Empirical Likelihood Method Research Articles

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACTIn this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.

Jackknife empirical likelihood for comparing two Gini indices

AbstractThe focus of this paper is to derive the jackknife empirical likelihood for the difference of two Gini indices. For independent data we propose a novel U‐statistic, which allows direct utilization of the jackknife empirical likelihood without involving a nuisance parameter. For paired data we established Wilks’ theorem for the profile likelihood after maximization over the nuisance parameter. Simulation studies show that our method is comparable to existing empirical likelihood methods in terms of coverage accuracy, but obtains much shorter intervals. The proposed methods are illustrated via analyzing a real data set.The Canadian Journal of Statistics44: 102–119; 2016 © 2016 Statistical Society of Canada

A hybrid parametric and empirical likelihood model for evaluating interactions in case-control Studies.

The case-control design provides an effective way to collect covariate information conditioning on subjects' disease status. The standard logistic regression model can be used to model the interaction between two covariates under such a design, but the prospective logistic regression method might not be the most efficient one when certain appropriate constraints can be imposed on the covariate distribution. We develop a hybrid approach for the statistical inference of the interaction under the case-control design. We use a parametric model to characterize the conditional distribution of one covariate given the another covariate in the control population, while leaving the distribution of the later covariate to be fully nonparametric. A maximum hybrid parametric and empirical likelihood method is adopted for the evaluation of all parameters. The estimator and the associated test derived from the proposed semiparametric model are suitable for evaluating the interaction between two covariates of various types (discrete or continuous). Asymptotic results for both the estimators and the test statistics were established, and the advantages of the proposed method over the existing ones are demonstrated through simulation results and a real data example.

Doubly robust and locally efficient estimation with missing outcomes

We consider parametric regression where the outcome is subject to miss- ingness. To achieve the semiparametric efficiency bound, most existing estimation methods require the correct modeling of certain second moments of the data, which can be very challenging in practice. We propose an estimation procedure based on the conditional empirical likelihood (CEL) method. Our method does not require us to model any second moments. We study the CEL-based inverse probability weighted (CEL-IPW) and augmented inverse probability weighted (CEL-AIPW) estimators in detail. Under some regularity conditions and the missing at ran- dom (MAR) mechanism, the CEL-IPW estimator is consistent if the missingness mechanism is correctly modeled, and the CEL-AIPW estimator is consistent if either the missingness mechanism or the conditional mean of the outcome is cor- rectly modeled. When both quantities are correctly modeled, the CEL-AIPW es- timator attains the semiparametric efficiency bound without modeling any second moments. The asymptotic distributions are derived. Numerical implementation through nested optimization routines using the Newton-Raphson algorithm is dis- cussed. We study the problem of parametric regression when the outcome is subject to missingness. The central interest is the estimation and inference of the re- gression coefficients. In practice there are many reasons that can lead to missing outcomes: budget or technique restrictions, subjects' failure to comply with the protocol, or simply the study design. Missing data usually bring big challenges to estimation and inference, as the application of statistical methods developed for data without missing values can lead to biased estimation and misleading conclusions. In addition to the outcome and covariates, we assume that some auxiliary variables are available for all subjects. Although the auxiliary variables are not of direct statistical interest, they may help to explain the missingness mecha- nism, and thus reduce the impact of missing data on estimation and inference.

Empirical Likelihood Inference for First-Order Random Coefficient Integer-Valued Autoregressive Processes

We apply the empirical likelihood method to estimate the variance of random coefficient in the first-order random coefficient integer-valued autoregressive (RCINAR(1)) processes. The empirical likelihood ratio statistic is derived and some asymptotic theory for it is presented. Furthermore, a simulation study is presented to demonstrate the performance of the proposed method.

Stress testing correlation matrix: a maximum empirical likelihood approach

ABSTRACTStress testing correlation matrix is a challenging exercise for portfolio risk management. Most existing methods directly modify the estimated correlation matrix to satisfy stress conditions while maintaining positive semidefiniteness. The focus lies on technical optimization issues but the resultant stressed correlation matrices usually lack statistical interpretations. In this article, we suggest a novel approach using Empirical Likelihood method to modify the probability weights of sample observations to construct a stressed correlation matrix. The resultant correlations correspond to a stress scenario that is nearest to the observed scenario in a Kullback–Leibler divergence sense. Besides providing a clearer statistical interpretation, the proposed method is non-parametric in distribution, simple in computation and free from subjective tunings. We illustrate the method through an application to a portfolio of international assets.

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.

Exponentially tilted empirical distribution function for ranked set samples

We study nonparametric estimation of the distribution function (DF) of a continuous random variable based on a ranked set sampling design using the exponentially tilted (ET) empirical likelihood method. We propose ET estimators of the DF and use them to construct new resampling algorithms for unbalanced ranked set samples. We explore the properties of the proposed algorithms. For a hypothesis testing problem about the underlying population mean, we show that the bootstrap tests based on the ET estimators of the DF are asymptotically normal and exhibit a small bias of order O(n−1). We illustrate the methods and evaluate the finite sample performance of the algorithms under both perfect and imperfect ranking schemes using a real data set and several Monte Carlo simulation studies. We compare the performance of the test statistics based on the ET estimators with those based on the empirical likelihood estimators.

Jackknife empirical likelihood confidence interval for the Gini index

Jackknife empirical likelihood for the Gini index is derived. Adjusted jackknife empirical likelihood and bootstrap calibration are further investigated. The resulting interval estimators are comparable to existing empirical likelihood methods in terms of coverage accuracy, but yield much shorter intervals.

Two-sample extended empirical likelihood for estimating equations

We propose a two-sample extended empirical likelihood for inference on the difference between two p -dimensional parameters defined by estimating equations. The standard two-sample empirical likelihood for the difference is Bartlett correctable but its domain is a bounded subset of the parameter space. We expand its domain through a composite similarity transformation to derive the two-sample extended empirical likelihood which is defined on the full parameter space. The extended empirical likelihood has the same asymptotic distribution as the standard one and can also achieve the second-order accuracy of the Bartlett correction. We include two applications to illustrate the use of two-sample empirical likelihood methods and to demonstrate the superior coverage accuracy of the extended empirical likelihood confidence regions.

Spatial median depth-based robust adjusted empirical likelihood

Empirical likelihood (EL) based inference for parameters defined by general estimating equations of Qin and Lawless [(1994), ‘Empirical Likelihood and General Estimating Equations’,The Annals of Statistics, 22, 300–325] remains an active research topic. However, the performance of the EL method can be hindered by non-robustness and empty set problems. In this paper, we propose a robust adjusted empirical likelihood (RAEL) to address these two problems simultaneously. The resulting RAEL ratio statistic is shown to have inherited the asymptotic properties of both the robust empirical likelihood and the adjusted empirical likelihood. The finite-sample performance of the proposed method is illustrated by simulation and two real-data examples are also presented.

Empirical Likelihood for Outlier Detection and Estimation in Autoregressive Time Series

Identification and estimation of outliers in time series is proposed by using empirical likelihood methods. Theory and applications are developed for stationary autoregressive models with outliers distinguished in the usual additive and innovation types. Some other useful outlier types are considered as well. A simulation experiment is used for studying the behaviour of the empirical likelihood‐based method in finite samples and indicates that the proposed methods are preferable when dealing with the non‐Gaussian data. Our simulations suggest that the usual sequential procedure for multiple outlier detection is suitable also for the methods based on empirical likelihood.

Efficient parameter estimation via Gaussian copulas for quantile regression with longitudinal data

Specifying a correlation matrix is challenging in quantile regression with longitudinal data. A naive method is simply to adopt an independence working model. However, the efficiency of parameter estimates may be lost. We propose constructing a working correlation matrix via Gaussian copula which can handle or incorporate general serial dependence. A suit of unbiased estimating functions can be obtained by assuming the Gaussian copula with different correlation matrices, and the empirical likelihood method can then combine these unbiased estimating functions. Furthermore, the induced smoothing approach is applied to the discontinuous estimating functions to reduce computation burdens. The asymptotic normality of the resulting estimators is established. Simulation studies indicate that the proposed method is superior to the alternative estimating functions especially when the working correlation matrix is misspecified. Finally, a real dataset from forced expiratory volume study is used to illustrate the proposed method.

Jackknife empirical likelihood for linear transformation models with right censoring

A class of linear transformation models with censored data was proposed as a generalization of Cox models in survival analysis. This paper develops inference procedure for regression parameters based on jackknife empirical likelihood approach. We can show that the limiting variance is not necessary to estimate and the Wilk’s theorem can be obtained. The jackknife empirical likelihood method benefits from the simpleness in optimization using jackknife pseudo-value. In our simulation studies, the proposed method is compared with the traditional empirical likelihood and normal approximation methods in terms of coverage probability and computational cost.

Empirical likelihood-based inference in Poisson autoregressive model with conditional moment restrictions

In this paper, we employ the empirical likelihood method to estimate the unknown parameters in Poisson autoregressive model in the presence of auxiliary information. It is shown that our approach proposed, compared to the maximum likelihood estimator, the least squares estimator, and the weighted least squares estimator, yields more efficient estimators. Some simulation studies are also conducted to investigate the finite sample performance of the proposed method.

Interval estimation for a measure of tail dependence

Systemic risk concerns extreme co-movement of several financial variables, which involves characterizing tail dependence. The coefficient of tail dependence was proposed by Ledford and Tawn (1996, 1997) to distinguish asymptotic independence and asymptotic dependence. Recently a new measure based on the conditional Kendall’s tau was proposed by Asimit et al. (2015) to measure the tail dependence and to distinguish asymptotic independence and asymptotic dependence. For effectively constructing a confidence interval for this new measure, this paper proposes a smooth jackknife empirical likelihood method, which does not need to estimate any additional quantities such as asymptotic variance. A simulation study shows that the proposed method has a good finite sample performance.

Jackknife empirical likelihood inference for the mean absolute deviation

In statistics mean absolute deviation plays an important role in measuring spread of a data. In this paper, we focus on using the jackknife, the adjusted and the extended jackknife empirical likelihood methods to construct confidence intervals for the mean absolute deviation of a random variable. The empirical log-likelihood ratio statistic is derived whose asymptotic distribution is a standard chi-square distribution. The results of simulation study show the comparison of the average length and coverage probability by using jackknife empirical likelihood methods and normal approximation method. The proposed adjusted and extended jackknife empirical likelihood methods perform better than other methods, in particular for skewed distributions. We use real data sets to illustrate the proposed jackknife empirical likelihood methods.

Empirical entropy for right censored data

The maximum entropy method has been widely used in many fields, such as statistical mechanics, economics, etc. Its crucial idea is that when we make inference based on partial information, we must use the distribution with maximum entropy subject to whatever is known. In this paper, we investigate the empirical entropy method for right censored data and use simulation to compare the empirical entropy method with the empirical likelihood method. Simulations indicate that the empirical entropy method gives better coverage probability than that of the empirical likelihood method for contaminated and censored lifetime data.

Accelerated failure time model with quantile information

The censored linear regression model, also referred to as the accelerated failure time model, is a useful alternative to the popular Cox model in the analysis of censored survival data. In this paper, we combine the quantile information with censored least-squares normal equations to get estimators with smaller estimated standard error for regression parameters. An inverse probability-weighted method is proposed to construct unbiased estimating equations with censored data and the lack of smoothness of the objective equations is overcome by replacing them with smooth approximations. The proposed estimators are established based on the empirical likelihood method and generalized method of moments, respectively, and their asymptotic properties are studied under some regular conditions. We also conduct some simulation experiments to investigate the finite-sample properties of the proposed estimators. The Stanford Heart Transplant data are used to illustrate the proposed estimating method.

Empirical likelihood methods for discretely observed Gaussian moving averages

This paper is concerned with parametric estimation, model specification and autocorrelation diagnosis for stationary moving averages driven by a Wiener process. By incorporating the analysis of the spectral densities of the discretely observed trajectory, empirical likelihood methods based on moment conditions are developed to the dependent sequences in this paper for estimation and test. Theoretical properties of the empirical likelihood estimator for parameters are provided. Empirical likelihood ratio tests for model specification of the moving averages are proposed by means of the bootstrap strategy. Simulation and empirical case studies are carried out to confirm the effectiveness of the proposed estimation and test.