Maximum Empirical Likelihood Research Articles

Smoothed empirical likelihood estimation and automatic variable selection for an expectile high-dimensional model

We consider a linear model which can have a large number of explanatory variables, the errors with an asymmetric distribution or the values of the explained variable are missing at random. In order to take in account these several situations, we consider the non parametric empirical likelihood (EL) estimation method. Because a constraint in EL contains an indicator function then a smoothed function instead of the indicator will be considered. Two smoothed expectile maximum EL methods are proposed, one of which will automatically select the explanatory variables. For each of the methods we obtain the convergence rate of the estimators and their asymptotic normality. The smoothed expectile empirical log-likelihood ratio process follow asymptotically a chi-square distribution and moreover the adaptive LASSO smoothed expectile maximum EL estimator satisfies the sparsity property which guarantees the automatic selection of zero model coefficients. In order to implement these methods, we propose four algorithms.

Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (AUC) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining AUC and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of (AUC,J) and construct an asymptotically valid confidence region for (AUC,J). Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of AUC0 and J0 with the null hypothesis H0:AUC≤AUC0 or J≤J0 against the alternative hypothesis Ha:AUC>AUC0 and J>J0. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.

Empirical likelihood inference for time-varying coefficient autoregressive models

In this paper, we propose time-varying coefficient autoregressive models (TVARMs) of [Formula: see text]th order as a solution for capturing the nonlinear structure of multiple explanatory variables. The stationary [Formula: see text]-mixing sequence of autoregressive variables is considered. We apply empirical likelihood approach to investigate confidence regions of the autoregressive coefficients of the TVARMs, and establish the nonparametric version of Wilks’ theorem of the proposed empirical log-likelihood ratio. Further, the maximum empirical likelihood estimator of the autoregressive coefficients and its asymptotic distribution are also obtained. Besides, to determine the order of autoregression, penalized empirical likelihood approach is developed with the help of smoothly clipped absolute deviation penalty and the resultant penalized estimators have the oracle property. Simulation studies and an analysis of monthly Chinese price index dataset are used to demonstrate the effectiveness of our proposed methods.

Comparison of estimation and prediction methods for a zero-inflated geometric INAR(1) process with random coefficients

This study explores zero-inflated count time series models used to analyze data sets with characteristics such as overdispersion, excess zeros, and autocorrelation. Specifically, we investigate the ZIGINAR RC ( 1 ) process, a first-order stationary integer-valued autoregressive model with random coefficients and a zero-inflated geometric marginal distribution. Our focus is on examining various estimation and prediction techniques for this model. We employ estimation methods, including Whittle, Taper Spectral Whittle, Maximum Empirical Likelihood, and Sieve Bootstrap estimators for parameter estimation. Additionally, we propose forecasting approaches, such as median, Bayesian, and Sieve Bootstrap methods, to predict future values of the series. We assess the performance of these methods through simulation studies and real-world data analysis, finding that all methods perform well, providing 95% highest predicted probability intervals that encompass the observed data. While Bayesian and Bootstrap methods require more time for execution, their superior predictive accuracy justifies their use in forecasting.

Quantile difference estimation with censoring indicators missing at random.

In this paper, we define estimators of distribution functions when the data are right-censored and the censoring indicators are missing at random, and establish their strong representations and asymptotic normality. Besides, based on empirical likelihood method, we define maximum empirical likelihood estimators and smoothed log-empirical likelihood ratios of two-sample quantile difference in the presence and absence of auxiliary information, respectively, and prove their asymptotic distributions. Simulation study and real data analysis are conducted to investigate the finite sample behavior of the proposed methods.

Maximum Augmented Empirical Likelihood Estimation of Categorical Marginal Models for Large Sparse Contingency Tables.

Categorical marginal models (CMMs) are flexible tools for modelling dependent or clustered categorical data, when the dependencies themselves are not of interest. A major limitation of maximum likelihood (ML) estimation of CMMs is that the size of the contingency table increases exponentially with the number of variables, so even for a moderate number of variables, say between 10 and 20, ML estimation can become computationally infeasible. An alternative method, which retains the optimal asymptotic efficiency of ML, is maximum empirical likelihood (MEL) estimation. However, we show that MEL tends to break down for large, sparse contingency tables. As a solution, we propose a new method, which we call maximum augmented empirical likelihood (MAEL) estimation and which involves augmentation of the empirical likelihood support with a number of well-chosen cells. Simulation results show good finite sample performance for very large contingency tables.

Statistical inference for the two-sample problem under likelihood ratio ordering, with application to the ROC curve estimation.

The receiver operating characteristic (ROC) curve is a powerful statistical tool and has been widely applied in medical research. In the ROC curve estimation, a commonly used assumption is that larger the biomarker value, greater severity the disease. In this article, we mathematically interpret "greater severity of the disease" as "larger probability of being diseased." This in turn is equivalent to assume the likelihood ratio ordering of the biomarker between the diseased and healthy individuals. With this assumption, we first propose a Bernstein polynomial method to model the distributions of both samples; we then estimate the distributions by the maximum empirical likelihood principle. The ROC curve estimate and the associated summary statistics are obtained subsequently. Theoretically, we establish the asymptotic consistency of our estimators. Via extensive numerical studies, we compare the performance of our method with competitive methods. The application of our method is illustrated by a real-data example.

Empirical likelihood in single-index quantile regression with high dimensional and missing observations

Based on empirical likelihood method, we investigate statistical inference in partially linear single-index quantile regression with high dimensional linear and single-index parameters when the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. In particular, applying B-spline approximation to the unknown link function, we establish asymptotic normality of bias-corrected empirical likelihood ratio function and maximum empirical likelihood estimators of the parameters. Variable selection is considered by using the SCAD penalty. Meanwhile, we propose a penalized empirical likelihood ratio statistic to test hypothesis, and prove its asymptotically chi-square distribution under the null hypothesis. Also, simulation study and a real data analysis are conducted to evaluate the performance of the proposed methods.

Semi-parametric inference on Gini indices of two semi-continuous populations under density ratio models

Summary The Gini index is a popular inequality measure with many applications in social and economic studies. This paper studies semi-parametric inference on the Gini indices of two semi-continuous populations. We characterise the distribution of each semi-continuous population by a mixture of a discrete point mass at zero and a continuous skewed positive component. A semi-parametric density ratio model is then employed to link the positive components of the two distributions. We propose the maximum empirical likelihood estimators of the two Gini indices and their difference, and further investigate the asymptotic properties of the proposed estimators. The asymptotic results enable us to construct confidence intervals, and perform hypothesis tests for the two Gini indices and their difference. The proposed method is also applicable to cases without excessive zero values. The superiority of our proposed method over some existing methods is shown theoretically and numerically. Two real-data applications are presented for illustration.

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.

Semiparametric modelling of two-component mixtures with stochastic dominance.

In this work, we studied a two-component mixture model with stochastic dominance constraint, a model arising naturally from many genetic studies. To model the stochastic dominance, we proposed a semiparametric modelling of the log of density ratio. More specifically, when the log of the ratio of two component densities is in a linear regression form, the stochastic dominance is immediately satisfied. For the resulting semiparametric mixture model, we proposed two estimators, maximum empirical likelihood estimator (MELE) and minimum Hellinger distance estimator (MHDE), and investigated their asymptotic properties such as consistency and normality. In addition, to test the validity of the proposed semiparametric model, we developed Kolmogorov–Smirnov type tests based on the two estimators. The finite-sample performance, in terms of both efficiency and robustness, of the two estimators and the tests were examined and compared via both thorough Monte Carlo simulation studies and real data analysis.Supplementary InformationThe online version contains supplementary material available at 10.1007/s10463-022-00835-5.

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.

Statistical methods for predicting the spatial abundance of reef fish species

Understanding the spatial distribution of organism abundance is fundamental to assessing and managing ecological populations. Marine species can be difficult and logistically challenging and expensive to observe. This often results in spatial data containing low detection rates when sampling underwater, biasing spatial predictions from many modeling approaches. We propose a multistage statistical workflow that can use zero inflated sampling data to develop non-linear predictive spatial distributions of reef fish abundance. The workflow includes: (1) an individual-based discrete event simulation which generates simulated survey data under different abundance settings; (2) empirical maximum likelihood analysis to establish the relationship between survey data and abundance from the simulation; (3) a two-step random smoothing method to estimate reliable block spatial abundance around each survey station; (4) an ensemble of different machine learning models which use the estimated abundance from step three as input to compute a stable non-linear prediction of abundance across the entire study area (Gulf of Mexico). Applying our workflow greatly improved the ability to forecast abundance at small spatial scales. The ability to forecast at fine spatial scales is critical when working with species that are patchily distributed. This workflow can apply to many ecological populations to develop abundance maps even if sample data is not well distributed across the study area or is zero inflated.

Dimension-reduced empirical likelihood estimation and inference for M-estimators with nonignorable nonresponse

When the responses have nonignorable missing data, we consider a general semiparametric propensity model and propose empirical likelihood (EL) based statistical inference for M-estimators. The unknown parameters in the propensity are firstly identified and estimated by combining instrumental estimating equations and dimension-reduced kernel estimators. We propose three bias-corrected nonparametric estimating equations in conjunction with the EL procedure. The resulting maximum EL estimators are shown asymptotically equivalent, achieve the desirable asymptotic properties of unbiasedness and asymptotic normality. An adjusted EL ratio procedure for constructing accurate confidence regions is also proposed. The finite-sample performance of the proposed estimators for response mean, distribution function and quantile is studied through simulation, and an application to ACTG 175 data set is also presented.

Estimation for a hybrid model of functional and linear measurement errors regression with missing response

In this paper, we consider least squares estimation and empirical likelihood inference for partial functional linear models when the covariates in the non-functional linear component are measured with additive error and the responses are missing at random. Asymptotic properties of the proposed estimators for the parametric and nonparametric components are established. A class of empirical log-likelihood ratio functions of the parametric component and response mean are developed, and the corresponding maximum empirical likelihood estimators are constructed. We also prove that the empirical log-likelihood ratio functions are asymptotic standard chi-squared distribution. The results can be used to construct confidence intervals for the parametric component and response mean. The asymptotic distributions of the corresponding maximum empirical likelihood estimators are also established. Meanwhile, a simulation study is conducted to demonstrate the finite sample performance of the proposed procedure. A real data analysis is also used to illustrate our methods.

Semiparametric empirical likelihood inference with estimating equations under density ratio models

The density ratio model (DRM) provides a flexible and useful platform for combining information from multiple sources. In this paper, we consider statistical inference under two-sample DRMs with additional parameters defined through and/or additional auxiliary information expressed as estimating equations. We examine the asymptotic properties of the maximum empirical likelihood estimators (MELEs) of the unknown parameters in the DRMs and/or defined through estimating equations, and establish the chi-square limiting distributions for the empirical likelihood ratio (ELR) statistics. We show that the asymptotic variance of the MELEs of the unknown parameters does not decrease if one estimating equation is dropped. Similar properties are obtained for inferences on the cumulative distribution function and quantiles of each of the populations involved. We also propose an ELR test for the validity and usefulness of the auxiliary information. Simulation studies show that correctly specified estimating equations for the auxiliary information result in more efficient estimators and shorter confidence intervals. Two real examples are used for illustrations.

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.

Semiparametric inference on general functionals of two semicontinuous populations

In this paper, we propose new semiparametric procedures for inference on linear functionals in the context of two semicontinuous populations. The distribution of each semicontinuous population is characterized by a mixture of a discrete point mass at zero and a continuous skewed positive component. To utilize the information from both populations, we model the positive components of the two mixture distributions via a semiparametric density ratio model. Under this model setup, we construct the maximum empirical likelihood estimators of the linear functionals. The asymptotic normality of the proposed estimators is established and is used to construct confidence regions and perform hypothesis tests for these functionals. We show that the proposed estimators are more efficient than the fully nonparametric ones. Simulation studies demonstrate the advantages of our method over existing methods. Two real-data examples are provided for illustration.

Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.

Weighted empirical likelihood for heteroscedastic varying coefficient partially non‐linear models with missing data

In this article, a weighted empirical likelihood technique for constructing the empirical likelihood confidence regions is applied to study the heteroscedastic varying coefficient partially non‐linear models with missing response data. We first provide an estimator of the error variance based on the Nadaraya–Watson kernel estimation method. Then a weighted empirical log‐likelihood ratio of the unknown parameter is constructed based on the inverse probability weighted technique. The maximum empirical likelihood (MEL) estimator of the unknown parameter is obtained. Further, a weighted empirical log‐likelihood ratio of the varying coefficient function is introduced based on the MEL estimator and the inverse probability weighted method. The limiting distributions of the resulting statistics for both the unknown parameter and varying coefficient function are shown to have the standard chi‐squared distribution. A simulation study and a real data set example are undertaken to investigate the finite sample performance of the proposed methods.