Canadian Journal of Statistics (1)

42 The Canadian Journal of StatisticsVol. 43, No. 1, 2015, Pages 42–59La revue canadienne de statistique

Resampling calibrated adjustedempirical likelihoodLei WANG1, Jiahua CHEN2* and Xiaolong PU1

1School of Finance and Statistics, East China Normal University, Shanghai 200241, P.R. China2Department of Statistics, University of British Columbia, Vancouver, BC, Canada V6T 1Z4

Key words and phrases: Bartlett correction; bootstrap; coverage probability; empirical likelihood; emptyset problem; estimating equation.

MSC 2010: Primary 62G15

Abstract: Empirical-likelihood-based inference for parameters defined by the general estimating equations ofQin & Lawless (1994) remains an active research topic. When the sample size is small and/or the dimensionof the accompanying estimating equations is high, the resulting confidence regions often have a lowerthan nominal coverage probability. In addition, the empirical likelihood can be hindered by an empty setproblem. The adjusted empirical likelihood (AEL) tackles both problems simultaneously. However, the AELconfidence region with high-order precision relies on accurate estimation of the required level of adjustment.This has proved difficult, particularly in over-identified cases. In this article, we show that the general AELis Bartlett-correctable and propose a two-stage procedure for constructing accurate confidence regions. Anaive AEL is first employed to address the empty set problem, and it is then Bartlett-corrected through aresampling procedure. The finite-sample performance of the proposed method is illustrated by simulationsand an example. The Canadian Journal of Statistics 43: 42–59; 2015 © 2014 Statistical Society of Canada

Resume: L’inference parametrique basee sur la vraisemblance empirique telle que definie par les equationsd’estimation de Qin et Lawless (1994) demeure un sujet de recherche actif. Lorsque la taille d’echantillon estfaible ou que la dimension des equations d’estimation est elevee, les regions de confiance obtenues presententsouvent un taux de couverture inferieur a leur valeur nominale. De plus, le probleme de l’ensemble videpeut causer des difficultes. La vraisemblance empirique ajustee resout ces deux problemes simultanement.Les regions de confiance decoulant de ces resultats d’ordre superieur necessitent toutefois une estimationprecise du niveau d’ajustement requis, ce qui s’avere difficile, surtout dans les cas de surspecification. Danscet article, les auteurs montrent que la correction de Bartlett peut s’appliquer a la vraisemblance empiriqueajustee et proposent une procedure en deux etapes pour la construction de regions de confiance precises.Ils utilisent d’abord une version naıve de la vraisemblance empirique ajustee pour regler le probleme del’ensemble vide, puis ils appliquent la correction de Bartlett a l’aide d’une methode de reechantillonnage.Ils illustrent la perfomance de leur methode sur des echantillons finis par des simulations et un exemple. Larevue canadienne de statistique 43: 42–59; 2015 © 2014 Societe statistique du Canada

1. INTRODUCTION

Let X1, X2, . . . , Xn be independent and identically distributed (i.i.d.) d-dimensional randomvectors from a distribution F . The problem of interest is inference on the p-dimensional parametervector θ = θ(F ) defined to be the unique solution to a q-dimensional estimating equation,

EF

{g(X; θ)

} = 0, (1)

* Author to whom correspondence may be addressed.E-mail: [email protected]

© 2014 Statistical Society of Canada / Societe statistique du Canada

2015 RESAMPLING CALIBRATED AEL 43

where the expectation is taken under distribution F . The parameters are said to be just-identifiedif q = p and over-identified if q > p. The choice of the vector estimating function g(X; θ) isflexible and accommodates a wide range of scenarios. Examples can be found in Hansen (1982),Liang & Zeger (1986), Kitamura & Stutzer (1997), Imbens (2002) and Bravo (2004).

Empirical likelihood (EL; Owen, 1988; Qin & Lawless, 1994) is a broadly applicable platformfor constructing confidence regions for the parameters defined by (1). Unlike the confidenceregions constructed via normal approximation, the EL confidence regions are transformationinvariant, are range respecting, have a data-driven shape, and are free of the burden of estimatingscaling parameters (Owen, 1990). However, when the sample size is small and/or q is large,the coverage probabilities of the EL confidence regions are often lower than the nominal value(under-coverage problem; DiCiccio, Hall, & Romano, 1991; Owen, 2001; Liu & Chen, 2010). Inaddition, the EL may not be properly defined because of the so-called empty set problem (Tsao,2004; Chen, Variyath, & Abraham, 2008; Grendar & Judge, 2009; Tsao & Wu, 2013).

A number of approaches have been proposed to improve the accuracy of the EL confidenceregions and to address the empty set problem. Those for the former problem include bootstrapcalibration (Owen, 1988; Hall & Horowitz, 1996) and Bartlett correction (DiCiccio, Hall, &Romano, 1991; Chen & Cui, 2007). Those for the latter problem include the AEL of Chen,Variyath, & Abraham (2008) and Emerson & Owen (2009). When the level of adjustment isproperly chosen (Liu & Chen, 2010; Li et al., 2011; Chen & Liu, 2012), the AEL also improvesthe accuracy of the EL confidence regions.

Applying Bartlett correction to EL or tuning AEL requires the accurate estimation of a Bartlettcorrection factor b. This task is challenging in many situations. For instance, when the parametersare over-identified, the Bartlett correction factor has a lengthy analytical expression involvinghigh-order moments. In theory, replacing b by a

√n-consistent estimator retains the high-order

precision of the Bartlett correction. However, the actual accuracy seems to contradict this claim.Estimating b by some straightforward resampling procedures (Chen & Cui, 2007) helps, but theyoften lack stability and suffer from the empty set problem. For the asset pricing example givenlater, estimating b with satisfactory precision and stability was found difficult by both Liu & Chen(2010, pp. 1356) and Matsushita & Otsu (2013, pp. 342).

In this article, we propose a two-stage procedure to construct accurate confidence regions.The first stage completely addresses the empty set problem by employing an AEL with its level ofadjustment tuned as if the estimating functions are observations from a normal distribution. Thus,the proposed method does not rely on an accurate estimate of the required level of adjustment b

and substantially enhances the applicability of the AEL method. In the second stage, a bootstrapprocedure is applied to this AEL to estimate its Bartlett correction factor and further calibrate theAEL confidence region. Simulation results indicate that the confidence regions constructed by theproposed method have coverage probabilities comparable to or substantially more accurate thanthe original EL and other competitors.

The rest of this article is organized as follows. In Section 2, we review the empirical likelihoodmethod and its extensions. The Bartlett correction of the general AEL confidence regions and theproposed method are given in Section 3. Simulation results are given in Section 4. A real-dataexample is given in Section 5, and some conclusions are in Section 6. The technical details aregiven in the Appendix.

2. METHODOLOGY REVIEW

In this section, we define the empirical likelihood, give concrete descriptions of the empty setproblem, and discuss the accuracy of the confidence regions. Some existing approaches to thesetwo problems are introduced and discussed.

DOI: 10.1002/cjs The Canadian Journal of Statistics / La revue canadienne de statistique

44 WANG, CHEN AND PU Vol. 43, No. 1

2.1. Empirical LikelihoodLet x1, x2, . . . , xn be a set of i.i.d. observations from some distribution F satisfying (1). The ELof F is defined to be

Ln(F ) =n∏

i=1

pi,

where pi = F ({xi}) = Pr(Xi = xi). The profile EL function of θ is defined to be

Ln(θ) = sup{ n∏

i=1

pi : pi ≥ 0,

n∑i=1

pi = 1,

n∑i=1

pig(xi; θ) = 0}

, (2)

and the EL ratio function of θ is defined to be Rn(θ) = −2 log{nnLn(θ)

}. If the convex hull of

g(xi; θ) does not contain the q-dimensional vector 0, the set for pi in (2) is empty. In this case, weeither declare that Ln(·) is not well defined at θ or set Ln(θ) = 0. If Ln(θ) = 0 for all θ, we havethe empty set problem discussed by Grendar & Judge (2009) and many others. Even if the set ofLn(θ) �= 0 is not empty, we may have difficulty locating this set. Asymptotically, the empty setproblem occurs with probability tending to 0 as n → ∞ if the model is correctly specified. Yetthis result does not solve all the problems arising in applications.

Let Wn(θ) = Rn(θ) − infθ Rn(θ). When q ≥ p and under some general conditions (Chen &Cui, 2007), it is well known that

Pr{Wn(θ0) ≤ x

} = Pr{χ2

p ≤ x} + O(n−1) as n → ∞,

where θ0 is the unique solution to (1) and χ2p is the χ2-distributed random variable with p degrees

of freedom. An approximate (1 − α) EL confidence region of θ is

IEL(θ) = {θ : Wn(θ) ≤ χ2

p(1 − α)}, (3)

with χ2p(1 − α) being the (1 − α)th quantile of χ2

p.The accuracy of the above EL confidence region can be poor, particularly when the sample

size is small and/or the dimension of the estimating equation is high.

2.2. Bootstrap Empirical LikelihoodTo improve the precision of the coverage probability, we may calibrate the distribution of Wn(θ)via resampling. Owen (1988, 2001), Hall & Horowitz (1996) and Chen & Cui (2007) proposedresampling from the empirical distribution function (EDF), while Hall & Presnell (1999) andBrown & Newey (2002) proposed resampling from the fitted distribution F through (2).

One bootstrap resampling procedure, proposed by Owen (2001), is as follows. Let {x∗i }ni=1 be

independent random vectors sampled from the EDF of the data {xi}ni=1 and compute W∗n (θn) =

R∗n(θn) − infθ R∗

n(θ), where θn is the maximum empirical likelihood estimator based on {xi}ni=1,and R∗

n(θ) is the EL ratio function based on the bootstrap sample {x∗i }ni=1. Repeat this procedure B

times and obtain W∗1n (θn), W∗2

n (θn), . . . , W∗Bn (θn). Let C∗(1 − α) be the (1 − α)th sample quantile

of {W∗νn (θn)}Bν=1. The (1 − α) bootstrap EL (BEL) confidence region is defined to be

IBEL(θ) = {θ : Wn(θ) < C∗(1 − α)}.

Clearly, the empty set problem remains a serious challenge with this BEL.

The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 10.1002/cjs


2.3. Bartlett-Corrected Empirical LikelihoodThe precision of the EL confidence regions can also be improved via Bartlett correction. DiCiccio,Hall, & Romano (1991) first proved that the EL is Bartlett-correctable for the population mean,Chen (1993) for linear regression coefficients, and Chen & Cui (2007) for parameters defined by(1). That is, there exists a constant b such that the Bartlett-corrected EL (BcEL) confidence region

IBcEL(θ) = {θ : Wn(θ)/(1 + b/n) ≤ χ2

p(1 − α)}

has coverage probability accuracy O(n−2).When q = p, the Bartlett correction factor b depends on the first four moments of g(X; θ0).

Let V (θ) = Var{g(X; θ)

}be the covariance matrix, and V (θ0) = Pdiag{η1, η2, . . . , ηp}PT be

an eigenvalue decomposition for some P such that PPT = I and η1, η2, . . . , ηp are eigen-values of V (θ0). Let Y = PTg(X; θ0). For any positive integers 1 ≤ r, s, . . . , t ≤ p, defineαrs...t = E{YrYs . . . Y t}, with Yt being the tth component of the vector Y . The Bartlett correctionfactor b is given by

b = 1p

{12

∑r,s

αrrss

αrrαss− 1

3

∑r,s,t

αrstαrst

αrrαssαtt

}. (4)

In applications, b is often replaced by some√

n-consistent estimator without invalidating thehigh-order asymptotic results. When q > p, however, b has a rather lengthy expression involvingmany terms and high-order moments (Chen & Cui, 2007). As a result, the corresponding momentestimate has poor precision and is unstable.

2.4. Adjusted Empirical LikelihoodBoth BEL and BcEL suffer from the empty set problem, especially when the parameters are over-identified. The convention of defining Rn(θ) = ∞ in this situation fails to provide informationon whether a θ-value is grossly unfit for the data or only slightly off an appropriate value.

To solve the empty set problem and to guarantee a meaningful likelihood value at any θ, Chen,Variyath, & Abraham (2008) proposed an adjusted empirical likelihood. Denote gi(θ) = g(xi; θ)for i = 1, 2, . . . , n and gn(θ) = n−1 ∑n

i=1 gi(θ). For any given θ, let gn+1(θ) = −agn(θ) for somepositive constant a. The profile AEL of θ is then defined to be

Ln(θ; a) = sup{ n+1∏

i=1

pi : pi ≥ 0,

n+1∑i=1

pi = 1,

n+1∑i=1

pigi(θ) = 0}

(5)

and the corresponding AEL ratio function of θ is defined as Rn(θ; a) = −2 log{

(n +1)n+1Ln(θ; a)

}. That is, the adjusted empirical likelihood is the usual empirical likelihood formed

on the augmented data set {g1(θ), . . . , gn(θ), gn+1(θ)} obtained by adding the pseudo-observationgn+1(θ). Note that pi = a/{n(1 + a)} for i = 1, . . . , n and pn+1 = 1/(1 + a) satisfy the restric-tions in (5), which implies that Ln(θ; a) is well defined at all θ. Let

Wn(θ; a) = Rn(θ; a) − infθ

Rn(θ; a). (6)

Chen, Variyath, & Abraham (2008) proved that inference based on Wn(θ; a) shares all first-orderasymptotic properties with Wn(θ). Furthermore, when a = b/2 with b being the Bartlett correctionfactor, Liu & Chen (2010) showed that

Pr{Wn(θ0; b/2) ≤ x

} = Pr{χ2

p ≤ x} + O(n−2) as n → ∞.



Subsequently, the AEL confidence region

IAEL(θ) = {θ : Wn(θ; b/2) ≤ χ2

p(1 − α)}

has the same high-order precision as that of the BcEL.One setback of the AEL is that Rn(θ; a) is a bounded function of θ. It is evident that the

distorted weights pi given earlier form a feasible solution to (5) for any value of θ. Subsequently,the resulting confidence regions may contain the entire parameter space, as pointed out by Emerson& Owen (2009). To fix this problem, Emerson & Owen (2009) proposed adding two pseudo-observations so that the weights are always balanced. For instance, one could set gn+1(θ) = −a

and gn+1(θ) = 2gn(θ) + a for some appropriately chosen constant a that is allowed to depend onθ. The augmented data set {g1(θ), g2(θ), . . . , gn+2(θ)} has the same sample mean gn as that beforethe augmentation. The new method overcomes both the empty set problem and the aforementionedsetback of the AEL.

Building on this idea, Chen & Huang (2013) proposed another modification to retain theBartlett correctability of the AEL. Let Sn(θ) = (n − 1)−1 ∑n

i=1{gi(θ) − gn(θ)

}{gi(θ) − gn(θ)

}T

and

K(θ) = {1 + 0.1 · gn(θ)T{Sn(θn)}−1gn(θ)}−1. (7)

The level of adjustment is then modified to

aK(θ) (8)

so that the level of adjustment decreases as gn(θ) moves away from 0. There is no theory behind thechoice of the constant 0.1 in K(θ), but it worked well in simulation studies. When gn(θ) deviatesfrom 0 by one unit measured by S−1

n , this choice recommends a 10% reduction from the initiallevel of adjustment a. Because K(θ0) = 1 + Op(n−1), the high-order asymptotic conclusions forAEL are not altered at the true parameter value θ0 when the model is correct. In particular, whena = b/2, this AEL retains the high-order precision. The two-stage procedure that we proposerelies on a resampling procedure to improve the accuracy of the AEL used at the first stage. Thus,both the size of a and the tuning constant 0.1 are of secondary importance. After this modification,the AEL ratio function becomes Rn(θ; aK(θ)).

3. BARTLETT-CORRECTING THE ADJUSTED EMPIRICAL LIKELIHOOD

In theory, the AEL confidence region with second-order precision completely solves the under-coverage and empty set problems when the level of adjustment is correctly specified. However,when q > p, accurately estimating b is a serious challenge. This prevents the AEL confidenceregion from achieving its full potential.

To overcome this difficulty, we propose a two-stage procedure. Given any level of adjustmentaccording to (8), we find that the corresponding AEL is Bartlett-correctable. Based on this result,we recommend an adjustment level of a = τp/2, where τp is the Bartlett correction factor whenthe equations are from the p-dimensional standard normal distribution. A bootstrap procedure isthen introduced to estimate the Bartlett correction factor of this AEL. The first stage allows theAEL to avoid the empty set problem. We choose τp so that all the equations are somewhat normal.Since this is not the case, the second stage combines the Bartlett correction and the resamplingmethod to improve the precision.



We establish the Bartlett correctability under the following regularity conditions:

(C1) Var{g(X; θ0)} is positive definite and the rank of E{∂g(X; θ0)/∂θ} is p;

(C2) lim sup‖t‖→∞∣∣E exp{itTg(X; θ0)}∣∣ < 1 and E‖g(X; θ0)‖18 < ∞;

(C3) There exists a neighbourhood of θ0, N(θ0), and an integrable function h(x) such thatsupθ∈N(θ0) ‖∂3g(x; θ)/∂3θ‖ ≤ h(x).

Theorem 1. Suppose X1, . . . , Xn is a set of i.i.d. random variables with distribution F and θ

is defined by (1). Assume that the conditions (C1)–(C3) are satisfied by F and g. Let the AELratio function Wn(θ; aK(θ)) be defined by (6), b be the Bartlett correction factor for Wn(θ), andba = b − 2a. As n → ∞, we have

Pr{Wn(θ0; aK(θ0)) ≤ x

} = Pr{χ2

p ≤ x} − baxfp(x)n−1 + O(n−2), (9)

where fp(·) is the density function of the χ2p distribution.

If Wn(θ0; aK(θ0)) is calibrated by the chi-square distribution with p degrees of freedom then,as indicated by Theorem 1, the resulting confidence region has a coverage error at O(n−1) ingeneral. It is also clear that we may apply the Bartlett correction to Wn(θ0; aK(θ0)) to achieve ahigher coverage precision.

Theorem 2. Under the conditions of Theorem 1, we have

Pr{Wn(θ0; aK(θ0))/(1 + ba/n) ≤ x

} = Pr{χ2

p ≤ x} + O(n−2).

Let ba be a√

n-consistent estimator of ba, then

Pr{Wn(θ0; aK(θ0))/(1 + ba/n) ≤ x

} = Pr{χ2

p ≤ x} + O(n−2).

According to Theorem 2, if we choose a = b/2, then no additional correction is needed onWn(θ; (b/2)K(θ)) to achieve the high-order precision. However, the algebraic expression of b

can be so complex that it may not be practical to estimate its value accurately by the method ofmoments. Estimating b based on direct bootstrap of Wn(θ) is difficult because of the empty setproblem. Based on the above two theorems, we suggest first employing a tentative adjustmenta in Wn(θ; aK(θ)) to avoid the empty set problem and then making a Bartlett correction toWn(θ; aK(θ)) with a bootstrapping estimate of ba.

We suggest setting a = τp/2 with τp being the Bartlett correction factor for the p-dimensionalstandard normal distribution. According to (4), τp = 1 + p/2. Any other positive constant shouldwork similarly. The (1 − α) Bartlett-corrected AEL (BcAEL) confidence region is given by

IBcAEL(θ) = {θ : Wn(θ; (τp/2)K(θ))/(1 + bτp/n) ≤ χ2

p(1 − α)},

where bτp = b − τp is the Bartlett correction factor.To implement the BcAEL, we recommend the following bootstrap procedure to estimate bτp :

Step 1. Let {x∗i }ni=1 be i.i.d. vectors sampled from the EDF of the data {xi}ni=1 and compute

W∗n (θn; (τp/2)K(θn)) = R∗

n(θn; (τp/2)K(θn)) − infθ

R∗n(θ; (τp/2)K(θ)),

where R∗n(θ; (τp/2)K(θ)) is the AEL ratio function based on the bootstrap sample.



Step 2. Repeat Step 1 B times and obtain W∗1n (θn; (τp/2)K(θn)), . . ., W∗B

n (θn; (τp/2)K(θn)).Step 3. Estimate bτp by

bBτp

= n[ B∑

ν=1

W∗νn (θn; (τp/2)K(θn))/(Bp) − 1

]. (10)

After an estimate of bτp is obtained, the (1 − α) BcAEL confidence region is constructed as

{θ : Wn(θ; (τp/2)K(θ))/(1 + bB

τp/n) ≤ χ2

p(1 − α)}. (11)

4. SIMULATION STUDIES

In this section, we conduct simulation studies to examine the finite-sample performance of theBcAEL confidence regions and a number of potential competitors. In particular, we obtain thesimulated coverage probability (CP), average size (AS) and median size (MS) of the confidenceregions. We also compute the coefficient of variation (CV), defined to be the ratio of the standarddeviation to the average size of the confidence regions. A large CV value is an indication ofunstable performance. All the simulated results are based on 5,000 replications, and the numberof bootstrap replications is set to B = 200.

4.1. Confidence Regions for Population MeanA classical problem is the construction of confidence regions for the population mean μ based onn i.i.d. observations. The most widely used method is based on Hotelling’s T 2 (Hotelling, 1931):

T 2n (μ) = n(xn − μ)TS−1

n (xn − μ),

where xn and Sn are the sample mean and the sample variance–covariance matrix, respectively.If the population distribution is multivariate normal of dimension d and μ0 is the true parametervalue, then (n − d)T 2

n (μ0)/[d(n − 1)] has an F -distribution with d and n − d degrees of freedom.Hence, a (1 − α) Hotelling’s T 2 confidence region is given by

IT 2 (μ) = {μ : T 2

n (μ) ≤ [d(n − 1)/(n − d)]Fd,n−d(1 − α)},

where Fd,n−d(1 − α) is the (1 − α)th quantile of the F -distribution with d and n − d degrees offreedom. When d = 1, Hotelling’s T 2 statistic becomes the square of the well-known Student’st-statistic. We use Hotelling’s T 2 as a yardstick for competing methods.

We study the performance of the following six methods: (a) Hotelling’s T 2, denoted T 2; (b)the usual empirical likelihood, denoted EL; (c) the bootstrap empirical likelihood, denoted BEL;(d) the Bartlett-corrected empirical likelihood with a moment estimate of b, denoted BcEL; (e) theadjusted empirical likelihood with a = b/2 and a moment estimate of b (Chen & Huang, 2013),denoted AEL; (f) the Bartlett-corrected adjusted empirical likelihood BcAEL proposed in thisarticle.

In the univariate case (d = 1), we generated data from each of the following three distributions:(1) the standard normal distribution, denoted N(0, 1); (2) an exponential distribution with onedegree of freedom, denoted Exp(1); (3) a χ2 distribution with one degree of freedom, denotedχ2

1. For the sample sizes n = 20, 50 we constructed confidence intervals at the nominal levels of90%, 95%, and 99%.

In the bivariate case (d = 2), we used the simulation settings of Liu & Chen (2010). Weobtained confidence regions at the nominal levels 1 − α = 95% and 99%. The area of the



confidence region is approximated by partitioning the region into triangles, as in Chen & Huang(2013). For the sample sizes n = 20 and 50 we generated data from each of the following fourbivariate distributions with D ∼ Uniform(1, 2): (1) the bivariate standard normal distributionN(0, I2); (2) the distribution (X1, X2) where X1 ∼ N(0, D−2) and X2 ∼ (D, 1); (3) thedistribution (X1, X2) where X1 ∼ (D−1, 1) and X2 ∼ (D, 1); (4) the distribution (X1, X2)where X1 = D and X2 ∼ (D, 1).

Simulation results for d = 1 and 2 are presented in Tables 1–3.It is clear that Hotelling’s T 2 confidence regions are superior when the data are from a

normal population. The EL confidence regions have mild under-coverage in these cases. Theother confidence regions have comparable performance in terms of the precision of the coverageprobability and the mean and median sizes. The BEL and BcAEL, however, have larger CVvalues. Clearly, resampling has led to additional variation in these two methods. Thus, when thedata appear normal, there is no reason to use any method other than Hotelling’s T 2.

When the population distribution deviates from normal, the performance of T 2 deteriorates.The EL confidence regions have even lower coverage probabilities. In these cases, a correction ofsome form on the EL becomes useful when n = 20 and d = 1 or when d = 2 with both samplesizes. The proposed BcAEL is generally a strong competitor.

4.2. Confidence Regions with Just-Identified Estimating EquationsExample 1. The parameters are defined by the vector estimating function g(y, x; β) =x(y − xTβ) with the data generated from the linear regression model

yi = xTi β + εi, 1 ≤ i ≤ n,

with the error distribution for εi chosen as N(0, 1) in the first case and as an exponential dis-tribution with mean 1 in the second case. We used the fixed design points xi given by Chen(1993).

Example 2. The parameters are defined by the vector estimating function g(x; μ, σ2) =(x − μ, x2 − μ2 − σ2)T with the data x1, . . . , xn generated i.i.d. from N(μ, σ2). We refer tothis as the mean-variance model.

Under the linear regression model, Chen (1993) showed that the EL confidence region for β

is Bartlett-correctable. In the simulation, we followed Chen (1993) with the sample sizes n = 30and 50, p = 2, and β0 = (1, 1)T. For the second example, we chose (μ0, σ

20 )T = (0, 1)T with the

sample sizes n = 20 and 50. We constructed confidence regions at the nominal coverage levelsof 90% and 95% in both examples. The simulation results are in Table 4.

Under the linear regression model, the EL has low coverage probabilities in all cases. TheBcEL and AEL have low coverage probabilities when the error distribution is exponential. TheBEL and the proposed BcAEL generally have closer to nominal coverage probabilities. Of thesetwo methods, the BcAEL is more stable since it has lower CV values.

In terms of coverage rates, the two strong competitors are BEL and the proposed BcAEL.Of these two methods, BcAEL has a slightly more accurate coverage probability, and it is muchmore stable as indicated by its lower CV values.

In conclusion, the proposed BcAEL is preferred in the just-identified cases.

4.3. Confidence Regions with Over-Identified Estimating EquationsThis subsection examines the performance with over-identified parameters. To implement theBcEL and AEL, the estimate of b is obtained by the robust modification bootstrap method (Liu& Chen, 2010).



Table 1: Performances of the confidence intervals for univariate population mean.

90% 95% 99%n Methods

CP AS MS CV CP AS MS CV CP AS MS CV

X ∼ N(0, 1)

20 T 2 90.64, 0.765, 0.763, 0.163 95.30, 0.926, 0.923, 0.163 99.28, 1.265, 1.262, 0.163

EL 87.56, 0.717, 0.716, 0.162 92.98, 0.859, 0.858, 0.163 98.00, 1.141, 1.138, 0.166

BEL 89.04, 0.758, 0.750, 0.183 94.22, 0.921, 0.910, 0.192 98.64, 1.266, 1.233, 0.230

BcEL 88.72, 0.741, 0.739, 0.163 93.76, 0.891, 0.888, 0.165 98.24, 1.184, 1.179, 0.168

AEL 88.78, 0.742, 0.741, 0.163 93.84, 0.889, 0.882, 0.164 98.28, 1.183, 1.175, 0.167

BcAEL 90.06, 0.771, 0.765, 0.179 94.66, 0.925, 0.917, 0.183 98.62, 1.234, 1.219, 0.188

50 T 2 89.84, 0.472, 0.472, 0.102 95.20, 0.565, 0.565, 0.101 98.94, 0.754, 0.754, 0.101

EL 88.76, 0.464, 0.462, 0.101 94.08, 0.555, 0.554, 0.101 98.78, 0.737, 0.736, 0.102

BEL 89.36, 0.469, 0.467, 0.120 94.56, 0.561, 0.559, 0.124 98.50, 0.736, 0.729, 0.142

BcEL 89.34, 0.470, 0.469, 0.102 94.54, 0.563, 0.562, 0.101 98.88, 0.748, 0.747, 0.103

AEL 89.40, 0.470, 0.469, 0.102 94.58, 0.563, 0.562, 0.101 98.90, 0.748, 0.747, 0.103

BcAEL 90.34, 0.472, 0.471, 0.114 95.14, 0.565, 0.563, 0.115 98.96, 0.752, 0.748, 0.117

X ∼ Exp(1 )

20 T 2 87.26, 0.741, 0.710, 0.300 92.24, 0.897, 0.859, 0.299 96.84, 1.226, 1.175, 0.299

EL 85.54, 0.698, 0.665, 0.312 91.24, 0.836, 0.796, 0.315 96.30, 1.110, 1.052, 0.320

BEL 88.50, 0.812, 0.730, 0.489 93.54, 1.122, 0.912, 0.517 97.42, 1.516, 1.267, 0.600

BcEL 86.90, 0.728, 0.695, 0.320 92.24, 0.872, 0.831, 0.322 96.72, 1.158, 1.100, 0.330

AEL 86.96, 0.730, 0.697, 0.321 92.32, 0.876, 0.833, 0.324 96.80, 1.165, 1.105, 0.335

BcAEL 89.44, 0.836, 0.754, 0.448 93.64, 1.006, 0.902, 0.474 98.16, 1.348, 1.196, 0.548

50 T 2 88.44, 0.464, 0.455, 0.192 93.14, 0.556, 0.546, 0.192 97.70, 0.742, 0.728, 0.193

EL 88.16, 0.462, 0.453, 0.199 93.60, 0.555, 0.543, 0.202 98.28, 0.741, 0.724, 0.208

BEL 88.72, 0.482, 0.466, 0.243 93.70, 0.583, 0.561, 0.258 98.10, 0.788, 0.742, 0.301

BcEL 88.84, 0.469, 0.459, 0.205 94.10, 0.564, 0.550, 0.208 98.40, 0.753, 0.734, 0.215

AEL 88.80, 0.471, 0.460, 0.206 94.18, 0.566, 0.552, 0.209 98.42, 0.756, 0.737, 0.215

BcAEL 89.32, 0.488, 0.472, 0.238 94.28, 0.587, 0.566, 0.244 98.56, 0.786, 0.754, 0.253

X ∼ χ21

20 T 2 84.92, 1.022, 0.957, 0.386 89.62, 1.237, 1.158, 0.386 94.14, 1.691, 1.583, 0.386

EL 84.36, 0.968, 0.909, 0.387 90.48, 1.161, 1.086, 0.389 95.76, 1.543, 1.440, 0.395

BEL 88.00, 1.254, 1.041, 0.733 93.32, 1.612, 1.293, 0.739 97.50, 2.523, 1.951, 0.743

BcEL 85.62, 1.013, 0.942, 0.396 91.42, 1.215, 1.129, 0.399 96.18, 1.616, 1.487, 0.406

AEL 85.82, 1.018, 0.939, 0.398 91.54, 1.217, 1.125, 0.401 96.22, 1.631, 1.498, 0.414

BcAEL 89.46, 1.278, 1.103, 0.575 93.60, 1.542, 1.323, 0.595 97.36, 2.108, 1.761, 0.700

Continued.



Table 1: Continued.

50 T 2 87.86, 0.652, 0.633, 0.248 92.64, 0.781, 0.759, 0.248 96.80, 1.042, 1.012, 0.249

EL 87.84, 0.650, 0.631, 0.257 93.28, 0.782, 0.758, 0.260 98.34, 1.049, 1.014, 0.267

BEL 88.62, 0.710, 0.665, 0.345 93.40, 0.861, 0.810, 0.369 98.10, 1.205, 1.105, 0.425

BcEL 88.72, 0.668, 0.647, 0.265 93.84, 0.804, 0.777, 0.269 98.44, 1.080, 1.041, 0.277

AEL 88.76, 0.669, 0.647, 0.266 93.88, 0.805, 0.778, 0.270 98.50, 1.082, 1.042, 0.279

BcAEL 89.88, 0.716, 0.676, 0.327 94.34, 0.863, 0.813, 0.335 98.56, 1.163, 1.089, 0.350

CP, coverage probability; AS, average size; MS, median size; CV, coefficient of variation.

Example 3. The parameter is defined by the vector estimating function g(x; θ) = (x − θ, x2 −2θ2 − 1)T. This is an over-identified case because two equations are used to define one parameter.The data were generated from N(θ, θ2 + 1), as in Qin & Lawless (1994).

Example 4. We generated data from the asset pricing model of Matsushita & Otsu (2013),which is a multivariate version of Hall & Horowitz (1996). The parameters are defined by thevector estimating function

g(x; θ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

r(x, θ)x2r(x, θ)

(x3 − 1)r(x, θ)...

(xq − 1)r(x, θ)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

where r(x, θ) = exp{−4.5σ2 − θ(x1 + x2) + 3x2} − 1. We generated x = (x1, x2, . . . , xq) as avector with independent entries such that x1, x2 are from N(0, σ2) and x3, . . . , xq are from χ2

1.

In the Qin–Lawless example, we chose θ0 = 1 with the sample sizes n = 30 and 60 in thesimulation. In the asset pricing model, we fixed σ = 0.4. The current choice of r(x, θ) and thedata distributions make θ0 = 3 the unique solution to the defining vector estimating functionfor any σ > 0. We conducted simulations with the sample sizes n = 100, 200 and q = 2, 5. Weobtained confidence regions with nominal coverage levels 95% and 99%. The simulation resultsare presented in Table 5.

It can be seen that the coverage probabilities of the EL are quite low in both examples. TheBEL confidence intervals improve the EL in terms of the accuracy of the coverage probability.However, the empty set problem is particularly serious for BEL. The AEL confidence regionshave accurate coverage probabilities. However, the sizes and variations of the AEL confidenceregions are much larger. Overall, the proposed BcAEL is the best choice.

5. REAL DATA

We illustrate the proposed method with a real-data example from Larsen & Marx (2006). Fourteenmen without a history of coronary incidents were examined in a heart-disease study. Their weights(in pounds) and blood cholesterol levels (in mg/dl) were measured. Figure 1 shows these records,



Table 2: Performances of the confidence regions for bivariate population mean.

95% 99%n Methods

CP AS MS CV CP AS MS CV

Bivariate distribution (1)

20 T 2 94.30 1.115 1.094 0.231 98.60 1.886 1.851 0.231

EL 90.04 0.880 0.860 0.235 96.64 1.362 1.332 0.237

BEL 93.24 1.193 1.150 0.306 98.04 2.217 2.036 0.419

BcEL 92.10 0.955 0.933 0.237 97.20 1.477 1.443 0.240

AEL 92.46 0.968 0.946 0.238 97.40 1.513 1.476 0.242

BcAEL 94.20 1.112 1.079 0.268 98.46 1.757 1.702 0.273

50 T 2 94.98 0.400 0.396 0.144 99.10 0.636 0.631 0.144

EL 94.02 0.376 0.373 0.145 98.64 0.591 0.587 0.146

BEL 94.44 0.397 0.393 0.188 98.30 0.621 0.605 0.229

BcEL 94.52 0.391 0.388 0.146 98.84 0.614 0.610 0.147

AEL 94.52 0.391 0.389 0.146 98.86 0.615 0.611 0.147

BcAEL 94.78 0.400 0.395 0.166 98.96 0.629 0.622 0.168


20 T 2 93.20 0.972 0.931 0.324 97.98 1.644 1.575 0.324

EL 88.70 0.764 0.728 0.328 95.78 1.179 1.126 0.330

BEL 93.70 1.205 1.029 0.692 98.00 2.504 1.930 0.834

BcEL 90.66 0.832 0.792 0.334 96.44 1.284 1.219 0.336

AEL 91.18 0.847 0.805 0.336 96.68 1.324 1.252 0.342

BcAEL 94.36 1.052 0.953 0.440 98.26 1.685 1.507 0.516

50 T 2 94.18 0.355 0.350 0.204 98.20 0.564 0.556 0.204

EL 93.22 0.337 0.331 0.211 98.20 0.531 0.521 0.215

BEL 94.40 0.371 0.355 0.276 98.60 0.594 0.558 0.333

BcEL 93.92 0.352 0.345 0.216 98.44 0.555 0.544 0.220

AEL 93.98 0.353 0.346 0.216 98.48 0.557 0.545 0.220

BcAEL 94.76 0.370 0.356 0.250 98.66 0.584 0.562 0.256


together with 95% confidence regions for the mean based on Hotelling’s T 2, EL and BcAEL.It can be seen that the confidence region based on the proposed method, BcAEL, preserves thedata-driven shape. It has also expanded the EL to ensure the coverage probability.

6. CONCLUSION

In this article, we have shown that the AEL with any given level of adjustment can be Bartlett-corrected. Thus, the AEL can be used to address both the empty set and under-coverage problems.



Table 3: Performances of the confidence intervals for bivariate population mean.

95% 99%n Methods



20 T 2 89.14 1.108 1.024 0.434 95.18 1.875 1.732 0.434

EL 86.90 0.861 0.794 0.440 93.70 1.323 1.217 0.443

BEL 93.10 1.765 1.283 1.045 96.90 4.309 2.754 1.229

BcEL 88.40 0.942 0.863 0.449 94.76 1.445 1.323 0.452

AEL 88.82 0.964 0.880 0.455 95.02 1.507 1.366 0.467

BcAEL 92.66 1.387 1.151 0.672 96.86 2.371 1.832 0.926

50 T 2 91.96 0.412 0.395 0.284 96.96 0.655 0.629 0.284

EL 92.10 0.394 0.376 0.296 97.72 0.620 0.593 0.301

BEL 93.30 0.466 0.426 0.411 98.24 0.790 0.701 0.500

BcEL 92.66 0.415 0.396 0.304 97.96 0.654 0.622 0.310

AEL 92.68 0.416 0.397 0.306 98.04 0.657 0.625 0.313

BcAEL 94.00 0.455 0.424 0.365 98.76 0.719 0.667 0.375


20 T 2 92.72 0.387 0.374 0.291 97.70 0.655 0.633 0.291

EL 90.10 0.295 0.286 0.288 95.96 0.451 0.437 0.287

BEL 93.50 0.433 0.379 0.542 97.90 0.897 0.706 0.828

BcEL 91.46 0.318 0.308 0.291 96.50 0.486 0.471 0.290

AEL 91.64 0.323 0.312 0.293 96.62 0.498 0.481 0.294

BcAEL 93.76 0.392 0.367 0.376 98.36 0.619 0.573 0.421

50 T 2 94.14 0.141 0.138 0.182 98.60 0.224 0.220 0.182

EL 94.06 0.130 0.128 0.187 98.56 0.203 0.198 0.188

BEL 94.20 0.145 0.136 0.247 98.60 0.222 0.209 0.311

BcEL 94.52 0.135 0.132 0.190 98.76 0.210 0.206 0.192

AEL 94.56 0.135 0.132 0.191 98.78 0.211 0.206 0.193

BcAEL 94.88 0.141 0.136 0.227 99.08 0.219 0.212 0.230


However, particularly in over-identified cases, there is no simple and accurate method for esti-mating the Bartlett correction factor. We hence propose a two-stage procedure for constructingaccurate confidence regions. The simulation results show that the new method is competitive atconstructing confidence regions for the population mean and works better in just-identified andover-identified cases, compared with the EL and its variants. The real-data example also providesevidence that the proposed method has some advantages over the EL.



Table 4: Performances of the confidence regions for just-identified estimating equations.

90% 95%n Methods


Linear regression model

Normal error N(0, 1)

30 EL 84.68 0.187 0.180 0.287 90.86 0.245 0.236 0.286

BEL 89.00 0.244 0.231 0.355 94.04 0.342 0.321 0.369

BcEL 86.40 0.202 0.194 0.292 92.14 0.265 0.254 0.291

AEL 86.74 0.205 0.196 0.293 92.38 0.269 0.258 0.293

BcAEL 90.16 0.242 0.230 0.334 94.68 0.319 0.302 0.334

50 EL 86.60 0.071 0.069 0.225 92.70 0.093 0.091 0.225

BEL 89.34 0.079 0.077 0.265 94.36 0.107 0.103 0.278

BcEL 88.24 0.075 0.073 0.228 93.52 0.098 0.096 0.228

AEL 88.32 0.075 0.073 0.228 93.60 0.098 0.096 0.228

BcAEL 89.86 0.080 0.078 0.254 94.70 0.105 0.102 0.254

Exponential error Exp(1) − 1

30 EL 76.60 0.346 0.282 0.732 83.60 0.455 0.371 0.726

BEL 87.36 0.897 0.473 2.212 92.98 1.472 0.737 2.540

BcEL 78.90 0.382 0.309 0.750 85.38 0.502 0.408 0.743

AEL 79.28 0.386 0.315 0.743 85.60 0.512 0.416 0.743

BcAEL 87.96 0.724 0.469 1.160 92.18 1.010 0.623 1.238

50 EL 82.86 0.142 0.119 0.644 89.44 0.189 0.158 0.643

BEL 88.14 0.235 0.157 2.245 93.40 0.348 0.229 2.037

BcEL 84.94 0.155 0.128 0.665 90.44 0.206 0.170 0.666

AEL 85.00 0.154 0.129 0.658 90.44 0.205 0.171 0.664

BcAEL 89.16 0.221 0.159 1.036 93.48 0.295 0.212 1.140

Mean-variance model

20 EL 78.04 0.407 0.386 0.382 84.20 0.526 0.497 0.382

BEL 87.10 0.943 0.640 1.508 92.50 1.784 1.031 1.588

BcEL 80.00 0.442 0.418 0.395 85.54 0.571 0.539 0.395

AEL 80.34 0.452 0.426 0.402 85.74 0.588 0.553 0.406

BcAEL 88.16 0.859 0.659 1.118 91.76 1.301 0.873 1.307

50 EL 85.72 0.191 0.187 0.253 91.36 0.250 0.243 0.255

BEL 88.50 0.232 0.216 0.375 94.04 0.319 0.291 0.418

BcEL 86.74 0.201 0.196 0.263 92.04 0.263 0.256 0.264

AEL 86.84 0.202 0.197 0.265 92.20 0.265 0.257 0.267

BcAEL 89.86 0.235 0.219 0.362 93.96 0.308 0.287 0.364




Table 5: Performances of the confidence regions for over-identified estimating equations.

90% 95%n Methods


Qin–Lawless model

30 EL 84.76 0.563 0.548 0.221 90.42 0.673 0.655 0.218

BEL 88.88 0.709 0.637 0.446 93.00 0.890 0.793 0.454

BcEL 86.76 0.637 0.587 0.364 91.35 0.762 0.701 0.368

AEL 89.40 1.311 0.646 1.034 93.72 1.606 0.789 0.938

BcAEL 90.42 0.721 0.656 0.401 93.14 0.863 0.784 0.401

60 EL 86.78 0.411 0.407 0.141 92.12 0.491 0.486 0.141

BEL 88.68 0.443 0.427 0.233 93.02 0.537 0.525 0.243

BcEL 88.50 0.435 0.423 0.222 92.90 0.520 0.513 0.222

AEL 90.22 0.711 0.441 1.159 94.36 0.904 0.537 1.092

BcAEL 89.70 0.451 0.434 0.225 93.94 0.519 0.522 0.227

Asset pricing model

q = 2

100 EL 80.40 0.774 0.759 0.181 87.12 0.923 0.906 0.178

BEL 85.94 0.956 0.874 0.363 92.24 1.164 1.072 0.357

BcEL 84.24 0.883 0.831 0.285 90.28 1.050 0.990 0.280

AEL 87.86 2.087 0.949 1.054 92.50 2.587 1.206 0.934

BcAEL 88.58 0.972 0.892 0.347 92.92 1.154 1.064 0.341

200 EL 84.96 0.574 0.564 0.150 91.60 0.683 0.671 0.147

BEL 88.36 0.657 0.615 0.284 93.56 0.794 0.744 0.285

BcEL 87.22 0.628 0.602 0.240 92.68 0.747 0.717 0.235

AEL 89.26 1.376 0.639 1.283 93.96 1.751 0.775 1.153

BcAEL 89.40 0.664 0.622 0.277 93.70 0.789 0.741 0.270

q = 5

100 EL 72.90 0.729 0.713 0.183 80.76 0.870 0.850 0.184

BEL 87.42 1.140 1.027 0.383 92.70 1.413 1.274 0.386

BcEL 86.12 1.036 0.956 0.336 90.98 1.238 1.142 0.333

AEL 97.00 5.661 5.492 0.444 98.30 6.520 5.947 0.393

BcAEL 88.74 1.178 1.054 0.388 93.46 1.411 1.265 0.386

200 EL 79.02 0.538 0.531 0.134 84.94 0.641 0.633 0.133

BEL 87.38 0.711 0.662 0.290 93.02 0.864 0.806 0.294

BcEL 86.22 0.682 0.641 0.276 92.20 0.812 0.765 0.274

AEL 94.12 4.981 3.938 0.669 96.56 5.540 4.451 0.590

BcAEL 88.64 0.723 0.677 0.283 93.74 0.861 0.809 0.283




140 160 180 200 220 240 260

150

200

250

300

350

400

Weight

Blo

od c

hole

ster

ol

dataHotelling’s T2

ELBcAEL

Figure 1: The 95% confidence regions in the real data example.

APPENDIXWe provide sketch proofs of Theorems 1 and 2 for Wn(θ0; a) instead of Wn(θ0; aK(θ0)). BecauseK(θ0) = 1 + Op(n−1), the results remain valid for Wn(θ0; aK(θ0)).Proof of Theorem 1. Our proof is built on existing results on Bartlett correction with just-identifiedand over-identified estimating equations. In both cases and under conditions (C1)–(C3), the ELratio function Rn(θ0) has a signed root decomposition

Rn(θ0) = n(R1 + R2 + R3)T(R1 + R2 + R3) + Op(n−3/2),

with R1, R2, R3 having certain properties. For instance, Rj = Op(n−j/2) for j = 1, 2, 3. Theexact expressions are not important here but can be found in Chen & Cui (2007). According toTheorem 1 of Liu & Chen (2010), the AEL ratio function defined by (6) has

Rn(θ0; a) = n(R1 + R2 + Ra)T(R1 + R2 + R3a) + Op(n−3/2),

with R3a = R3 − n−1aR1.Denote Qn = √

n(R1 + R2 + R3a) and let κr,s,...,t(Qn) denote the joint cumulant of the rth,sth, . . . , tth components of Qn. Without loss of generality, suppose αrs = I(r = s) at θ = θ0.According to Liu & Chen (2010), for the just-identified case,

κr(Qn) = −n−1/2μr + O(n−3/2), κr,s(Qn) = I(r = s) + n−1γrs + O(n−2),

κr,s,t(Qn) = O(n−3/2), κr,s,t,u(Qn) = O(n−2),

where I(·) is the indicator function, μr = 16αrss, and

γrs = 12αrstt − 1

3αrtuαstu − 1

36αrstαtuu − 2aI(r = s).



Here, we have used the summation convention, according to which if an index occurs more thanonce in an expression, summation over the index is understood.

Let fQn (z) and φ(z) be the density functions of Qn and the p-variate standard normal distri-bution, respectively. By a formal Edgeworth expansion,

fQn (z) = {1 +

4∑i=1

n−i/2πi(z) + o(n−2)}φ(z),

with

πr1(z) = urzr, πr

2(z) = 12

(γrs + urus){

zrzs − I(r = s)},

and for some polynomials π3(z) and π4(z) that are of order no more than four, the former is oddand the latter is even.

The above expansion implies that

Pr{QT

nQn ≤ x} =

∫zTz≤x

{1 +

4∑i=1

n−i/2πi(z)}φ(z) dz + o(n−2).

Because π1(z) and π3(z) are odd functions, their integrations over the symmetry region are zero.In addition, for each i = 1, 2, . . . , p,

∫zTz<x

(z2i − 1)φ(z) dz = −2x

pfp(x),

where fp(·) is the density function of the χ2p distribution. Thus,

Pr{Rn(θ0; a) ≤ x

} = Pr{χ2

p ≤ x} + 1

2n−1

∫zTz<x

p∑r=1

(γrr + μrμr)(z2i − 1)φ(z) dz + O(n−2)

= Pr{χ2

p ≤ x} + 1

2n−1

p∑r=1

(γrr + μrμr)∫

zTz<x

(z21 − 1)φ(z) dz + O(n−2)

= Pr{χ2

p ≤ x} − n−1 x

pfp(x)

p∑r=1

(γrr + μrμr) + O(n−2)

= Pr{χ2

p ≤ x} − baxfp(x)n−1 + O(n−2), (A.1)

where ba = 1p

p∑r=1

(γrr + μrμr) = 1p

(12

p∑r,s

αrrss − 13

p∑r,s,t

αrstαrst) − 2a = b − 2a. This b is the

Bartlett correction factor given in DiCiccio, Hall, & Romano (1991). We note in particular thatthe remainder term in (A.1) is O(n−2) rather than O(n−3/2). See Barndorff-Nielsen & Hall (1988)for an account of this phenomenon.

For the over-identified case, the above proof remains applicable except for the expression forthe Bartlett correction factor b, which is now given in Chen & Cui (2007). This completes theproof of Theorem 1.



Proof of Theorem 2. According to (A.1), we have

Pr{Wn(θ0; a) ≤ x(1 + n−1ba)

}= Pr

{χ2

p ≤ x(1 + n−1ba)} − ba

[x(1 + n−1ba)

]fp

(x(1 + n−1ba)

)n−1 + O(n−2)

= Pr{χ2

p ≤ x(1 + n−1ba)} − baxfp

(x(1 + n−1ba)

)n−1 + O(n−2). (A.2)

Using the Taylor expansion, we have

Pr{χ2

p ≤ x(1 + n−1ba)} = Pr

{χ2

p ≤ x} + baxfp(x)n−1 + O(n−2) (A.3)

and

fp

(x(1 + n−1ba)

) = fp(x) + O(n−1). (A.4)

Substituting (A.3) and (A.4) into (A.2) gives

Pr{Wn(θ0; a) ≤ x(1 + n−1ba)

} = Pr{χ2

p ≤ x} + O(n−2).

According to DiCiccio, Hall, & Romano (1991), replacing the Bartlett correction factor ba by a√n-consistent estimator ba has no effect on the order of n−2. Thus,

Pr{Wn(θ0; a) ≤ x(1 + n−1ba)

} = Pr{χ2

p ≤ x} + O(n−2).

This completes the proof of Theorem 2.

ACKNOWLEDGEMENTSThe research is supported by the Natural Science and Engineering Research Council of Canada,the UBC Killam faculty research fellowship, the Postdoctoral Science Foundation of China andthe National Natural Science Foundation of China. The authors would like to thank the Editor,the Associate Editor and three anonymous referees for their insightful comments and suggestionson this article, which have led to significant improvements.

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Received 16 October 2013Accepted 30 August 2014


Canadian Journal of Statistics (1)

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