Pseudo‐empirical likelihood ratio confidence intervals for complex surveys

The authors show how an adjusted pseudo‐empirical likelihood ratio statistic that is asymptotically distributed as a chi‐square random variable can be used to construct confidence intervals for a finite population mean or a finite population distribution function from complex survey samples. They consider both non‐stratified and stratified sampling designs, with or without auxiliary information. They examine the behaviour of estimates of the mean and the distribution function at specific points using simulations calling on the Rao‐Sampford method of unequal probability sampling without replacement. They conclude that the pseudo‐empirical likelihood ratio confidence intervals are superior to those based on the normal approximation, whether in terms of coverage probability, tail error rates or average length of the intervals.     

Exact likelihood ratio and score confidence intervals for the binomial proportion

Many methods are available for computing a confidence interval for the binomial parameter, and these methods differ in their operating characteristics. It has been suggested in the literature that the use of the exact likelihood ratio (LR) confidence interval for the binomial proportion should be considered. This paper provides an evaluation of the operating characteristics of the two‐sided exact LR and exact score confidence intervals for the binomial proportion and compares these results to those for three other methods that also strictly maintain nominal coverage: Clopper‐Pearson, Blaker, and Casella. In addition, the operating characteristics of the two‐sided exact LR method and exact score method are compared with those of the corresponding asymptotic methods to investigate the adequacy of the asymptotic approximation. Copyright © 2013 John Wiley & Sons, Ltd.     

Smoothed empirical likelihood confidence intervals for quantile regression parameters with auxiliary information

This paper develops a smoothed empirical likelihood (SEL)-based method to construct confidence intervals for quantile regression parameters with auxiliary information. First, we define the SEL ratio and show that it follows a Chi-square distribution. We then construct confidence intervals according to this ratio. Finally, Monte Carlo experiments are employed to evaluate the proposed method.     

Mid-P confidence intervals based on the likelihood ratio for proportions estimated by group testing

Group testing has been used in many fields of study to estimate proportions. When groups are of different size, the derivation of exact confidence intervals is complicated by the lack of a unique ordering of the event space. An exact interval estimation method is described here, in which outcomes are ordered according to a likelihood ratio statistic. The method is compared with another exact method, in which outcomes are ordered by their associated MLE. Plots of the P‐value against the proportion are useful in examining the properties of the methods. Coverage provided by the intervals is assessed using several realistic grouptesting procedures. The method based on the likelihood ratio, with a mid‐P correction, is shown to give very good coverage in terms of closeness to the nominal level, and is recommended for this type of problem.     

Empirical likelihood confidence intervals for nonparametric functional data analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilk's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression models involving functional data. Our numerical results demonstrate improved performance of the empirical likelihood methods over normal approximation-based methods.     

Nested coordinate descent algorithms for empirical likelihood

Empirical likelihood (EL) as a nonparametric approach has been demonstrated to have many desirable merits. While it has intensive development in methodological research, its practical application is less explored due to the requirements of intensive optimizations. Effective and stable algorithms therefore are highly desired for practical implementation of EL. This paper bears the effort to narrow the gap between methodological research and practical application of EL. We try to tackle the computation problems, which are considered difficult by practitioners, by introducing a nested coordinate descent algorithm and one modified version to EL. Coordinate descent as a class of convenient and robust algorithms has been shown in the existing literature to be effective in optimizations. We show that the nested coordinate descent algorithms can be conveniently and stably applied in general EL problems. The combination of nested coordinate descent with the MM algorithm further simplifies the computation. The nested coordinate descent algorithms are a natural and perfect match with inferences based on profile estimation and variable selection in high-dimensional data. Extensive examples are conducted to demonstrate the performance of the nested coordinate descent algorithms in the context of EL.     

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Abstract

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.     

Comparison of empirical likelihood and its dual likelihood under density ratio model

Density ratio models (DRMs) are commonly used semiparametric models to link related populations. Empirical likelihood (EL) under DRM has been demonstrated to be a flexible and useful platform for semiparametric inferences. Since DRM-based EL has the same maximum point and maximum likelihood as its dual form (dual EL), EL-based inferences under DRM are usually made through the latter. A natural question comes up: is there any efficiency loss of doing so? We make a careful comparison of the dual EL and DRM-based EL estimation methods from theory and numerical simulations. We find that their point estimators for any parameter are exactly the same, while they may have different performances in interval estimation. In terms of coverage accuracy, the two intervals are comparable for non- or moderate skewed populations, and the DRM-based EL interval can be much superior for severely skewed populations. A real data example is analysed for illustration purpose.     

Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Effects of censoring on the robustness of exponential-based confidence intervals for median lifetime

Statistical procedures for constructing confidence intervals for median lifetime often rest on a distributional assumption for failure times.This paper explores the interplay between censoring levels and robustness for two construction procedures based on exponential lifetime, subject to general right-censoring. Data are simulated from nearby Weibull distributions. As expected, the simulations indicate that when the exponential assumption is not satisfied, observed coverage by the confidence intervals may differ substantially from the specified coverage level. The marked improvement in the robustness properties of the intervals as the level of censoring increases suggests questions for future research.     

A comparative study of confidence intervals to assess biosimilarity from analytical data

Assessment of analytical similarity of tier 1 quality attributes is based on a set of hypotheses that tests the mean difference of reference and test products against a margin adjusted for standard deviation of the reference product. Thus, proper assessment of the biosimilarity hypothesis requires statistical tests that account for the uncertainty associated with the estimations of the mean differences and the standard deviation of the reference product. Recently, a linear reformulation of the biosimilarity hypothesis has been proposed, which facilitates development and implementation of statistical tests. These statistical tests account for the uncertainty in the estimation process of all the unknown parameters. In this paper, we survey methods for constructing confidence intervals for testing the linearized reformulation of the biosimilarity hypothesis and also compare the performance of the methods. We discuss test procedures using confidence intervals to make possible comparison among recently developed methods as well as other previously developed methods that have not been applied for demonstrating analytical similarity. A computer simulation study was conducted to compare the performance of the methods based on the ability to maintain the test size and power, as well as computational complexity. We demonstrate the methods using two example applications. At the end, we make recommendations concerning the use of the methods.     

Empirical likelihood inference for parameters in a partially linear errors-in-variables model

In this paper, we consider the application of the empirical likelihood method to a partially linear model with measurement errors in the non-parametric part. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter by using the empirical log-likelihood ratio function, and the resulting estimator is shown to be asymptotically normal. Some simulations and an application are conducted to illustrate the proposed method.     

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.     

Generalized empirical likelihood inference in partial linear regression model for longitudinal data

In this paper, empirical likelihood inference for longitudinal data within the framework of partial linear regression models are investigated. The proposed procedures take into consideration the correlation within groups without involving direct estimation of nuisance parameters in the correlation matrix. The empirical likelihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence intervals. A nonparametric version of Wilk's theorem for the limiting distribution of the empirical likelihood ratio is derived. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. The finite sample behaviour of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial data set.     

Empirical likelihood confidence intervals for the mean of a population containing many zero values

If a population contains many zero values and the sample size is not very large, the traditional normal approximation‐based confidence intervals for the population mean may have poor coverage probabilities. This problem is substantially reduced by constructing parametric likelihood ratio intervals when an appropriate mixture model can be found. In the context of survey sampling, however, there is a general preference for making minimal assumptions about the population under study. The authors have therefore investigated the coverage properties of nonparametric empirical likelihood confidence intervals for the population mean. They show that under a variety of hypothetical populations, these intervals often outperformed parametric likelihood intervals by having more balanced coverage rates and larger lower bounds. The authors illustrate their methodology using data from the Canadian Labour Force Survey for the year 2000.     

On robustness of usual confidence region under transformation misspecification

Robustness of confidence region for linear model parameters following a misspecified transformation of dependent variable is studied. It is shown that when error standard deviation is moderate to large the usual confidence region is robust against transformation misspecification. When error standard deviation is small the usual confidence region could be very conservative for structured models and slightly liberal for unstructured models. However, the conservativeness in structured case can be controlled if the transformation is selected with the help of data rather than prior information since this is the case when data is able to provide a very accurate estimate of transformation.     

Empirical likelihood for generalized linear models with fixed and adaptive designs

Empirical likelihood inference for generalized linear models with fixed and adaptive designs is considered. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter and the resulting estimator is shown to be asymptotically normal. Some simulations are conducted to illustrate the proposed method.     

On the construction of exact lower confidence bounds on the distance between any two ranked location parameters of K populations

In this paper, an exact lower confidence bound on the distance between any two ranked location parameters of k populations is derived. This confidence bound contains the confidence bounds of Kim (1986) and of Chen, Xiong and Lam (1993) as special cases.