Empirical Likelihood Ratio Statistic Research Articles

A Penalized Empirical Likelihood Approach for Estimating Population Sizes under the Negative Binomial Regression Model

In capture–recapture experiments, the presence of overdispersion and heterogeneity necessitates the use of the negative binomial regression model for inferring population sizes. However, within this model, existing methods based on likelihood and ratio regression for estimating the dispersion parameter often face boundary and nonidentifiability issues. These problems can result in nonsensically large point estimates and unbounded upper limits of confidence intervals for the population size. We present a penalized empirical likelihood technique for solving these two problems by imposing a half-normal prior on the population size. Based on the proposed approach, a maximum penalized empirical likelihood estimator with asymptotic normality and a penalized empirical likelihood ratio statistic with asymptotic chi-square distribution are derived. To improve numerical performance, we present an effective expectation-maximization (EM) algorithm. In the M-step, optimization for the model parameters could be achieved by fitting a standard negative binomial regression model via the R basic function glm.nb(). This approach ensures the convergence and reliability of the numerical algorithm. Using simulations, we analyze several synthetic datasets to illustrate three advantages of our methods in finite-sample cases: complete mitigation of the boundary problem, more efficient maximum penalized empirical likelihood estimates, and more precise penalized empirical likelihood ratio interval estimates compared to the estimates obtained without penalty. These advantages are further demonstrated in a case study estimating the abundance of black bears (Ursus americanus) at the U.S. Army’s Fort Drum Military Installation in northern New York.

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.

Smoothed empirical likelihood for optimal cut point analysis

In diagnostic studies, a continuous biomarker is often dichotomized for the diagnosis of binary disease status. Various criteria have been studied for the cut point selection of the continuous biomarker in receiver operating characteristic (ROC) analysis, in particular, the Youden index, the closest-to-(0,1) index, and the concordance probability index. Recently, Wang, Tian, and Zhao (2017) established a Wilks theorem for a smoothed empirical likelihood ratio statistic of Youden index. However, it is not directly useful for statistical inference compared to the cut point. In addition, the optimal cut point may vary with different criteria. In this article, we study smoothed empirical likelihood for optimal cut point selection with Youden index, closest-to-(0,1) criterion, and concordance probability. We develop confidence estimation for the optimal cut points based on the smoothed empirical likelihood ratio statistics. We examine the empirical performance by extensive simulation studies. We also illustrate the method with a real dataset.

Adjusted empirical likelihood for probability density functions under strong mixing samples

To obtain a profile empirical likelihood ratio statistic is essentially to address a constrained maximization problem. However, when sample size is small, the optimal function may have no numerical solution, as the convex hull of the sample points may not contain the zero vector 0 as its interior point. The adjusted empirical likelihood (AEL) method introduced by Chen, Variyath, and Abraham (2008) is very useful to solve this problem. The innovation of this article is that we extend the AEL method to probability density functions (p.d.f.) under dependent samples, as there are few literatures using this method under dependent samples. It is shown that the AEL ratio statistic for a p.d.f. is asymptotically χ 2 -type distributed under a mixing sample, which is used to obtain an AEL-based confidence interval for the p.d.f. Our simulations show that the AEL is faster to compute than unadjusted empirical likelihood and the coverage probability based on the AEL method is superior to the other two usual methods, namely the unadjusted empirical likelihood and the normal approximation methods, particularly under small sample sizes.

Empirical Likelihood for a First-Order Generalized Random Coefficient Integer-Valued Autoregressive Process.

In this paper, the authors consider the empirical likelihood method for a first-order generalized random coefficient integer-valued autoregressive process. The authors establish the log empirical likelihood ratio statistic and obtain its limiting distribution. Furthermore, the authors investigate the point estimation, confidence regions and hypothesis testing for the parameters of interest. The performance of empirical likelihood method is illustrated by a simulation study and a real data example.

Empirical likelihood inference for monotone index model

This paper proposes an empirical likelihood inference method for monotone index models. We construct the empirical likelihood function based on a modified score function developed by Balabdaoui et al. (Scand J Stat 46:517–544, 2019), where the monotone link function is estimated by isotonic regression. It is shown that the empirical likelihood ratio statistic converges to a weighted chi-squared distribution. We suggest inference procedures based on an adjusted empirical likelihood statistic that is asymptotically pivotal, and a bootstrap calibration with recentering. A simulation study illustrates usefulness of the proposed inference methods.

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.

Empirical likelihood without population covariance matrix

In this paper, we show that the log empirical likelihood ratio statistic for the population mean converges in distribution to achi-squared distribution when the population is in the generalized domain of attraction of multivariate normal law.

Empirical likelihood for spatial dynamic panel data models with spatial lags and spatial errors

We study spatial dynamic panel data models with both spatial lags and spatial errors. The empirical likelihood (EL) ratio statistics are constructed for the parameters of the models. It is shown that the limiting distributions of the EL ratio statistics are chi-square distributions, which are used to construct confidence regions for the parameters of the models. A simulation study is conducted to compare the performances of the EL based and the normal approximation (NA) based confidence regions. Simulation results show that the EL can be computationally easier than the NA method to implement in practice.

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 for spatial dynamic panel data models.

Spatial dynamic panel data (SDPD) models have received great attention in economics in recent 10 years. Existing approaches for the estimation and test of SDPD models are quasi-maximum likelihood (QML) approach and generalized method of moments (GMM). In this article, we introduce the empirical likelihood (EL) method to the statistical inference for SDPD models. The EL ratio statistics are constructed for the parameters of spatial dynamic panel data models. It is shown that the limiting distributions of the empirical likelihood ratio statistics are chi-squared distributions, which are used to construct confidence regions for the parameters of the models. Simulation results show that the EL based confidence regions outperform the normal approximation based confidence regions.

A comprehensive distribution‐free scheme for tri‐aspect surveillance of complex processes

AbstractDistribution‐free charting schemes for process monitoring are more robust and reliable than their parametric counterparts when the process distributions are unknown and complicated. Most of the existing control charts are uni‐aspect schemes because they can detect a shift in only one aspect of the process distribution, like location or scale. Research on schemes for simultaneous surveillance of location and scale parameters has been very active in recent years, leading to many bi‐aspect schemes. These schemes do not explicitly deal with the shape of the process distribution. However, a shift in the shape parameter with or without a location or scale shift can occur in practice, especially in production or time‐to‐event processes. Note that there are nonparametric schemes for detecting general shifts based on goodness‐of‐fit test statistics or empirical likelihood ratio statistics, which cannot isolate if there is a shift in location, scale, shape, or combination of these parameters. This article introduces a new distribution‐free process monitoring scheme that can detect a shift in the location, scale or shape parameters, or any combinations of them. The new scheme is based on a combined statistic designed via Euclidean distance of the standardized Wilcoxon statistic for location, the standardized Ansari–Bradley statistic for scale, and the standardized Savage‐type statistic for shape. We discuss the implementation design using the average run‐length as the performance metric and investigate the proposed scheme's in‐control performance. It is shown that the new charting scheme is in‐control robust irrespective of the underlying process distribution and therefore applicable to monitor any univariate continuous processes. An out‐of‐control performance comparison study of the new scheme with many existing schemes shows that the new scheme is preferable to the existing schemes. We illustrate the proposed chart with real data to monitor a passenger train's arrival delays in Italy's regional route. The proposed scheme is comprehensive because it integrates the follow‐up procedure via integrated sub‐charts for classifying the signaling component.

Empirical Likelihood for Partially Linear Single-Index Models under Negatively Associated Errors

In this paper, the authors consider the application of the blockwise empirical likelihood method to the partially linear single-index model when the errors are negatively associated, which often exist in sequentially collected economic data. Thereafter, the blockwise empirical likelihood ratio statistic for the parameters of interest is proved to be asymptotically chi-squared. Hence, it can be directly used to construct confidence regions for the parameters of interest. A few simulation experiments are used to illustrate our proposed method.

Empirical likelihood inference for threshold autoregressive conditional heteroscedasticity model

This paper considers the parameter estimation problem of a first-order threshold autoregressive conditional heteroscedasticity model by using the empirical likelihood method. We obtain the empirical likelihood ratio statistic based on the estimating equation of the least squares estimation and construct the confidence region for the model parameters. Simulation studies indicate that the empirical likelihood method outperforms the normal approximation-based method in terms of coverage probability.

Estimation and testing of multivariate random coefficient autoregressive model based on empirical likelihood

This paper studies the empirical likelihood method for a multivariate first-order random coefficient autoregressive (RCAR(1)) model. Based on the modified least square score equation, empirical likelihood method is used to test the randomness of coefficients and estimate the unknown parameters. First, the limiting distribution of the log empirical likelihood ratio statistic is established. Second, the maximum empirical likelihood estimator is derived and its asymptotic properties are established. Finally, the performances of the estimator are compared with the modified least square estimator via simulation. Furthermore, simulation result shows that the test has the correct size and very good power for all cases considered.

Personal Privacy Protection via Irrelevant Faces Tracking and Pixelation in Video Live Streaming

To date, the privacy-protection intended pixelation tasks are still labor-intensive and yet to be studied. With the prevailing of video live streaming, establishing an online face pixelation mechanism during streaming is an urgency. In this paper, we develop a new method called Face Pixelation in Video Live Streaming (FPVLS) to generate automatic personal privacy filtering during unconstrained streaming activities. Simply applying multi-face trackers will encounter problems in target drifting, computing efficiency, and over-pixelation. Therefore, for fast and accurate pixelation of irrelevant people's faces, FPVLS is organized in a frame-to-video structure of two core stages. On individual frames, FPVLS utilizes image-based face detection and embedding networks to yield face vectors. In the raw trajectories generation stage, the proposed Positioned Incremental Affinity Propagation (PIAP) clustering algorithm leverages face vectors and positioned information to quickly associate the same person's faces across frames. Such frame-wise accumulated raw trajectories are likely to be intermittent and unreliable on video level. Hence, we further introduce the trajectory refinement stage that merges a proposal network with the two-sample test based on the Empirical Likelihood Ratio (ELR) statistic to refine the raw trajectories. A Gaussian filter is laid on the refined trajectories for final pixelation. On the video live streaming dataset we collected, FPVLS obtains satisfying accuracy, real-time efficiency, and contains the over-pixelation problems.

An Efficient Multiple Imputation Approach for Estimating Equations with Response Missing at Random and High-Dimensional Covariates

Empirical-likelihood-based inference for parameters defined by the general estimating equations of Qin and Lawless (1994) remains an active research topic. When the response is missing at random (MAR) and the dimension of covariate is not low, the authors propose a two-stage estimation procedure by using the dimension-reduced kernel estimators in conjunction with an unbiased estimating function based on augmented inverse probability weighting and multiple imputation (AIPW-MI) methods. The authors show that the resulting estimator achieves consistency and asymptotic normality. In addition, the corresponding empirical likelihood ratio statistics asymptotically follow central chi-square distributions when evaluated at the true parameter. The finite-sample performance of the proposed estimator is studied through simulation, and an application to HIV-CD4 data set is also presented.

Inference for Long-memory Time Series Models Based on Modified Empirical Likelihood

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, it experiences the undercoverage issue which causes the coverage probabilities of parameters lower than the given nominal levels, especially for small sample sizes. In this paper, we propose a modified empirical likelihood which combines the advantages of the adjusted empirical likelihood and the transformed empirical likelihood to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model for the purpose of improving coverage probabilities.Asymptotic null distribution of the test statistic has been established as the standard chi-square distribution with the degree of freedom one. Simulations have been conducted to investigate the performance of the proposed method as well as the comparisons of other existing methods to illustrate that the proposed method can provide better coverage probabilities especially for small sample sizes.

Empirical likelihood ratio for two-sample compound Poisson processes under infinite second moment

In this article, the author focus on the simple and yet very important case of making inference on the difference of two population means using the empirical likelihood approach of two sample compound Poisson process under infinite second moment. It is shown that the log empirical likelihood ratio statistic for the difference of two population means converges in distribution to as The simulation results show that the empirical likelihood ratio method is applicable under infinite second moment.

Maximum likelihood abundance estimation from capture-recapture data when covariates are missing at random.

In capture-recapture experiments, individual covariates may be subject to missingness, especially when the number of captures is small. When the covariate information is missing at random, the inverse probability weighting method and the multiple imputation method are widely used to obtain point estimators of the abundance. These estimators are then used to construct Wald-type confidence intervals. However, such intervals may have seriously inaccurate coverage probabilities. In this paper, we propose a maximum empirical likelihood (EL) estimation approach for the abundance in the presence of missing covariates. We show that the maximum EL estimator is asymptotically normal, and that the EL ratio statistic for the abundance has a chi-square limiting distribution with one degree of freedom. Simulations indicate that the proposed estimator has a smaller mean square error than existing estimators, and the proposed EL ratio confidence interval usually has more accurate coverage probabilities than the existing Wald-type confidence intervals. We illustrate the proposed method by analyzing data collected in Hong Kong for the yellow-bellied prinia, a birdspecies.