Semiparametric empirical likelihood inference with estimating equations under density ratio 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.

SummaryIn this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function. The finite‐sample performance of the method is evaluated through a simulation study.

ABSTRACTBased on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.

This work proposes a Bayesian approach for the analysis of a semiparametric density ratio model, a model useful for the integration of data from multiple sources. The proposed Bayesian analysis uses a non‐parametric likelihood and a transformed Gaussian prior for the “non‐parametric part” of the model. The former choice guarantees the validity of the Bayesian analysis in contrast to some semiparametric Bayesian analyses that rely on empirical likelihoods whereas the latter choice allows the representation of an expected smoothness property. We describe a Markov chain Monte Carlo algorithm to fit this model which was found to empirically display good convergence behaviour. The model is illustrated with the analysis of motor vibration data obtained from three different locations on a motor. The Canadian Journal of Statistics 45: 274–289; 2017 © 2017 Statistical Society of Canada

In this article, we employ the empirical likelihood method to construct confidence intervals for nonparametric regression functions with functional stationary ergodic data. Under some wild conditions, it is proved that the empirical likelihood ratio statistic is asymptotically chi-squared distributed.

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.

Receiver operating characteristic (ROC) curve is often used to study and compare two-sample problems in medicine. When more information may be available on one treatment than the other, one can improve estimator of ROC curve if the auxiliary population information is taken into account. The authors show that the empirical likelihood method can be naturally adapted to make efficient use of the auxiliary information to such problems. The authors propose a smoothed empirical likelihood estimator for ROC curve with some auxiliary information in medical studies. The proposed estimates are more efficient than those ROC estimators without any auxiliary information, in the sense of comparing asymptotic variances and mean squared error (MSE). Some asymptotic properties for the empirical likelihood estimation of ROC curve are established. A simulation study is presented to demonstrate the performance of the proposed estimators.

EMLK2D: a computer program for spatial estimation using empirical maximum likelihood kriging

Testing homogeneity for multiple nonnegative distributions with excess zero observations

This paper addresses the problem of hypothesis test on response mean with various inequality constraints in the presence of covariates when response data are missing at random. The various hypotheses include to test single point, two points, set of inequalities as well as two-sided set of inequalities of the response mean. The test statistics is constructed by the weighted-corrected empirical likelihood function of the response mean based on the approach of weighted-corrected imputation for the response variable. We investigate limiting distributions and asymptotic powers of the proposed empirical likelihood ratio test statistics with auxiliary information. The results show that the test statistics with auxiliary information is more efficient than that without auxiliary information. A simulation study is undertaken to investigate the finite sample performance of the proposed method.

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

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.

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

For panel data models with spatial errors, the empirical likelihood ratio statistics are constructed for the parameters of the 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. A simulation study is conducted to show the performance of the proposed method.

Empirical likelihood for partially linear proportional hazards models with growing dimensions