Empirical likelihood based inference for semiparametric varying coefficient partially linear models with error-prone linear covariates

The purpose of this paper is to incorporate semiparametric alternatives to maximum likelihood estimation and inference in the context of unordered multinomial response data when in practice there is often insufficient information to specify the parametric form of the function linking the observables to the unknown probabilities. We specify the function linking the observables to the unknown probabilities using a very general flexible class of functions belonging to the Pearson system of cumulative distribution equations. In this setting we consider the observations as arising from a multinomial distribution characterized by one of the CDFs in the Pearson system. Given this situation, it is possible to utilize the concept of unbiased estimating functions (EFs), combined with the concept of empirical likelihood (EL) to define an (empirical) likelihood function for the parameter vector based on a nonparametric representation of the sample’s PDF. This leads to the concept of maximum empirical likelihood (MEL) estimation and inference, which is analogous to parametric maximum likelihood methods in many respects.

By introducing the notion of “empirical likelihood function of observing sums”, unequally-spaced time series data and longitudinal data generated from Levy processes can be analyzed. Characteristic function is further incorporated to handle the situations where the density function of the increments is difficult to obtain. In the situations where both characteristic function and the density function are available, it is shown through the simulation examples that the proposed empirical maximum likelihood method does not suffer from significant information loss comparing to the maximum likelihood estimation method. The performances of the proposed method is tested for both equally-spaced and unequally-spaced observations.

In this paper, a partially linear model under longitudinal data is considered. In order to take into consideration the within-subject correlation structure of the repeated measurements, an empirical likelihood incorporating the correlation structure is developed. The asymptotic normality of the maximum empirical likelihood estimates of the regression coefficients is obtained. It also can be shown that the proposed empirical likelihood ratio is asymptotically standard chi-square. The results can be used directly to construct the asymptotic confidence regions of the regression coefficients. The convergence rate of the baseline function is derived.

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 semi-varying coefficient models for panel data with fixed effects

Efficient empirical-likelihood-based inferences for the single-index model

This article studies the empirical likelihood method for long‐memory time series models. By virtue of the Whittle likelihood, one obtains a score function that can be viewed as an estimating equation of the parameters of a fractional integrated autoregressive moving average (ARFIMA) model. This score function is used to obtain an empirical likelihood ratio which is shown to be asymptotically chi‐square distributed. Confidence regions for the parameters are constructed based on the asymptotic distribution of the empirical likelihood ratio. Bartlett correction and finite sample properties of the empirical likelihood confidence regions are examined.

Empirical likelihood for single-index models

In this paper, we consider how to yield a robust empirical likelihood estimation for regression models. After introducing modal regression, we propose a novel empirical likelihood method based on modal regression estimation equations, which has the merits of both robustness and high inference efficiency compared with the least square based methods. Under some mild conditions, we show that Wilks’ theorem of the proposed empirical likelihood approach continues to hold. Advantages of empirical likelihood modal regression as a nonparametric approach are illustrated by constructing confidence intervals/regions. Two simulation studies and a real data analysis confirm our theoretical findings.

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.

GEL Estimation for Semi-Strong Non-Linear GARCH with Robust Empirical Likelihood Inference

Empirical Likelihood for Partially Linear Models

Summary This paper is concerned with empirical likelihood inference on the population mean when the dimension $p$ and the sample size $n$ satisfy $p/n\rightarrow c\in [1,\infty)$. As shown in Tsao (2004), the empirical likelihood method fails with high probability when $p/n>1/2$ because the convex hull of the $n$ observations in $\mathbb{R}^p$ becomes too small to cover the true mean value. Moreover, when $p> n$, the sample covariance matrix becomes singular, and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio. To deal with these two challenges, we propose a new strategy of adding two artificial data points to the observed data. We establish the asymptotic normality of the proposed empirical likelihood ratio test. The proposed test statistic does not involve the inverse of the sample covariance matrix. Furthermore, its form is explicit, so the test can easily be carried out with low computational cost. Our numerical comparison shows that the proposed test outperforms some existing tests for high-dimensional mean vectors in terms of power. We also illustrate the proposed procedure with an empirical analysis of stock data.

In this article we study inference in parametric–nonparametric errors-in-covariables regression models using an empirical likelihood approach based on validation data. It is shown that the asymptotic behavior of the proposed estimator depends on the ratio of the sizes of the primary sample and the validation sample, respectively. Unlike cases without measurement errors, the limit distribution of the estimator is no longer tractable and cannot be used for constructing confidence regions. Monte Carlo approximations are employed to simulate the limit distribution. To increase the coverage accuracy of confidence regions, two adjusted empirical likelihood estimators are recommended, which in the limit have a standard chi-squared distribution. A simulation study is carried out to compare the proposed methods with other existing methods. The new methods outperform the least squares method, and one of them works better than simulation–extrapolation (SIMEX) estimation, even when the restrictive model assumptions needed for SIMEX are satisfied. An application to a real dataset illustrates our new approach.

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.