Empirical likelihood based inference for the derivative of the nonparametric regression function

Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.

Empirical likelihood ratio confidence interval estimation of best linear combinations of biomarkers

The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data.

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.

Measuring the difference between two Gini indexes is important in comparing the inequality between two groups. This paper systematically provides confidence intervals for this difference. First, normal approximation- and bootstrap-based confidence intervals are provided for the difference. Second, empirical likelihood (EL)-based confidence intervals are proposed according to a ratio statistic that is proven to have a weighted Chi-squared distribution with one degree of freedom. Third, two calibration approaches are established to improve the undercoverage issue of the EL method; these two approaches are augmented EL and bootstrap-calibrated EL. Monte Carlo simulations show that bootstrap-calibrated EL generally outperforms other methods in constructing confidence intervals. In the simulations, these methods are compared with the permutation method to test the equality of two Gini indexes. Lastly, these methods are applied to real data.

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.

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 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.

Summary High-dimensional statistical inference with general estimating equations is challenging and remains little explored. We study two problems in the area: confidence set estimation for multiple components of the model parameters, and model specifications tests. First, we propose to construct a new set of estimating equations such that the impact from estimating the high-dimensional nuisance parameters becomes asymptotically negligible. The new construction enables us to estimate a valid confidence region by empirical likelihood ratio. Second, we propose a test statistic as the maximum of the marginal empirical likelihood ratios to quantify data evidence against the model specification. Our theory establishes the validity of the proposed empirical likelihood approaches, accommodating over-identification and exponentially growing data dimensionality. Numerical studies demonstrate promising performance and potential practical benefits of the new methods.

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.

For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. Owen showed that empirical likelihood ratio statistics for various parameters $\theta(F)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods. Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.

For regression analysis of doubly truncated data, we propose two empirical likelihood (EL) inference approaches, called non-smooth EL and non-smooth Jackknife EL (JEL), to make inference about regression parameters based on the generalized estimating equations of existing weighted rank estimators. The limiting distributions of non-smooth log-EL and log-JEL ratios statistics are derived and non-smooth EL, and JEL confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed approaches with the random weighting (RW) approach. The simulation results demonstrate that the non-smooth EL and JEL confidence intervals have better performances than the RW confidence intervals based on coverage probability and average length of confidence intervals of regression parameters when the dependent variable is subject to the double truncation. A real data example is provided to illustrate the proposed approaches.

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.