Empirical Likelihood Estimator Research Articles

An approximated exponentially tilted empirical likelihood estimator of moment condition models

. This study proposes an approximated estimator for moment condition models. Our estimator approximates the exponentially tilted empirical likelihood (ETEL) estimator in Schennach (2007) by using a second-order approximation of implied probabilities (IPs) for the exponential tilting (ET). The resulting approximated ETEL estimator generates positive IPs. As the nested optimization of ETEL and EL is avoided, it is computationally simple. It is second-order asymptotically equivalent to the EL estimator. In particular, it has the same O(n −1) bias as EL. It also has the same O(n −2) variance as EL. Like ET and ETEL estimators, it is n convergent under model misspecification, while the EL estimator may not be.

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.

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.

Orthogonality-based bias-corrected empirical likelihood inference for partial linear varying coefficient EV models with longitudinal data

In this paper, we study the model estimation for the partial linear varying coefficient errors-in-variables (EV) models with longitudinal data. Based on the empirical likelihood and quadratic inference functions, we propose an orthogonality-based bias-corrected empirical likelihood estimation method using the QR decomposition method of matrix. The proposed method can handle the measurement errors of covariates and unknown within-subject correlation simultaneously, estimate the parametric and nonparametric components separately. Under some regularity conditions, the resulting estimator of the parametric component is asymptotic normal and the estimators of the nonparametric varying coefficients achieve the optimal convergence rate. Furthermore, we conduct simulation studies and a real data analysis to compare the performance of the proposed method with existing methods in finite samples. The results show that the feasibility of the proposed method is obvious.

Sampling Importance Resampling Algorithm with Nonignorable Missing Response Variable Based on Smoothed Quantile Regression

The presence of nonignorable missing response variables often leads to complex conditional distribution patterns that cannot be effectively captured through mean regression. In contrast, quantile regression offers valuable insights into the conditional distribution. Consequently, this article places emphasis on the quantile regression approach to address nonrandom missing data. Taking inspiration from fractional imputation, this paper proposes a novel smoothed quantile regression estimation equation based on a sampling importance resampling (SIR) algorithm instead of nonparametric kernel regression methods. Additionally, we present an augmented inverse probability weighting (AIPW) smoothed quantile regression estimation equation to reduce the influence of potential misspecification in a working model. The consistency and asymptotic normality of the empirical likelihood estimators corresponding to the above estimating equations are proven under the assumption of a correctly specified parameter working model. Furthermore, we demonstrate that the AIPW estimation equation converges to an IPW estimation equation when a parameter working model is misspecified, thus illustrating the robustness of the AIPW estimation approach. Through numerical simulations, we examine the finite sample properties of the proposed method when the working models are both correctly specified and misspecified. Furthermore, we apply the proposed method to analyze HIV—CD4 data, thereby exploring variations in treatment effects and the influence of other covariates across different quantiles.

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.

Augmented two-step estimating equations with nuisance functionals and complex survey data

Summary Statistical inference in the presence of nuisance functionals with complex survey data is an important topic in social and economic studies. The Gini index, Lorenz curves, and quantile shares are among the commonly encountered examples. The nuisance functionals are usually handled by a plug-in nonparametric estimator and the main inferential procedure can be carried out through a two-step generalized empirical likelihood method. Unfortunately, the resulting inference is not efficient and the nonparametric version of the Wilks’ theorem breaks down even under simple random sampling. We propose an augmented estimating equations method with nuisance functionals and complex surveys. The second step augmented estimating functions obey the Neyman orthogonality condition and automatically handle the impact of the first step plug-in estimator, and the resulting estimator of the main parameters of interest is invariant to the first step method. More importantly, the generalized empirical likelihood-based Wilks’ theorem holds for the main parameters of interest under the design-based framework for commonly used survey designs, and the maximum generalized empirical likelihood estimators achieve the semiparametric efficiency bound. Performances of the proposed methods are demonstrated through simulation studies and an application using the dataset from the New York City Social Indicators Survey.

Robust penalized empirical likelihood estimation method for linear regression

Maximum likelihood estimation is a popular method for parameter estimation in regression models. However, since in some data sets it may not be possible to make any distributional assumptions on the error term, the likelihood method cannot be used to estimate the parameters of interest. For those data sets, one can use the empirical likelihood estimation method to estimate the parameters of a linear regression model. The aim of this study is to propose a robust penalized empirical likelihood estimation method to estimate the regression parameters and select significant variables, simultaneously, for data scenarios for which a well-defined likelihood function may not be available. This will be achieved by combining a robust empirical estimation method and the bridge penalty function. We investigate the asymptotic properties of the proposed estimator and explore the finite sample behaviour with a simulation study and a real data example.

A comparison of different methods to adjust survival curves for confounders.

Treatment specific survival curves are an important tool to illustrate the treatment effect in studies with time-to-event outcomes. In non-randomized studies, unadjusted estimates can lead to biased depictions due to confounding. Multiple methods to adjust survival curves for confounders exist. However, it is currently unclear which method is the most appropriate in which situation. Our goal is to compare forms of inverse probability of treatment weighting, the G-Formula, propensity score matching, empirical likelihood estimation and augmented estimators as well as their pseudo-values based counterparts in different scenarios with a focus on their bias and goodness-of-fit. We provide a short review of all methods and illustrate their usage by contrasting the survival of smokers and non-smokers, using data from the German Epidemiological Trial on Ankle-Brachial-Index. Subsequently, we compare the methods using a Monte-Carlo simulation. We consider scenarios in which correctly or incorrectly specified models for describing the treatment assignment and the time-to-event outcome are used with varying sample sizes. The bias and goodness-of-fit is determined by taking the entire survival curve into account. When used properly, all methods showed no systematic bias in medium to large samples. Cox regression based methods, however, showed systematic bias in small samples. The goodness-of-fit varied greatly between different methods and scenarios. Methods utilizing an outcome model were more efficient than other techniques, while augmented estimators using an additional treatment assignment model were unbiased when either model was correct with a goodness-of-fit comparable to other methods. These "doubly-robust" methods have important advantages in every considered scenario.

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.

Portfolio Efficiency Tests with Conditioning Information-Comparing GMM and GEL Estimators.

We evaluate the use of generalized empirical likelihood (GEL) estimators in portfolio efficiency tests for asset pricing models in the presence of conditional information. The use of conditional information is relevant to portfolio management as it allows for checking whether asset allocations are efficiently exploiting all the information available in the market. Estimators from the GEL family present some optimal statistical properties, such as robustness to misspecifications and better properties in finite samples. Unlike generalized method of moments (GMM) estimators, the bias for GEL estimators does not increase with the number of moment conditions included, which is expected in conditional efficiency analysis. Due to these better properties in finite samples, our main hypothesis is that portfolio efficiency tests using GEL estimators may have better properties in terms of size, power, and robustness. Using Monte Carlo experiments, we show that GEL estimators have better performance in the presence of data contaminations, especially under heavy tails and outliers. Extensive empirical analyses show the properties of the estimators for different sample sizes and portfolio types for two asset pricing models.

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.

Improved semiparametric estimation of the proportional rate model with recurrent event data.

Owing to its robustness properties, marginal interpretations, and ease of implementation, the pseudo-partial likelihood method proposed in the seminal papers of Pepe and Cai and Lin etal. has become the default approach for analyzing recurrent event data with Cox-type proportional rate models. However, the construction of the pseudo-partial score function ignores the dependency among recurrent events and thus can be inefficient. An attempt to investigate the asymptotic efficiency of weighted pseudo-partial likelihood estimation found that the optimal weight function involves the unknown variance-covariance process of the recurrent event process and may not have closed-form expression. Thus, instead of deriving the optimal weights, we propose to combine a system of pre-specified weighted pseudo-partial score equationsvia the generalized method of moments and empirical likelihood estimation. We show that a substantial efficiency gain can be easily achieved without imposing additional model assumptions. More importantly, the proposed estimation procedures can be implemented with existing software. Theoretical and numerical analyses show that the empirical likelihood estimator is more appealing than the generalized method of moments estimator when the sample size is sufficiently large. An analysis of readmission risk in colorectal cancer patients is presented to illustrate the proposedmethodology.

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.

Empirical likelihood inference for longitudinal data with covariate measurement errors: An application to the LEAN study

Measurement errors usually arise during the longitudinal data collection process. Ignoring the effects of measurement errors will lead to invalid estimates. The Lifestyle Education for Activity and Nutrition (LEAN) study was designed to assess the effectiveness of intervention for enhancing weight loss over nine months. The covariates systolic blood pressure (SBP) and diastolic blood pressure (DBP) were measured at baseline, month 4, and month 9. At each assessment time, there were two replicate measurements for SBP and DBP. The replicate measurement errors of SBP follow different distributions, as does DBP. To account for the distributional difference of replicate measurement errors, a new method for analyzing longitudinal data with replicate covariate measurement errors is developed based on the empirical likelihood method. The asymptotic properties of the proposed estimator are established under some regularity conditions. The confidence region for the parameters of interest can be constructed based on the chi-squared approximation without estimating the covariance matrix. Additionally, the proposed empirical likelihood estimator is asymptotically more efficient than the estimator of Lin et al. (2018). Extensive simulations demonstrate that the proposed method can eliminate the effects of measurement errors in the covariate and has a high estimation efficiency. The proposed method indicates the significant effect of the intervention on BMI in the LEAN study.

Culling the Herd of Moments with Penalized Empirical Likelihood

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly agnostic about moment selection. While a large pool of valid moments can potentially improve estimation efficiency, in the meantime a few invalid ones may undermine consistency. This article investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and establish its oracle property with consistent detection of invalid moments. The PEL estimator is asymptotically normally distributed, and a projected PEL procedure further eliminates its asymptotic bias and provides more accurate normal approximation to the finite sample behavior. Simulation exercises demonstrate excellent numerical performance of these methods in estimation and inference.

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.

A higher-order correct fast moving-average bootstrap for dependent data

We develop the theory of a novel fast bootstrap for dependent data. Our scheme deploys i.i.d. resampling of smoothed moment indicators. We characterize the class of parametric and semiparametric estimation problems for which the method is valid. We show the asymptotic refinements of the new procedure, proving that it is higher-order correct under mild assumptions on the time series, the estimating functions, and the smoothing kernel. We illustrate the applicability and the advantages of our procedure for M-estimation, generalized method of moments, and generalized empirical likelihood estimation. In a Monte Carlo study, we consider an autoregressive conditional duration model and we compare our method with other extant, routinely-applied first- and higher-order correct methods. The results provide numerical evidence that the novel bootstrap yields higher-order accurate confidence intervals, while remaining computationally lighter than its higher-order correct competitors. A real-data example on dynamics of trading volume of US stocks illustrates the empirical relevance of our method.

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.