Empirical Likelihood Approach Research Articles

Estimating the size of a closed population by modeling latent and observed heterogeneity.

The paper extends the empirical likelihood (EL) approach of Liu etal. to a new and very flexible family of latent class models for capture-recapture data also allowing for serial dependence on previous capture history, conditionally on latent type and covariates. The EL approach allows to estimate the overall population size directly rather than by adding estimates conditional to covariate configurations. A Fisher-scoring algorithm for maximum likelihood estimation is proposed and a more efficient alternative to the traditional EL approach for estimating the non-parametric component is introduced; this allows us to show that the mapping between the non-parametric distribution of the covariates and the probabilities of being never captured is one-to-one and strictly increasing. Asymptotic results are outlined, and a procedure for constructing profile likelihood confidence intervals for the population size is presented. Two examples based on real data are used to illustrate the proposed approach and a simulation study indicates that, when estimating the overall undercount, the method proposed here is substantially more efficient than the one based on conditional maximum likelihood estimation, especially when the sample size is not sufficiently large.

Empirical likelihood in a partially linear single-index model with censored response data

An empirical likelihood (EL) approach for a partial linear single-index model with censored response data is studied. A bias-corrected EL ratio is proposed, and the asymptotic chi-squared distribution of this ratio is obtained. The result can be directly used to construct the confidence regions of the regression parameters. The estimators of regression parameters and link function are constructed, and their asymptotic distributions are obtained. Also, a confidence band of the link function is constructed. The proposed method has two main features: The first feature is that the EL ratio is calibrated directly from within, instead of multiplying an adjustment factor by an EL ratio, which reflects the nature of EL. The second feature is avoiding undersmoothing of nonparametric functions, thus ensuring that the n-consistency of the parameter estimator. As a byproduct, the EL and estimation of a single-index model with censored response data are studied. The performance of the bias-corrected EL is evaluated by the simulation studies. The proposed method is illustrated with an example of a real data analysis.

Empirical-Likelihood-Based Inference for Partially Linear Models

Partially linear models find extensive application in biometrics, econometrics, social sciences, and various other fields due to their versatility in accommodating both parametric and nonparametric elements. This study aims to establish statistical inference for the parametric component effects within these models, employing a nonparametric empirical likelihood approach. The proposed method involves a projection step to eliminate the nuisance nonparametric component and utilizes an empirical-likelihood-based technique, along with the Bartlett correction, to enhance the coverage probability of the confidence interval for the parameter of interest. This method demonstrates robustness in handling normally and non-normally distributed errors. The proposed empirical likelihood ratio statistic converges to a limiting chi-square distribution under certain regulations. Simulation studies demonstrate that this method provides better inference in terms of coverage probabilities compared to the conventional normal-approximation-based method. The proposed method is illustrated by analyzing the Boston housing data from a real study.

Distributed Empirical Likelihood Approach to Integrating Unbalanced Datasets

Distributed Empirical Likelihood Approach to Integrating Unbalanced Datasets

Adjusting for Selection Bias in Nonprobability Samples by Empirical Likelihood Approach

Abstract Large amount of data are today available, that are easier and faster to collect than survey data, bringing new challenges. One of them is the nonprobability nature of these big data that may not represent the target population properly and hence result in highly biased estimators. In this article two approaches for dealing with selection bias when the selection process is nonignorable are discussed. The first one, based on the empirical likelihood, does not require parametric specification of the population model but the probability of being in the nonprobability sample needed to be modeled. Auxiliary information known for the population or estimable from a probability sample can be incorporated as calibration constraints, thus enhancing the precision of the estimators. The second one is a mixed approach based on mass imputation and propensity score adjustment requiring that the big data membership is known throughout a probability sample. Finally, two simulation experiments and an application to income data are performed to evaluate the performance of the proposed estimators in terms of robustness and efficiency.

An Empirical Likelihood Approach to Reduce Selection Bias in Voluntary Samples

How to construct the pseudo-weights in voluntary samples is an important practical problem in survey sampling. The problem is quite challenging when the sampling mechanism for the voluntary sample is allowed to be non-ignorable. Under the assumption that the sample participation model is correctly specified, we can compute a consistent estimator of the model parameter and construct the propensity score estimator of the population mean. We propose using the empirical likelihood method to construct the final weights for voluntary samples by incorporating the bias calibration constraints and the benchmarking constraints. Linearization variance estimation of the proposed method is developed. A toy example is also presented to illustrate the idea and the computational details. A limited simulation study is also performed to evaluate the performance of the proposed methods. AMS subject classification: 62D10, 63D05

Robust empirical likelihood inference for partially linear varying coefficient models with longitudinal data

This paper presents a robust empirical likelihood procedure based on the exponential squared loss (ESL) function and leverage-based weights for the partially linear varying coefficient model with longitudinal data. The proposed method simultaneously solves the problems of correlation structure of longitudinal data and the existence of outliers, and achieves robustness and efficiency by introducing an appropriate data-driven tuning parameter. More importantly, profit from the QR decomposition technique, our method allows the parametric and nonparametric parts of the models to be estimated separately, which can avoid the mutual influence between them and make the implementation easier. Under some mild conditions, the large sample theoretical properties of the robust empirical likelihood approach are established. Simulation studies and a real data analysis are also carried out to assess and illustrate the finite sample performance.

Accounting for Non‐ignorable Sampling and Non‐response in Statistical Matching

SummaryData for statistical analysis is often available from different samples, with each sample containing measurements on only some of the variables of interest. Statistical matching attempts to generate a fused database containing matched measurements on all the target variables. In this article, we consider the use of statistical matching when the samples are drawn by informative sampling designs and are subject to not missing at random non‐response. The problem with ignoring the sampling process and non‐response is that the distribution of the data observed for the responding units can be very different from the distribution holding for the population data, which may distort the inference process and result in a matched database that misrepresents the joint distribution in the population. Our proposed methodology employs the empirical likelihood approach and is shown to perform well in a simulation experiment and when applied to real sample data.

Empirical likelihood in single-index partially functional linear model with missing observations

In this paper, we focus on the empirical likelihood in the single-index partially functional linear model. When the response variables or/and part of the covariates are missing at random, we construct the empirical likelihood ratio of the parameter in the model based on B-spline approximation for the link and slope functions, and define maximum empirical likelihood (MEL) estimator of the parameter. Under suitable assumptions, the asymptotic distributions of the proposed empirical log-likelihood ratio and MEL estimator are established. At the same time, based on penalized empirical likelihood (PEL) approach, we define the PEL estimator of the parameter and investigate variable selection of the model. A simulation study is done to evaluate the finite sample performance for the proposed methods.

Pseudo empirical likelihood inference for nonprobability survey samples

AbstractIn this article, we first provide an overview of two major developments on complex survey data analysis: the empirical likelihood methods and statistical inference with nonprobability survey samples. We highlight the important research contributions to the field of survey sampling in general and the two topics in particular by Canadian survey statisticians. We then propose new inferential procedures for analyzing nonprobability survey samples through the pseudo empirical likelihood approach. The proposed methods lead to point estimators asymptotically equivalent to those discussed in the recent literature but with more desirable features on confidence intervals such as range‐respecting and data‐driven orientation. Results from a simulation study demonstrate the superiority of the proposed methods in dealing with binary response variables.

Unconditional empirical likelihood approach for analytic use of public survey data

AbstractModeling survey data often requires having the knowledge of design and weighting variables. With public‐use survey data, some of these variables may not be available for confidentiality reasons. The proposed approach can be used in this situation, as long as calibrated weights and variables specifying the strata and primary sampling units are available. It gives consistent point estimation and a pivotal statistics for testing and confidence intervals. The proposed approach does not rely on with‐replacement sampling, single‐stage, negligible sampling fractions, or noninformative sampling. Adjustments based on design effects, eigenvalues, joint‐inclusion probabilities or bootstrap, are not needed. The inclusion probabilities and auxiliary variables do not have to be known. Multistage designs with unequal selection of primary sampling units are considered. Nonresponse can be easily accommodated if the calibrated weights include reweighting adjustment for nonresponse. We use an unconditional approach, where the variables and sample are random variables. The design can be informative.

Empirical likelihood confidence interval for difference-in-differences estimator with panel data

This paper develops an empirical likelihood approach to construct the confidence interval for the average treatment effect on the treated under the difference-in-differences framework. The empirical likelihood function is constructed based on a doubly robust moment function of the parameter of interest. Under some regularity conditions, we show that the proposed method retains the nonparametric Wilks property of empirical likelihood.

Semiparametric empirical likelihood inference for abundance from one-inflated capture-recapture data.

Abundance estimation from capture-recapture data is of great importance in many disciplines. Analysis of capture-recapture data is often complicated by the existence of one-inflation and heterogeneity problems. Simultaneously taking these issues into account, existing abundance estimation methods are usually constructed on the basis of conditional likelihood under one-inflated zero-truncated count models. However, the resulting Horvitz-Thompson-type estimators may be unstable, and the resulting Wald-type confidence intervals may exhibit severe undercoverage. In this paper, we propose a semiparametric empirical likelihood (EL) approach to abundance estimation under one-inflated binomial and Poisson regression models. To facilitate the computation of the EL method, we develop an expectation-maximization algorithm. We also propose a new score test for the existence of one-inflation and prove its asymptotic normality. Our simulation studies indicate that compared with existing estimators, the proposed score test is more powerful and the maximum EL estimator has a smaller mean square error. The advantages of our approaches are further demonstrated by analyses of prinia data from Hong Kong and drug user data fromBangkok.

An application of the interval estimation for the At-Risk-of-Poverty Rate assessment

In the document [Eurostat (Your Key to European Statistics) 2020], At-Risk-of-Poverty Rate (ARPR in short) is defined as the percentage of population with an income not exceeding 60% of the general population median income. Extensive and thorough research on the estimation of this measure has been conducted since its introduction. For example, in the paper of [Zieliński 2009a] a non-parametric, distribution-free confidence interval for ARPR has been constructed. An example of application of the confidence interval proposed by [Zieliński 2009a] has been given in [Zieliński 2009b]. Some other interesting approach regarding the interval estimation of ARPR has been proposed in [Luo and Qin 2017], where the authors introduced new concepts of the interval estimation for the so-called Low-Income Proportion (LIP) measure, which is a generalization of ARPR. The LIP measure and thus, the ARPR parameter in particular, are important indexes describing the inequality in an income distribution. Based on the construction of the point smoothed kernel estimate for LIP, [Luo and Qin 2017] established a smoothed jackknife empirical likelihood approach leading to the introduction of some new non-parametric confidence intervals for the LIP measure and consequently, for the ARPR index as well. In our work, we aim to apply the most interesting ideas of LIP and ARPR point and interval estimation for data consisting of observations concerning an equalised disposable income of households in Poland from 2003. We also discuss the accuracy and adequacy of the empirical results relating to the ARPR interval estimation, obtained by the implementation of the constructed confidence intervals.

Smoothed quantile regression with nonignorable dropouts

In this paper, we adopt a three-stage estimation procedure and statistical inference methods for quantile regression (QR) based on empirical likelihood (EL) approach with nonignorable dropouts. In the first stage, we consider a parametric model on the dropout propensity of response and handle the parameter identifiability issue by using nonresponse instrument. With the estimated dropout propensity, in the second stage the inverse probability weighting and kernel smoothing methods are applied to construct the bias-corrected and smoothed generalized estimating equations for nonignorable dropouts. In the third stage, borrowing the matrix expansion idea of quadratic inference function, we obtain the proposed estimators that can accommodate the within-subject correlations and improve the estimation efficiency simultaneously. A class of improved estimators and their confidence regions for QR coefficient are derived. Further, the penalized EL method and algorithm for variable selection are investigated. Simulation studies and a real example on HIV-CD4 data set are also provided to show the performance of the proposed estimators.

A semi-parametric empirical likelihood approach for conditional estimating equations under endogenous selection

The estimation and inference for conditional estimating equations models with endogenous selection, are considered. The approach takes into account possible endogenous selection which may lead to a selection bias. It can be used for a wide range of statistical models not covered by the model-based sampling theory. Endogeneity can be either part of the selection or within the covariates. It is particularly well suited for models with unknown heteroscedasticity, uncontrolled confounders and measurement errors. It will not be necessary to model the relationship between the endogenous covariates and the instrumental variables, which offers major advantages over two-stage least-squares. The approach proposed has the advantage of being based on a fixed number of constraints determined by the size of the parameter.

Testing conditional moment restriction models using empirical likelihood

Summary An empirical likelihood test is proposed for parameters of models defined by conditional moment restrictions, such as models with nonlinear endogenous covariates, with and without heteroscedastic errors and non-separable transformation models. The number of empirical likelihood constraints is given by the size of the parameter, unlike alternative semi-parametric approaches. We show that the empirical likelihood ratio test is asymptotically pivotal, without explicit studentization. A simulation study shows that the observed size is close to the nominal level, unlike alternative empirical likelihood approaches. It also offers a major advantage over two-stage least-squares, because the relationship between endogenous and instrumental variables does not need to be known. An empirical likelihood model specification test is also proposed.

Second order expansions of estimators in nonparametric moment conditions models with weakly dependent data

This paper considers estimation of nonparametric moment conditions models with weakly dependent data. The estimator is based on a local linear version of the generalized empirical likelihood approach, and is an alternative to the popular local linear generalized method of moment estimator. The paper derives uniform convergence rates and pointwise asymptotic normality of the resulting local linear generalized empirical likelihood estimator. The paper also develops second order stochastic expansions (under a standard undersmoothing condition) that explain the better finite sample performance of the local linear generalized empirical likelihood estimator compared to that of the efficient local linear generalized method of moments estimator, and can be used to obtain (second order) bias corrected estimators. Monte Carlo simulations and an empirical application illustrate the competitive finite sample properties and the usefulness of the proposed estimators and second order bias corrections.

Bayesian empirical analysis of the proportional hazards model for right-censored failure time data

Statistical analysis of right-censored failure time data has been extensively discussed in the literature as such data often occur in many fields. In this article, we propose a new Bayesian-based empirical likelihood approach for the problem under the proportional hazards model. The new method allows one to take into account the existing prior information among other advantages, and for the implementation of the method, a Metropolis–Hasting algorithm is developed. To assess the performance of the proposed approach, a simulation study is conducted and suggests that it works well. The method is applied to a set of kidney dialysis data.

Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

ABSTRACT Due to cost-effectiveness and high efficiency, two-phase case-control sampling has been widely used in epidemiology studies. We develop a semi-parametric empirical likelihood approach to two-phase case-control data under the logistic regression model. We show that the maximum empirical likelihood estimator has an asymptotically normal distribution, and the empirical likelihood ratio follows an asymptotically central chi-square distribution. We find that the maximum empirical likelihood estimator is equal to Breslow and Holubkov (1997)'s maximum likelihood estimator. Even so, the limiting distribution of the likelihood ratio, likelihood-ratio-based interval, and test are all new. Furthermore, we construct new Kolmogorov–Smirnov type goodness-of-fit tests to test the validation of the underlying logistic regression model. Our simulation results and a real application show that the likelihood-ratio-based interval and test have certain merits over the Wald-type counterparts and that the proposed goodness-of-fit test is valid.