $0.50$ | $0.30$
20 | $I_{2}$ | 0.878 | 0.851 | 0.795 | 0.659 |  | 0.074 | 0.062 | 0.054 | 0.044
| $I_{1}$ | 0.881 | 0.863 | 0.820 | 0.701 |  | 0.074 | 0.062 | 0.056 | 0.048
40 | $I_{2}$ | 0.889 | 0.878 | 0.859 | 0.800 |  | 0.053 | 0.046 | 0.042 | 0.039
| $I_{1}$ | 0.891 | 0.884 | 0.874 | 0.833 |  | 0.053 | 0.046 | 0.043 | 0.041
60 | $I_{2}$ | 0.897 | 0.892 | 0.877 | 0.839 |  | 0.044 | 0.037 | 0.035 | 0.034
| $I_{1}$ | 0.898 | 0.897 | 0.888 | 0.863 |  | 0.044 | 0.038 | 0.035 | 0.035
80 | $I_{2}$ | 0.895 | 0.888 | 0.884 | 0.853 |  | 0.038 | 0.033 | 0.031 | 0.030
| $I_{1}$ | 0.896 | 0.892 | 0.892 | 0.871 |  | 0.038 | 0.033 | 0.031 | 0.031
Table 5: The coverage ratio and average width of confidence intervals for ${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})$, under the assumptions $Y\sim$ Weibull (1, 10), 30% censoring proportion and $1-\alpha=0.90$. Sample Size $n$ | Method | Coverage Ratio |  | Average Width
---|---|---|---|---
|  | $P(Y\geq t_{0})$ |  | $P(Y\geq t_{0})$
|  | $0.90$ | $0.70$ | $0.50$ | $0.30$ |  | $0.90$ | $0.70$ | $0.50$ | $0.30$
20 | $I_{2}$ | 0.864 | 0.833 | 0.760 | 0.605 |  | 0.079 | 0.065 | 0.055 | 0.043
| $I_{1}$ | 0.872 | 0.851 | 0.793 | 0.659 |  | 0.079 | 0.065 | 0.058 | 0.048
40 | $I_{2}$ | 0.887 | 0.872 | 0.846 | 0.777 |  | 0.057 | 0.048 | 0.045 | 0.041
| $I_{1}$ | 0.891 | 0.882 | 0.867 | 0.818 |  | 0.057 | 0.049 | 0.046 | 0.043
60 | $I_{2}$ | 0.892 | 0.888 | 0.870 | 0.822 |  | 0.046 | 0.040 | 0.037 | 0.036
| $I_{1}$ | 0.895 | 0.895 | 0.884 | 0.851 |  | 0.046 | 0.040 | 0.038 | 0.037
80 | $I_{2}$ | 0.892 | 0.888 | 0.878 | 0.845 |  | 0.040 | 0.035 | 0.033 | 0.032
| $I_{1}$ | 0.895 | 0.895 | 0.887 | 0.869 |  | 0.040 | 0.035 | 0.033 | 0.033

The following are noted from Tables 4 and 5.

(1) As the sample size $n$ increases, all of the coverage proportions increase
and are close to the nominal levels.

(2) The coverage proportions of $I_{1}$ are much better than that of $I_{2}$.

(3) In 15/32 of the cases, the average width of $I_{2}$ is slightly shorter
than that of $I_{1}$. In 17/32 of the cases, $I_{2}$ and $I_{1}$ have the same
average width.

##  6 Appendix

###### Proof of Lemma 3.2.

Put $\eta_{n}=o_{p}(1)$. By assumptions, for any $\varepsilon>0$ and
$\delta>0$, there exist $M>0$, $b<b_{H}$ and $n_{0}>1$ such that for $n\geq
n_{0}$, $P(|V_{n}|\geq M)\leq\delta$, $P(|h_{n}(b)|\geq\varepsilon/M)\leq
P(|h(b)|\geq\varepsilon/M)+\delta/2\leq\delta$ and
$P(|\eta_{n}|\geq\varepsilon)<\delta$ . It follows that for $n\geq n_{0}$,

$\displaystyle P(|S_{n}|\geq 2\varepsilon)$ $\displaystyle\leq$ $\displaystyle
P(|\eta_{n}|\geq\varepsilon)+P(|V_{n}h_{n}(b)|\geq\varepsilon)$
$\displaystyle\leq$
$\displaystyle\delta+P(|V_{n}h_{n}(b)|\geq\varepsilon,|h_{n}(b)|\leq\varepsilon/M)+\delta$
$\displaystyle\leq$ $\displaystyle P(|V_{n}|\geq M)+2\delta$
$\displaystyle\leq$ $\displaystyle 3\delta.$

∎

###### Proof of Lemma 4.2.

The differences $W_{ni}-W_{i}$ in eq. (4.4) and (4.1) can be expressed in
terms of

$\displaystyle\gamma_{i}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}-\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})},$
$\displaystyle\eta_{i}=\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i}),$
$\displaystyle\nu_{i}=\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s),$

as

$W_{ni}-W_{i}=\gamma_{i}+\eta_{i}-\nu_{i}.$

Applying an elementary inequality $(a+b+c)^{2}\leq 3(a^{2}+b^{2}+c^{2})$, we
obtain

$(W_{ni}-W_{i})^{2}=(\gamma_{i}+\eta_{i}-\nu_{i})^{2}\leq
3(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}).$ (6. 1)

The lemma will be proven by showing that the sample means of
$\gamma^{2}_{i},\eta_{i}^{2}$ and $\nu_{i}^{2}$ tend to zero in probability.
The proofs will be presented in (A), (B) and (C) below.

(A) The sample mean of $\gamma_{i}^{2}$ is $o_{p}(1)$.

Proof: Let $G_{n}(x)$ be the K-M estimator defined in (2.3) and $b<b_{H}$.
Then as $n\to\infty$,

$U_{n}=\sup_{s\leq b}\frac{|G_{n}(s-)-G(s)|}{\overline{G}_{n}(s)}=o_{p}(1),\ \
V_{n}=\sup_{s\leq\max\\{Z_{i}\\}}\frac{|G_{n}(s-)-G(s)|}{\overline{G}_{n}(s-)}=O_{p}(1).$
(6. 2)

See Zhou (1992). To apply this result, we shall in the following proof split
the integrals into two intervals $[0,b]$ and $(b,b_{H}]$ accordingly.

For any $b<b_{H}$, using (2.2), we have

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\gamma_{i}^{2}$ $\displaystyle=$
$\displaystyle\int\Big{(}\frac{\xi(s)}{\overline{G}_{n}(s-)}-\frac{\xi(s)}{\overline{G}(s)}\Big{)}^{2}\,\text{d}H_{n}^{1}(s)$
(6. 3) $\displaystyle\leq$ $\displaystyle
U_{n}^{2}\int_{0}^{b}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)+V_{n}^{2}\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)$
$\displaystyle=$ $\displaystyle o_{p}(1)O_{p}(1)+O_{p}(1)h_{n}(b)=o_{p}(1),\
\text{as}\ n\to\infty,$

where $U_{n}^{2}$ and $V_{n}^{2}$ are given by (6.2) and

$\displaystyle
h_{n}(b)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s).$
(6. 4)

Recall that $h_{n}(b)$ is explained in (3.12). It was shown that $h_{n}(b)$
satisfies the conditions in Lemma 3.2. The proof follows by invoking Lemma
3.2.

(B) The sample mean of $\eta_{i}^{2}$ is $o_{p}(1)$.

Proof: For $b<b_{H}$, define

$T_{n}(b,t]=\int_{b}^{t}\frac{\psi_{n}^{2}(s)}{\overline{H}_{n}^{2}(s-)}\,\text{d}H^{0}_{n}(s),\
\
S_{n}(b,t]=\int_{b}^{t}\frac{\psi^{2}(s)}{\overline{H}^{2}(s)}\,\text{d}H^{0}_{n}(s).$

Observe that

$\psi_{n}^{2}(s)=\Big{(}\int_{u\geq
s}\xi(u)\,\text{d}F_{n}(u)\Big{)}^{2}\leq\overline{F}_{n}(s-)\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u),$

and $F_{n}$ and $G_{n}$ have no common jumps. It follows that

$\displaystyle T_{n}(b,b_{H}]$ $\displaystyle\leq$
$\displaystyle\int_{b}^{b_{H}}\Big{(}\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u)\Big{)}\overline{F}_{n}^{2}(s-)\frac{\,\text{d}G_{n}(s)}{\overline{H}_{n}^{2}(s-)}$
(6. 5) $\displaystyle\leq$ $\displaystyle\int_{b}^{b_{H}}\Big{(}\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u)\Big{)}\,\text{d}\Big{(}\frac{1}{\overline{G}_{n}(s)}\Big{)}$
$\displaystyle\leq$ $\displaystyle\lim_{s\to
b_{H}}\frac{1}{\overline{G}_{n}(s)}\int_{s}^{b_{H}}\xi^{2}(u)\,\text{d}F_{n}(u)+\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}_{n}(s-)}\,\text{d}F_{n}(s)$
$\displaystyle\leq$ $\displaystyle
2\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}_{n}(s-)}\,\text{d}F_{n}(s)$
$\displaystyle\leq$ $\displaystyle
2\int_{b}^{b_{H}}\Big{(}\frac{1}{\overline{G}_{n}(s-)}-\frac{1}{\overline{G}(s)}+\frac{1}{\overline{G}(s)}\Big{)}^{2}\xi^{2}(s)\,\text{d}H_{n}^{1}(s)$
$\displaystyle=$ $\displaystyle o_{p}(1)+O_{p}(1)h_{n}(b).$

The first inequality follows from (2.6) the second and the third from
integration by parts, the fifth from (2.5) and the last equality from
$(a+b)^{2}\leq 2(a^{2}+b^{2})$, (6.3) and (6.4).

By the same token, we conclude that
$S_{n}(b,b_{H}]=o_{p}(1)+O_{p}(1)h_{n}(b)$.

Write

$\xi=\xi^{+}-\xi^{-},\ \psi_{n}(x)=\int_{s\geq
x}\xi^{+}(s)\,\text{d}F_{n}(s)-\int_{s\geq x}\xi^{-}(s)\,\text{d}F_{n}(s),$

where $\xi^{+}$ and $\xi^{-}$ are the positive and negative part of $\xi$.
Define monotone functions:

$\psi_{n}^{\pm}(x)=\int_{s\geq x}\xi^{\pm}(s)\,\text{d}F_{n}(s),\
\psi^{\pm}(x)=\int_{s\geq x}\xi^{\pm}(s)\,\text{d}F(s).$

$\psi_{n}^{\pm}(x)$ converges to $\psi^{\pm}(x)$ almost surely for
$x\in[0,b_{H}]$ as shown by Stute & Wang (1993). Furthermore, by Lemma 4.1,
the convergence is uniform on $[0,b_{H}]$.

From these we conclude the uniform convergence of $\psi_{n}$ to $\psi$,

$\displaystyle\sup_{0\leq x\leq b_{H}}|\psi_{n}(x)-\psi(x)|=o(1),\
\text{a.s.}.$ (6. 6)

Therefore, for $b<b_{H}$,

$\sup_{s\leq
b}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\to
0,\ \text{a.s.}.$

Applying $(a+b)^{2}\leq 2a^{2}+2b^{2}$ and Lemma 3.2, we have

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\eta_{i}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{b}+\int_{b}^{b_{H}}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\,\text{d}H^{0}_{n}(s)$
(6. 7) $\displaystyle\leq$
$\displaystyle\int_{0}^{b}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\,\text{d}H^{0}_{n}(s)+2T_{n}(b,b_{H}]+2S_{n}(b,b_{H}]$
$\displaystyle=$ $\displaystyle
o_{p}(1)+O_{p}(1)h_{n}(b)+O_{p}(1)h_{n}(b)=o_{p}(1).$

(C) The sample mean of $\nu_{i}^{2}$ is $o_{p}(1)$.

Proof: Write, for $0\leq a<t$,

$B_{n}(a,t]=\int_{a}^{t}\frac{\psi_{n}(s)\,\text{d}H_{n}^{0}(s)}{\overline{H}_{n}^{2}(s-)},\
\ D(a,t]=\int_{a}^{t}\frac{\psi(s)\,\text{d}H^{0}(s)}{\overline{H}^{2}(s)}.$

Then, for $b<b_{H}$, we have

$\displaystyle\Delta_{n}^{2}$ $\displaystyle\equiv$
$\displaystyle\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)=\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}(-\overline{H}_{n}(t))$
$\displaystyle\leq$
$\displaystyle\overline{H}_{n}(b)B_{n}^{2}(b,b]+2\int_{b}^{b_{H}}\overline{H}_{n}(t-)B_{n}(b,t]\frac{\psi_{n}(t)}{\overline{H}_{n}^{2}(t-)}\,\text{d}H_{n}^{0}(t)$
$\displaystyle\leq$ $\displaystyle
0+2\Big{(}\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)\Big{)}^{1/2}\Big{(}\int_{b}^{b_{H}}\frac{\psi_{n}^{2}(t)}{\overline{H}_{n}^{2}(t-)}\,\text{d}H_{n}^{0}(t)\Big{)}^{1/2}$
$\displaystyle=$ $\displaystyle\Delta_{n}2[T_{n}(b,b_{H})]^{1/2}.$

The second term in the first inequality is obtained using the Lebesgue-
Stieltjes integration by parts.

Applying (6.5), we get

$\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)=\Delta_{n}^{2}\leq
4T_{n}(b,b_{H}]=o_{p}(1)+O_{p}(1)h_{n}(b).$

Similarly, for $S(b,b_{H}]=\text{E}S_{n}(b,b_{H}]$, we have

$\int_{b}^{b_{H}}D^{2}(b,t]\,\text{d}H_{n}(t)\leq
4S(b,b_{H}]+o_{p}(1)=o_{p}(1)+O_{p}(1)h_{n}(b).$

Applying the uniform convergence of $\psi_{n}$ to $\psi$ (see(6.6)), we
conclude that for any $b<b_{H}$, with probability 1, $B_{n}(0,t]\to D(0,t]$
uniformly on $[0,b]$. It follows that $B_{n}(0,b]\to D(0,b]$, and

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\nu_{i}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{b}+\int_{b}^{b_{H}}\big{(}B_{n}(0,t]-D(0,t]\big{)}^{2}\,\text{d}H_{n}(t)$
$\displaystyle=$ $\displaystyle
o_{p}(1)+\int_{b}^{b_{H}}\big{(}B_{n}(0,b]-D(0,b]+B_{n}(b,t]-D(b,t]\big{)}^{2}\,\text{d}H_{n}(t)$
$\displaystyle\leq$ $\displaystyle
o_{p}(1)+4\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)+4\int_{b}^{b_{H}}D^{2}(b,t)\,\text{d}H_{n}(t)$
$\displaystyle=$ $\displaystyle
o_{p}(1)+O_{p}(1)h_{n}(b)+O_{p}(1)h_{n}(b)=o_{p}(1).$

Now, we prove result (2) of the lemma. Introduce
$A_{ni}=W_{i}{\overline{H}}_{n}(Z_{i}-){\overline{H}}(Z_{i})$. From (6.1) we
get

$(V_{ni}-A_{ni})^{2}\leq
3(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}){\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i}).$

Similar to (6.3) and (6.7), we have

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\gamma_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\Big{(}\frac{\xi(s)}{\overline{G}_{n}(s-)}-\frac{\xi(s)}{\overline{G}(s)}\Big{)}^{2}{\overline{H}}^{2}_{n}(s-){\overline{H}}^{2}(s)\,\text{d}H_{n}^{1}(s)=o(1),\
\text{a.s.},$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\eta_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}{\overline{H}}^{2}_{n}(s-){\overline{H}}^{2}(s)\,\text{d}H^{0}_{n}(s)=o(1),\
\text{a.s.}.$

Since for any $b<b_{H}$, with probability 1,

$|B_{n}(0,t]-D(0,t]|{\overline{H}_{n}(t-)}{\overline{H}}(t)\to 0$

uniformly on $[0,b]$, and

$\sup_{t\leq b_{H}}|B_{n}(0,t]-D(0,t]|{\overline{H}_{n}(t-)}{\overline{H}}(t)$

is bounded by some constant, it follows that

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\nu_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\big{[}(B_{n}(0,t]-D(0,t]){\overline{H}}_{n}(t-){\overline{H}}(t)\big{]}^{2}\,\text{d}H_{n}(t)=o(1),\
\text{a.s.}.$

Now we get

$\frac{1}{n}\sum_{i=1}^{n}(V_{ni}-A_{ni})^{2}\leq
3\frac{1}{n}\sum\limits_{i=1}^{n}(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}){\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})=o(1),\
\text{a.s.}.$

At last, we have

$\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}(V_{ni}-V_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2}{n}\sum\limits_{i=1}^{n}(V_{ni}-A_{ni})^{2}+\frac{2}{n}\sum\limits_{i=1}^{n}(A_{ni}-V_{i})^{2}$
$\displaystyle=$ $\displaystyle
o(1)+\frac{2}{n}\sum\limits_{i=1}^{n}(H(Z_{i})-H_{n}(Z_{i}-))^{2}W_{i}^{2}\overline{H}^{2}(Z_{i})$
$\displaystyle=$ $\displaystyle o(1),\ \text{a.s.}.$

∎

###### Proof of Lemma 4.3.

Since $W_{i}$ are i.i.d. random variables with zero mean and finite variance
$\sigma^{2}$, hence $\max\limits_{1\leq i\leq n}|W_{i}|=o_{p}(\sqrt{n})$. It
follows from Lemma 4.2 that

$\displaystyle\max\limits_{1\leq i\leq n}|W_{ni}|$ $\displaystyle\leq$
$\displaystyle\Big{(}\max\limits_{1\leq i\leq
n}|W_{ni}-W_{i}|^{2}\Big{)}^{1/2}+\max\limits_{1\leq i\leq n}|W_{i}|$ (6. 8)
$\displaystyle\leq$
$\displaystyle\sqrt{n}\Big{(}\frac{1}{n}\sum_{i=1}^{n}(W_{ni}-W_{i})^{2}\Big{)}^{1/2}+o_{p}(\sqrt{n})$
$\displaystyle=$ $\displaystyle o_{p}(\sqrt{n}).$

Note that $W_{ni}^{2}$ is bounded by

$W^{2}_{i}+(W_{i}-W_{ni})^{2}-2|W_{i}(W_{i}-W_{ni})|\leq W_{ni}^{2}\leq
W_{i}^{2}+(W_{ni}-W_{i})^{2}+2|W_{i}(W_{i}-W_{ni})|.$

By Lemma 4.2, we get

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\sigma^{2}+o_{p}(1).$

The last result follows from Theorem 3.1, $\xi(x)=g(x,{\theta}_{0})$ and
(4.6).

∎

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# MixUp Training Leads to Reduced Overfitting and Improved Calibration for the
Transformer Architecture

Wancong Zhang
Courant Institute of
Mathematical Sciences
New York University
<EMAIL_ADDRESS>
&Ieshan Vaidya
Center for Data Science
New York University
<EMAIL_ADDRESS>

###### Abstract

MixUp is a computer vision data augmentation technique that uses convex
interpolations of input data and their labels to enhance model generalization
during training. However, the application of MixUp to the natural language
understanding (NLU) domain has been limited, due to the difficulty of
interpolating text directly in the input space. In this study, we propose
MixUp methods at the Input, Manifold, and sentence embedding levels for the
transformer architecture, and apply them to finetune the BERT model for a
diverse set of NLU tasks. We find that MixUp can improve model performance, as
well as reduce test loss and model calibration error by up to $50\%$.

##  1 Introduction

MixUp is a data augmentation technique originated from computer vision, where
convex interpolations of input images and their corresponding labels are used
as additional training source. It has empirically been shown to improve the
performance of image classifiers, and leads to decision boundaries that
transition linearly between classes, providing a smoother estimate of
uncertainty (Zhang et al., 2018).

Adapting MixUp to language is non-trivial and not well explored. Unlike with
images, one can not interpolate texts in the input space directly. Several
studies get around this issue by interpolating sentence or word embeddings,
and show that MixUp can improve model performance for sentence classification
(Guo et al., 2019; Sun et al., 2020).

We investigate whether MixUp can be extended to a wider range of NLU tasks
including sentiment classification, multiclass classification, sentence
acceptability, natural language inference (NLI), and question answering (QA).
In addition, we propose a more diverse set of MixUp methods at the 1) Input,
2) Manifold, and 3) sentence-embedding levels for BERT (Devlin et al., 2019),
and compare their performance for different task types at different resource
settings. At last, we study the effect of MixUp on BERT’s calibration, which
tells us about the quality of a model’s predictive uncertainty (Guo et al.,
2017).

We find that 1) MixUp improves model performance over baselines for IMDb and
AGNews, particularly at lower resource settings. 2) MixUp can significantly
reduce the test loss and calibration error of BERT by up to $50\%$ without
sacrificing performance. 3) While MixUp of sentence embeddings can be
universally applied to a diverse set of NLU tasks, MixUp methods that
interpolate individual tokens, such as Input and Manifold MixUp, are more
appropriate for simpler tasks that do not necessarily require syntactic
knowledge (eg. sentiment analysis).

##  2 Background

###  2.1 MixUp

In MixUp, convex interpolations of training input and their labels are used
instead of the original data for training, with the effect of improved
generalization and robustness against adversarial attacks. The following
describes the MixUp process:

$\begin{gathered}\widetilde{x}=\lambda x_{i}+(1-\lambda)x_{j}\\\
\widetilde{y}=\lambda y_{i}+(1-\lambda)y_{j}\end{gathered}$ (1)

where $x_{i},x_{j}$ are the input vectors, $y_{i},y_{j}$ their one-hot coded
labels, and $\lambda\in[0,1]$ is sampled from the Beta distribution:
$\lambda\sim\beta(\alpha,\alpha)$ during every mini-batch, where $\alpha$ is a
hyperparameter. Lower $\alpha$ leads to more even proportions between
$\lambda$ and $1-\lambda$.

The application of MixUp can also be extended to the hidden representations.
This is referred to as Manifold MixUp (Verma et al., 2019).

###  2.2 MixUp for Language

Guo et al. (2019) investigate two strategies to apply MixUp to language using
an RNN architecture: 1) sentence-level MixUp interpolates the sentence
embeddings of two sentences, while 2) word-level MixUp interpolate individual
word embeddings at each position between two sentences. Both strategies have
been shown to improve accuracy and regularize the training for simple sentence
classification tasks such as sentiment analysis.

In a similar work done concurrently at the time of our experiments, Sun et al.
(2020) applied sentence embedding MixUp to finetune transformers on GLUE tasks
(Wang et al., 2018), and reported improved predictive performance for most
tasks.

While (Sun et al., 2020) focused on sentence embedding MixUp as the
augmentation strategy and predictive performance as the evaluation metric, we
evaluated a more diverse set of MixUp methods, including those operating on
the input and manifold levels, on additional metrics including negative log-
likelihood and model calibration. Unlike the previous work, we did not find
significant predictive improvement for the shared GLUE tasks. On the other
hand, we showed that MixUp can significantly improve model calibration and
reduce overfitting for transformers, and that Input and Manifold MixUp methods
work better than sentence embedding MixUp for content-based tasks.

###  2.3 Model Calibration

As neural networks become more widely used in high risk fields such as medical
diagnosis and autonomous vehicles, the quality of their predictive uncertainty
becomes an important feature. Models should not only be accurate, but also
know when they are likely to be wrong.

To quantitatively evaluate a model’s predictive uncertainty, Guo et al. (2017)
defines calibration as the degree to which a model’s predictive scores are
indicative of the actual likelihood of correctness. Thulasidasan et al. (2019)
show that MixUp can improve calibration for CNN based models. Desai and
Durrett (2020) evaluate the calibration of pretrained transformers and finds
it to substantially deteriorate under data shift.

##  3 Methods

###  3.1 Architecture

We used the bert-base-uncased pretrained model based on Hugging Face’s
implementation (Wolf et al., 2019).

###  3.2 Training

Baseline models are trained using Empirical Risk Minimization (no MixUp) with
cross entropy loss. Hyperparameters are given in Appendix C.

MixUp models are trained using the MixUp methods discussed below, with
otherwise identical architecture, hyperparameters, and training dataset (prior
to augmentation).

Results are reported for the epoch with the best performance metric (accuracy
or MCC).

###  3.3 Datasets

We choose to test on a diverse set of tasks in terms of both difficulties and
task types. In particular:

  * •

IMDb: Sentiment classification for positive and negative movie reviews (Maas
et al., 2011)

  * •

AGNews: Multi-class classification for 4 topics of News articles. (Zhang et
al., 2015)

  * •

CoLA: Classification for grammatically correct/incorrect sentences (Wang et
al., 2018)

  * •

RTE: Natural language inference task, with entailment and no-entailment
classes. (Wang et al., 2018)

  * •

BoolQ: Question-answering task that matches a short passage with a yes / no
question about the passage. (Wang et al., 2019)

Task | Metric | Train | Dev | Test
---|---|---|---|---
IMDb | Accuracy | 25000 | 3200 | 21800
AGNews | Accuracy | 120000 | 2400 | 5200
CoLA | Matt Corr | 8551 | 1043 | 1064
RTE | Accuracy | 2491 | 278 | 3000
BoolQ | Accuracy | 9427 | 3270 | 3245
Table 1:  Datasets sizes (unit: sentence)

We simulated low resource settings for IMDb and AGNews by training only using
a random sample of 32 training examples.

###  3.4 Metrics

We evaluate our models based on 1) Accuracy, 2) Cross Entropy Loss, and 3)
Expected Calibration Error (ECE).

Classifiers capable of reliably forecasting their accuracy are considered to
be well-calibrated. For instance, a calibrated classifier should be correct
$80\%$ of the time on examples to which it assigns $80\%$ confidence (the
winning probability). Let the classifier’s confidence for its prediction
$\hat{Y}$ be $C$. Then, the Expected Calibration Error is the expectation of
the absolute difference between the accuracy at a given confidence level and
the actual confidence level:

$\begin{gathered}\mathrm{E_{C}}[|P(Y=\hat{Y}|C=c)-c|]\end{gathered}$ (2)

Details on how to empirically estimate this quantity can be found in Appendix
A.

###  3.5 MixUp for Transformers

We investigate three variants of MixUp that can be applied to the transformer
architecture, in particular BERT: 1) CLS MixUp, 2) input-token MixUp, and 3)
Manifold MixUp. As we shall see, the last is a generalization of the former
two.

####  3.5.1 CLS MixUp

Unlike in computer vision, where inputs are preprocessed to the same
dimensions, it is not straightforward to interpolate texts in the input space
directly. Instead, we can perform MixUp on the sentence embeddings, which in
the specific case of BERT are represented by the pooled CLS tokens from the
final layer. Formally, given a pair of training input sentences $x_{1},x_{2}$,
their one hot label vectors $y_{1},y_{2}$, and the sentence embeddings
$f(x_{1}),f(x_{2})$ generated using the BERT encoder, we interpolate their
embeddings and the labels:

$\begin{gathered}\widetilde{x}=\lambda\>f(x_{1})+(1-\lambda)\>f(x_{2})\\\
\widetilde{y}=\lambda y_{1}+(1-\lambda)y_{2}\\\
\widetilde{y_{p}}=\mathrm{Classifier}(\widetilde{x})\end{gathered}$ (3)

MixUp samples, in this case embeddings and labels, are generated by
interpolating each sample in the batch with another randomly selected sample.
We feed the interpolated embeddings into the classifier (while discarding the
unmixed embeddings, thus maintaining the same batch size) and minimize the
cross entropy loss between the predictions $\tilde{y_{p}}$ and the
interpolated labels $\tilde{y}$ in a given batch by summing their losses and
backpropagating it through the entire computational graph.

We repeat this process stochastically for every batch, where a new $\lambda$
is sampled every time (Eq 1).

####  3.5.2 Input MixUp

Though less intuitive, we can also interpolate individual input token
embeddings at each index between two sentences.

Given sentences $x_{1}=\left[x_{1}^{1},x_{1}^{2},\cdots,x_{1}^{m}\right]$,
$x_{2}=\left[x_{2}^{1},x_{2}^{2},\cdots,x_{2}^{n}\right]$ and their one hot
label vectors $y_{1}$ and $y_{2}$, where $m\geq n$ and $x^{i}$ represents the
$i^{\mathrm{th}}$ input token embedding, we pad $x_{1}$ and $x_{2}$ to the
same length using the SEP token from BERT vocabulary (see Appendix B for an
evaluation of different padding options). We can then generate a new sentence
$\widetilde{x}$ by taking a MixUp of individual input token embeddings from
each position:

$\begin{gathered}\widetilde{x}^{i}=\lambda x_{1}^{i}+(1-\lambda)x_{2}^{i}\ ;\
i\in(1,...,m)\\\
\widetilde{x}=[\widetilde{x}^{1},\widetilde{x}^{2},...,\widetilde{x}^{m}]\\\
\widetilde{y}=\lambda y_{1}+(1-\lambda)y_{2}\\\
\widetilde{y_{p}}=\mathrm{Classifier}(f(\widetilde{x}))\end{gathered}$ (4)

Where $f(x)$ is the sentence embedding returned by the BERT encoder given
input tokens $x$.

####  3.5.3 Manifold MixUp

Verma et al. (2019) demonstrated that applying MixUp to the hidden
represenations, in addition to the inputs, yields additional benefit in
reducing test errors in computer vision. Given that BERT consists of multiple
layers, we also investigate the effect of this technique on its training.

We proceed as follows: for each batch, select a random layer $k$ from a set of
eligible layers $S$ in the BERT with $n$ encoder layers. This set may include
the input layer $0$ as well as the pooled sentence embedding layer $n+1$. We
apply MixUp on the hidden representation embeddings, before feeding the
interpolated embeddings to the next layer to continue with training. The loss
is computed using the logits and the interpolated labels (Figure 1).

When $S$ only contains $0$, Manifold MixUp reduces to Input MixUp. When $S$
only contains $n+1$, it reduces to CLS MixUp.

Figure 1: Schematics of Manifold MixUp. We mix the layer $k$ embeddings and
the one-hot labels of two training samples using the same proportions
$\lambda$ and $(1-\lambda)$ (See eq 1).

##  4 Results

Our IMDB (Table 2) and AGNews (Table 3) results indicate that Manifold, Input,
and CLS MixUp methods improve model robustness and calibration across a range
of data regimes; the improvements are more significant when the amount of
training data is scarce, which matches our expectation since MixUp can be seen
as a form of data augmentation. Manifold MixUp offers the best improvement for
model calibration; in fact, the Manifold MixUp model trained on merely 32
samples is equally or more calibrated in comparison to the baseline model
trained using the full training set.

Both sentiment analysis on IMDB and news category classification on AGNews are
relatively simple tasks that do not necessarily require the model to have
syntactic knowledge; instead, they are likely to be solved by only inspecting
the sentiment or classification of the individual words. For example, a movie
review with more negative words than positive ones is likely to be negative
overall. Therefore, we expect Input and Manifold MixUp, which interpolate
embeddings at the token level, to work well here; indeed, they outperform
baseline and CLS MixUp in these simpler scenarios.

Size | Model | Acc | Loss | ECE
---|---|---|---|---
Full | Base | $91.8\pm 0.1$ | $61\pm 8$ | $7.7\pm 0.1$
| CLS | $92.0\pm 0.1$ | $29\pm 1$ | $5.8\pm 0.2$
| Input | $92.1\pm 0.1$ | $37\pm 2$ | $6.8\pm 0.2$
| Mani | $91.8\pm 0.1$ | $27\pm 1$ | $4.4\pm 0.6$
$32$ | Base | $68.2\pm 1.6$ | $180\pm 19$ | $29.0\pm 1.5$
| CLS | $69.6\pm 1.7$ | $86\pm 4$ | $21.0\pm 2.8$
| Input | $70.8\pm 1.2$ | $61\pm 7$ | $8.9\pm 2.3$
| Mani | $72.3\pm 1.0$ | $55\pm 2$ | $3.4\pm 0.9$
Table 2:  MixUp experiments for IMDB using various models (baseline, cls, input, manifold), train sizes. Standard errors over five runs.  Size | Model | Acc | Loss | ECE
---|---|---|---|---
Full | Base | $93.7\pm 0.1$ | $36\pm 7$ | $5.0\pm 0.6$
| CLS | $93.7\pm 0.1$ | $28\pm 1$ | $4.3\pm 0.3$
| Input | $93.8\pm 0.1$ | $25\pm 1$ | $4.1\pm 0.3$
| Man | $93.7\pm 0.1$ | $23\pm 5$ | $3.0\pm 0.2$
$32$ | Base | $81.9\pm 0.6$ | $72\pm 9$ | $9.9\pm 1.9$
| CLS | $83.4\pm 0.6$ | $55\pm 2$ | $7.2\pm 0.7$
| Input | $82.9\pm 0.3$ | $53\pm 2$ | $6.2\pm 2.4$
| Mani | $82.3\pm 0.4$ | $52\pm 1$ | $4.8\pm 1.8$
Table 3:  MixUp experiments for AGNews using various train sizes. Standard
errors over five runs.

Unlike IMDB and AGNews, the remaining tasks CoLA (acceptability), RTE (NLI),
and BoolQ (QA) rely on syntactic knowledge. Therefore, we would expect Input
and Manifold MixUp to work relatively poorly for these tasks, since they can
disrupt sentence syntax (eg. by interpolating nouns with verbs). This is
empirically confirmed when we applied the three MixUp methods to training on
the CoLA dataset (Table 4), where the Input and Manifold MixUp models under-
perform baseline.

Task | Model | MCC/Acc | Loss | ECE
---|---|---|---|---
CoLA | Base | $52.5\pm 0.6$ | $56\pm 4$ | $11.5\pm 0.3$
| CLS | $52.8\pm 0.6$ | $48\pm 1$ | $9.7\pm 0.1$
| Input | $50.8\pm 0.8$ | $48\pm 8$ | $10.1\pm 1.4$
| Mani | $51.0\pm 1.0$ | $45\pm 2$ | $9.0\pm 0.9$
RTE | Base | $63.8\pm 1.4$ | $125\pm 20$ | $27.6\pm 1.9$
| CLS | $63.8\pm 1.0$ | $80\pm 4$ | $20.8\pm 1.3$
BoolQ | Base | $73.4\pm 0.3$ | $170\pm 7$ | $24.8\pm 0.3$
| CLS | $73.1\pm 0.3$ | $89\pm 1$ | $21.8\pm 0.3$
Table 4:  MixUp experiments for CoLA, RTE, and BoolQ. CoLA is evaluated using
Test Matthew Correlation Coefficient (MCC), others using Test Accuracy.
Standard errors over five runs.

For the three more advanced tasks, while CLS MixUp exerts negligible effect on
the model’s predictive performance, it still significantly reduces test loss
and improves calibration without sacrificing performance. The evolution of
test losses across training steps for MixUp and baseline models are plotted in
Appendix D, demonstrating the ability of MixUp to reduce model overfitting. We
note that in contrast to our experiments, Sun et al. (2020) were able to
obtain improvements using CLS MixUp for CoLA and RTE using a different
training regimen, where MixUp is only applied during the last half of training
epochs.

##  5 Conclusion

In this work, we propose CLS, Input, and Manifold MixUp methods for the
transformer architecture and apply them to a range of NLU tasks including
sentiment analysis, multi-class classification, acceptability, NLI, and QA. We
find that MixUp can improve the predictive performance of simpler tasks. More
generally, MixUp can significantly reduce model overfitting and improve model
calibration.

## References

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##  Appendix A Calibration Metrics

As in the work of Guo et al. (2017), we calculate the expected calibration
error (ECE) of our model as follows: Predictions are grouped into $15$
interval bins of equal size ordered by prediction confidence. Let $B_{m}$ be
the set of samples that fall into bin $m$. The accuracy and confidence of
$B_{m}$ are defined as:

$\mathrm{acc}(B_{m})=\frac{1}{|B_{m}|}\sum_{i\in B_{m}}1(\hat{y_{i}}=y_{i})$
$\mathrm{conf}(B_{m})=\frac{1}{|B_{m}|}\sum_{i\in B_{m}}\hat{p_{i}}$

where $\hat{p_{i}}$ is the confidence (winning score) of sample $i$. The
Expected Calibration Error (ECE) is then defined as:

$\mathrm{ECE}=\sum^{M}_{m=1}\frac{|B_{m}|}{n}\Big{|}\mathrm{acc}(B_{m})-\mathrm{conf}(B_{m})\Big{|}$

##  Appendix B Padding Options for Input and Manifold MixUp

In the previous MixUp study in language (Guo et al., 2019), all sentences are
padded to the same length prior to training using Input MixUp. We believe that
while sentences should be padded to the same length, it ought to be done using
the minimum amount of padding tokens possible, as the introduction of
excessive padding tokens can disrupt performance. Instead of uniformly padding
all sentences to length of the longest sentence, we propose to pad pairs of
sentences about to be interpolated during training to the length of the longer
sentence.

We consider SEP, unused0, PAD tokens from the BERT vocabulary to be potential
padding candidates. We empirically evaluate the effect of using each of these
paddings for Input MixUp, both when using our proposal (pair) and the one in
which all sentences are padded to the length of the longest sequence (max).

We performed padding experiments on SST-2 dataset (Wang et al., 2018) due to
its short training sentence lengths, which leaves more room for padding. We
simulate low resource setting by using a random sample of 64 training samples
to increase task difficulty, thereby making it easier to detect performance
differences.

Our results (Table 5) confirm that pair-wise padding works better than no
padding at all as well as the uniform padding of sentences to the maximum
length, while the choice of SEP results in the best performance.

Type | Accuracy
---|---
No Pad | $78.38\pm 3.2$
unused (pair) | $80.22\pm 1.2$
SEP (pair) | $\boldsymbol{82.00\pm 1.4}$
PAD (pair) | $81.20\pm 1.0$
unused (max) | $68.14\pm 5.3$
SEP (max) | $69.21\pm 5.6$
PAD (max) | $73.61\pm 3.3$
Table 5:  The effect of different padding strategies on finetuning BERT on the
SST-2 dataset using Input MixUp.

##  Appendix C Hyperparameters

Hyperparameters for training downstream tasks are shown in Table 6.

| LR | BS | EP | DR | MSL | $\alpha$
---|---|---|---|---|---|---
IMDB | $1E-5$ | $16$ | $10$ | 0.1 | $128$ | $1.00$
AGNews | $2E-5$ | $16$ | $10$ | 0.1 | $128$ | $1.00$
CoLA | $2E-5$ | $16$ | $5$ | 0.1 | $128$ | $0.75$
RTE | $3E-5$ | $16$ | $5$ | 0.1 | $128$ | $1.00$
BoolQ | $2E-5$ | $16$ | $10$ | 0.1 | $256$ | $1.00$
Table 6:  Hyperparameters for training downstream tasks. LR: Learning Rate.
BS: Batch Size. EP: Training Epochs. DR: Dropout rate for BERT and classifier.
MSL: Maximum Sequence Length. $\alpha$: hyperparameter for beta distribution
used in MixUp.

##  Appendix D Evolution of test loss

Evolution of test losses across training steps for CoLA, RTE, and BoolQ (2)
are plotted. MixUp reduces overfitting by allowing the model to converge to a
much lower test loss.

Figure 2: Test losses obtained by the baseline models versus CLS MixUp models
across training steps