Two-sample empirical likelihood method for difference between coefficients in linear regression model

The empirical likelihood method is proposed to construct the confidence regions for the difference in value between coefficients of two-sample linear regression model. Unlike existing empirical likelihood procedures for one-sample linear regression models, as the empirical likelihood ratio function is not concave, the usual maximum empirical likelihood estimation cannot be obtained directly. To overcome this problem, we propose to incorporate a natural and well-explained restriction into likelihood function and obtain a restricted empirical likelihood ratio statistic (RELR). It is shown that RELR has an asymptotic chi-squared distribution. Furthermore, to improve the coverage accuracy of the confidence regions, a Bartlett correction is applied. The effectiveness of the proposed approach is demonstrated by a simulation study.     

A note on the two-sample mean problem based on jackknife empirical likelihood

In this article, we employ the jackknife empirical likelihood (JEL) method to construct the confidence regions for the difference of the means of two d-dimensional samples. Compared with traditional EL for the two-sample mean problem, JEL is extremely simpler to use in practice and is more effective in computing. Based on the JEL ratio test, a version of Wilks’ theorem is developed. Furthermore, to improve the coverage accuracy of confidence regions, a Bartlett correction is applied. The effectiveness of the proposed method is demonstrated by a simulation study and a real data analysis.     

SOME RESAMPLING PROCEDURES UNDER SYMMETRY

This paper investigates the properties of bootstrap and related methods assuming that the underlying distribution is symmetric but otherwise unknown. In particular it studies the percentile-t, nonparametric tilting and empirical likelihood and finds that the performance of percentile-t and non-parametric tilting methods can be improved by incorporating the symmetry into the resampling procedure. However, for symmetric empirical likelihood, the Bartlett correctability no longer holds, although use of bootstrap calibration restores the good coverage properties typically associated with Bartlett correction. This surprising result shows that Bartlett correctability is a very delicate property.     

Empirical likelihood for quantile regression models with longitudinal data

We develop two empirical likelihood-based inference procedures for longitudinal data under the framework of quantile regression. The proposed methods avoid estimating the unknown error density function and the intra-subject correlation involved in the asymptotic covariance matrix of the quantile estimators. By appropriately smoothing the quantile score function, the empirical likelihood approach is shown to have a higher-order accuracy through the Bartlett correction. The proposed methods exhibit finite-sample advantages over the normal approximation-based and bootstrap methods in a simulation study and the analysis of a longitudinal ophthalmology data set.     

Extending the empirical likelihood by domain expansion

We extend the empirical likelihood beyond its domain by expanding its contours nested inside the domain with a similarity transformation. The extended empirical likelihood achieves two objectives at the same time: escaping the “convex hull constraint” on the empirical likelihood and improving the coverage accuracy of the empirical likelihood ratio confidence region to $O(n^{-2})$ . The latter is accomplished through a special transformation which matches the extended empirical likelihood with the Bartlett corrected empirical likelihood. The extended empirical likelihood ratio confidence region retains the shape of the original empirical likelihood ratio confidence region. It also accommodates adjustments for dimension and small sample size, giving it good coverage accuracy in large and small sample situations. The Canadian Journal of Statistics 41: 257–274; 2013 © 2013 Statistical Society of Canada     

Empirical likelihood for the varying‐coefficient single‐index model

In this article the author investigates the application of the empirical‐likelihood‐based inference for the parameters of varying‐coefficient single‐index model (VCSIM). Unlike the usual cases, if there is no bias correction the asymptotic distribution of the empirical likelihood ratio cannot achieve the standard chi‐squared distribution. To this end, a bias‐corrected empirical likelihood method is employed to construct the confidence regions (intervals) of regression parameters, which have two advantages, compared with those based on normal approximation, that is, (1) they do not impose prior constraints on the shape of the regions; (2) they do not require the construction of a pivotal quantity and the regions are range preserving and transformation respecting. A simulation study is undertaken to compare the empirical likelihood with the normal approximation in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. A real data example is given to illustrate the proposed approach. The Canadian Journal of Statistics 38: 434–452; 2010 © 2010 Statistical Society of Canada     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Generalized Empirical Likelihood Inference in Generalized Linear Models for Longitudinal Data

In this article, the generalized linear model for longitudinal data is studied. A generalized empirical likelihood method is proposed by combining generalized estimating equations and quadratic inference functions based on the working correlation matrix. It is proved that the proposed generalized empirical likelihood ratios are asymptotically chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. In addition, the maximum empirical likelihood estimates of parameters are obtained, and their asymptotic normalities are proved. Some simulations are undertaken to compare the generalized empirical likelihood and normal approximation-based method in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. An example of a real data is used for illustrating our methods.     

Bartlett corrections in beta regression models

We consider the issue of performing accurate small-sample testing inference in beta regression models, which are useful for modeling continuous variates that assume values in (0,1), such as rates and proportions. We derive the Bartlett correction to the likelihood ratio test statistic and also consider a bootstrap Bartlett correction. Using Monte Carlo simulations we compare the finite sample performances of the two corrected tests to that of the standard likelihood ratio test and also to its variant that employs Skovgaard's adjustment; the latter is already available in the literature. The numerical evidence favors the corrected tests we propose. We also present an empirical application.     

Approximate jackknife empirical likelihood method for estimating equations

It is known that the profile empirical likelihood method based on estimating equations is computationally intensive when the number of nuisance parameters is large. Recently, Li, Peng, & Qi (2011) proposed a jackknife empirical likelihood method for constructing confidence regions for the parameters of interest by estimating the nuisance parameters separately. However, when the estimators for the nuisance parameters have no explicit formula, the computation of the jackknife empirical likelihood method is still intensive. In this paper, an approximate jackknife empirical likelihood method is proposed to reduce the computation in the jackknife empirical likelihood method when the nuisance parameters cannot be estimated explicitly. A simulation study confirms the advantage of the new method. The Canadian Journal of Statistics 40: 110–123; 2012 © 2012 Statistical Society of Canada     

Level-specific correction for nonparametric likelihoods

The popular empirical likelihood method not only has a convenient chi-square limiting distribution but is also Bartlett correctable, leading to a high-order coverage precision of the resulting confidence regions. Meanwhile, it is one of many nonparametric likelihoods in the Cressie–Read power divergence family. The other likelihoods share many attractive properties but are not Bartlett correctable. In this paper, we develop a new technique to achieve the effect of being Bartlett correctable. Our technique is generally applicable to pivotal quantities with chi-square limiting distributions. Numerical experiments and an example reveal that the method is successful for several important nonparametric likelihoods.     

Censored median regression and profile empirical likelihood

We implement profile empirical likelihood-based inference for censored median regression models. Inference for any specified subvector is carried out by profiling out the nuisance parameters from the “plug-in” empirical likelihood ratio function proposed by Qin and Tsao. To obtain the critical value of the profile empirical likelihood ratio statistic, we first investigate its asymptotic distribution. The limiting distribution is a sum of weighted chi square distributions. Unlike for the full empirical likelihood, however, the derived asymptotic distribution has intractable covariance structure. Therefore, we employ the bootstrap to obtain the critical value, and compare the resulting confidence intervals with the ones obtained through Basawa and Koul’s minimum dispersion statistic. Furthermore, we obtain confidence intervals for the age and treatment effects in a lung cancer data set.     

Fast Covariance Estimation for Innovations Computed from a Spatial Gibbs Point Process

In this paper, we derive an exact formula for the covariance of two innovations computed from a spatial Gibbs point process and suggest a fast method for estimating this covariance. We show how this methodology can be used to estimate the asymptotic covariance matrix of the maximum pseudo‐likelihood estimator of the parameters of a spatial Gibbs point process model. This allows us to construct asymptotic confidence intervals for the parameters. We illustrate the efficiency of our procedure in a simulation study for several classical parametric models. The procedure is implemented in the statistical software R , and it is included in spatstat , which is an R package for analyzing spatial point patterns.     

Second‐order Accurate Confidence Regions Based on Members of the Generalized Power Divergence Family

Recently, a technique based on pseudo‐observations has been proposed to tackle the so‐called convex hull problem for the empirical likelihood statistic. The resulting adjusted empirical likelihood also achieves the high‐order precision of the Bartlett correction. Nevertheless, the technique induces an upper bound on the resulting statistic that may lead, in certain circumstances, to worthless confidence regions equal to the whole parameter space. In this paper, we show that suitable pseudo‐observations can be deployed to make each element of the generalized power divergence family Bartlett‐correctable and released from the convex hull problem. Our approach is conceived to achieve this goal by means of two distinct sets of pseudo‐observations with different tasks. An important effect of our formulation is to provide a solution that permits to overcome the problem of the upper bound. The proposal, which effectiveness is confirmed by simulation results, gives back attractiveness to a broad class of statistics that potentially contains good alternatives to the empirical likelihood.     

Quasi-likelihood from<Emphasis Type="Italic">M</Emphasis>-estimators: A numerical comparison with empirical likelihood

In this paper we compare two robust pseudo-likelihoods for a parameter of interest, also in the presence of nuisance parameters. These functions are obtained by computing quasi-likelihood and empirical likelihood from the estimating equations which define robustM-estimators. Application examples in the context of linear transformation models are considered. Monte Carlo studies are performed in order to assess the finite-sample performance of the inferential procedures based on quasi-and empirical likelihood, when the objective is the construction of robust confidence regions.     

Reduce computation in profile empirical likelihood method

Since its introduction by Owen (1988, 1990), the empirical likelihood method has been extensively investigated and widely used to construct confidence regions and to test hypotheses in the literature. For a large class of statistics that can be obtained via solving estimating equations, the empirical likelihood function can be formulated from these estimating equations as proposed by Qin and Lawless (1994). If only a small part of parameters is of interest, a profile empirical likelihood method has to be employed to construct confidence regions, which could be computationally costly. In this article the authors propose a jackknife empirical likelihood method to overcome this computational burden. This proposed method is easy to implement and works well in practice. The Canadian Journal of Statistics 39: 370–384; 2011 © 2011 Statistical Society of Canada     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

Robust empirical likelihood for linear models under median constraints

In this paper, we consider the application of the empirical likelihood for

linear models under median constraints in view of robustness. For two simple median constraints, it is shown that conditions to ensure the consistency of the empirical likelihood confidence regions can be surprisingly relaxed compared with the normal approach under L norm. However, the coverage accuracy of the empirical likelihood confidence regions based on simple median constrains cannot be improved because of the discontinuity of the constraints. Therefore, a smoothed version of median constraint is proposed and a general theory is established to ensure its validity.     

Statistical Inference in Partially Linear Varying-Coefficient Models with Missing Responses at Random

This article considers statistical inference for partially linear varying-coefficient models when the responses are missing at random. We propose a profile least-squares estimator for the parametric component with complete-case data and show that the resulting estimator is asymptotically normal. To avoid to estimate the asymptotic covariance in establishing confidence region of the parametric component with the normal-approximation method, we define an empirical likelihood based statistic and show that its limiting distribution is chi-squared distribution. Then, the confidence regions of the parametric component with asymptotically correct coverage probabilities can be constructed by the result. To check the validity of the linear constraints on the parametric component, we construct a modified generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Then, we extend the generalized likelihood ratio technique to the context of missing data. Finally, some simulations are conducted to illustrate the proposed methods.     

On Nonsmooth Estimating Functions via Jackknife Empirical Likelihood

In many applications, the parameters of interest are estimated by solving non‐smooth estimating functions with U‐statistic structure. Because the asymptotic covariances matrix of the estimator generally involves the underlying density function, resampling methods are often used to bypass the difficulty of non‐parametric density estimation. Despite its simplicity, the resultant‐covariance matrix estimator depends on the nature of resampling, and the method can be time‐consuming when the number of replications is large. Furthermore, the inferences are based on the normal approximation that may not be accurate for practical sample sizes. In this paper, we propose a jackknife empirical likelihood‐based inferential procedure for non‐smooth estimating functions. Standard chi‐square distributions are used to calculate the p‐value and to construct confidence intervals. Extensive simulation studies and two real examples are provided to illustrate its practical utilities.