Semiparametric Empirical Likelihood Research Articles

Two-step semiparametric empirical likelihood inference from capture–recapture data with missing covariates

Two-step semiparametric empirical likelihood inference from capture–recapture data with missing covariates

Approximate Bayesian computation with semiparametric density ratio model

Approximate Bayesian computation (ABC) is a likelihood-free inference method commonly employed for statistical inference in models with unknown or complex likelihood functions. ABC estimates the posterior distributions of model parameters by comparing summary statistics of the simulated and observed data. However, the selection of informative summary statistics can be challenging in practice. In this study, we propose a summary-statistic-free method called ABC with semiparametric empirical likelihood ratio (ABC-SELR). This method utilises the empirical likelihood within the framework of the semiparametric density ratio model to assess the simulated parameters via testing whether the simulated and observed data conform to the same distribution. By eliminating the need for choosing summary statistics, our approach avoids potential information loss. Furthermore, the proposed method captures the shared characteristics between the simulated and observed data, leading to improved efficiency. We demonstrate the accuracy and efficiency of ABC-SELR by comparing with existing ABC methods through numerical simulations.

Semiparametric empirical likelihood inference for abundance from one-inflated capture-recapture data.

Abundance estimation from capture-recapture data is of great importance in many disciplines. Analysis of capture-recapture data is often complicated by the existence of one-inflation and heterogeneity problems. Simultaneously taking these issues into account, existing abundance estimation methods are usually constructed on the basis of conditional likelihood under one-inflated zero-truncated count models. However, the resulting Horvitz-Thompson-type estimators may be unstable, and the resulting Wald-type confidence intervals may exhibit severe undercoverage. In this paper, we propose a semiparametric empirical likelihood (EL) approach to abundance estimation under one-inflated binomial and Poisson regression models. To facilitate the computation of the EL method, we develop an expectation-maximization algorithm. We also propose a new score test for the existence of one-inflation and prove its asymptotic normality. Our simulation studies indicate that compared with existing estimators, the proposed score test is more powerful and the maximum EL estimator has a smaller mean square error. The advantages of our approaches are further demonstrated by analyses of prinia data from Hong Kong and drug user data fromBangkok.

Semiparametric empirical likelihood inference with estimating equations under density ratio models

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.

A semi-parametric empirical likelihood approach for conditional estimating equations under endogenous selection

The estimation and inference for conditional estimating equations models with endogenous selection, are considered. The approach takes into account possible endogenous selection which may lead to a selection bias. It can be used for a wide range of statistical models not covered by the model-based sampling theory. Endogeneity can be either part of the selection or within the covariates. It is particularly well suited for models with unknown heteroscedasticity, uncontrolled confounders and measurement errors. It will not be necessary to model the relationship between the endogenous covariates and the instrumental variables, which offers major advantages over two-stage least-squares. The approach proposed has the advantage of being based on a fixed number of constraints determined by the size of the parameter.

Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

ABSTRACT Due to cost-effectiveness and high efficiency, two-phase case-control sampling has been widely used in epidemiology studies. We develop a semi-parametric empirical likelihood approach to two-phase case-control data under the logistic regression model. We show that the maximum empirical likelihood estimator has an asymptotically normal distribution, and the empirical likelihood ratio follows an asymptotically central chi-square distribution. We find that the maximum empirical likelihood estimator is equal to Breslow and Holubkov (1997)'s maximum likelihood estimator. Even so, the limiting distribution of the likelihood ratio, likelihood-ratio-based interval, and test are all new. Furthermore, we construct new Kolmogorov–Smirnov type goodness-of-fit tests to test the validation of the underlying logistic regression model. Our simulation results and a real application show that the likelihood-ratio-based interval and test have certain merits over the Wald-type counterparts and that the proposed goodness-of-fit test is valid.

Empirical likelihood meta-analysis with publication bias correction under Copas-like selection model

Meta-analysis is commonly used to synthesize multiple results from individual studies. However, its validation is usually threatened by publication bias and between-study heterogeneity, which can be captured by the Copas selection model. Existing inference methods under this model are all based on conditional likelihood and may not be fully efficient. In this paper, we propose a full likelihood approach to meta-analysis by integrating the conditional likelihood and a marginal semi-parametric empirical likelihood under a Copas-like selection model. We show that the maximum likelihood estimators (MLE) of all the underlying parameters have a jointly normal limiting distribution, and the full likelihood ratio follows an asymptotic central chi-square distribution. Our simulation results indicate that compared with the conditional likelihood method, the proposed MLEs have smaller mean squared errors and the full likelihood ratio confidence intervals have more accurate coverage probabilities. A real data example is analyzed to show the advantages of the full likelihood method over the conditional likelihood method.

Diagnostic test meta-analysis by empirical likelihood under a Copas-like selection model

The validation of diagnostic test meta-analysis is often threatened by publication bias, which can be commonly characterized by the Copas selection model. Under this model, conventional approaches to diagnostic meta-analysis are based on conditional likelihood. Since they may have efficiency loss, we propose a full likelihood diagnostic meta-analysis method by integrating the usual conditional likelihood and a marginal semi-parametric empirical likelihood. We show that the resulting maximum likelihood estimators (MLEs) have a jointly normal limiting distribution, and the resulting likelihood ratio follows a central chisquare limiting distribution. Our numerical studies indicate that the proposed MLEs often have smaller mean square errors than the conditional likelihood MLEs. The full likelihood ratio interval estimators generally have more accurate coverage probabilities than the conditional-likelihood-based Wald intervals. We re-study two real meta analyses on influenza and mental health respectively for illustration.

Two-step semiparametric empirical likelihood inference

In both parametric and certain nonparametric statistical models, the empirical likelihood ratio satisfies a nonparametric version of Wilks’ theorem. For many semiparametric models, however, the commonly used two-step (plug-in) empirical likelihood ratio is not asymptotically distribution-free, that is, its asymptotic distribution contains unknown quantities, and hence Wilks’ theorem breaks down. This article suggests a general approach to restore Wilks’ phenomenon in two-step semiparametric empirical likelihood inferences. The main insight consists in using as the moment function in the estimating equation the influence function of the plug-in sample moment. The proposed method is general; it leads to a chi-squared limiting distribution with known degrees of freedom; it is efficient; it does not require undersmoothing; and it is less sensitive to the first-step than alternative methods, which is particularly appealing for high-dimensional settings. Several examples and simulation studies illustrate the general applicability of the procedure and its excellent finite sample performance relative to competing methods.

Statistical inference for nonignorable missing-data problems: a selective review

ABSTRACTNonignorable missing data are frequently encountered in various settings, such as economics, sociology and biomedicine. We review statistical inference for nonignorable missing-data problems, including estimation, influence analysis and model selection. For estimation of mean functionals, we review semiparametric method and empirical likelihood (EL) approach. For estimation of parameters in exponential family nonlinear structural equation models, we introduce expectation-maximisation algorithm, Bayesian approach, and Bayesian EL method. For influence analysis, we investigate the case-deletion method and local influence analysis method from the frequentist and Bayesian viewpoints. For model selection, we present the modified Akaike information criterion and penalised method.

Likelihood Inference on Semiparametric Models: Average Derivative and Treatment Effect

Over the past few decades, much progress has been made in semiparametric modelling and estimation methods for econometric analysis. This paper is concerned with inference (i.e. confidence intervals and hypothesis testing) in semiparametric models. In contrast to the conventional approach based on t-ratios, we advocate likelihood-based inference. In particular, we study two widely applied semiparametric problems, weighted average derivatives and treatment effects, and propose semiparametric empirical likelihood and jackknife empirical likelihood methods. We derive the limiting behaviour of these empirical likelihood statistics and investigate their finite sample performance through Monte Carlo simulation. Furthermore, we extend the (delete-1) jackknife empirical likelihood toward the delete-d version with growing d and establish general asymptotic theory. This extension is crucial to deal with non-smooth objects, such as quantiles and quantile average derivatives or treatment effects, due to the well-known inconsistency phenomena of the jackknife under non-smoothness.

Semiparametric empirical likelihood tests in varying coefficient partially linear models with repeated measurements

Varying coefficient partially linear models are commonly used for analyzing data measured repeatedly, such as longitudinal data and panel data. In this paper, the testing problem for varying coefficient partially linear models with repeated measurements is investigated. Based on the empirical likelihood method, the test statistics are constructed for some testing problems. The Wilks phenomenon of these test statistics is proved, and then the rejection regions are constructed. Some simulation studies are undertaken to investigate the power of the empirical likelihood based testing method.

Semi-parametric Efficient Inference for Heteroscedastic Semivarying-coefficient Models

Semivarying-coefficient models with heteroscedastic errors are frequently used in statistical modeling. When the error is conditional heteroskedastic, Ahmad, et al. (2005) proposed a general series method to obtain an efficient estimation. In this article we study the heteroscedastic semi-varying coefficient models with a nonparametric variance function, not only use the semi-parametric efficient normal approximation method to derive a family of semi-parametric efficient estimator, but also use the semi-parametric efficient empirical likelihood method to construct the efficient empirical likelihood confidence regions. The proposed estimators retain the double robustness feature of semi-parametric efficient estimator.

Semiparametric empirical likelihood estimation for two-stage outcome-dependent sampling under the frame of generalized linear models

Epidemiologic studies use outcome-dependent sampling (ODS) schemes where, in addition to a simple random sample, there are also a number of supplement samples that are collected based on outcome variable. ODS scheme is a cost-effective way to improve study efficiency. We develop a maximum semiparametric empirical likelihood estimation (MSELE) for data from a two-stage ODS scheme under the assumption that given covariate, the outcome follows a general linear model. The information of both validation samples and nonvalidation samples are used. What is more, we prove the asymptotic properties of the proposed MSELE.

Estimating effect of environmental contaminants on women's subfecundity for the MoBa study data with an outcome-dependent sampling scheme.

Motivated by the need from our on-going environmental study in the Norwegian Mother and Child Cohort (MoBa) study, we consider an outcome-dependent sampling (ODS) scheme for failure-time data with censoring. Like the case-cohort design, the ODS design enriches the observed sample by selectively including certain failure subjects. We present an estimated maximum semiparametric empirical likelihood estimation (EMSELE) under the proportional hazards model framework. The asymptotic properties of the proposed estimator were derived. Simulation studies were conducted to evaluate the small-sample performance of our proposed method. Our analyses show that the proposed estimator and design is more efficient than the current default approach and other competing approaches. Applying the proposed approach with the data set from the MoBa study, we found a significant effect of an environmental contaminant on fecundability.

Semiparametric empirical likelihood confidence intervals for the difference of areas under two correlated ROC curves under density ratio model

The area under the receiver operating characteristic (ROC) curve, AUC, is commonly used to assess the ability of a diagnostic test to correctly classify individuals into diseased and nondiseased populations. When there are two diagnostic tests available, it is of interest to evaluate and compare their performances. Based on the difference of two placement values, we propose a two-sample empirical likelihood method for comparing AUCs of two ROC curves. The proposed empirical likelihood ratio statistic converges in distribution to a scaled chi-squared random variable. Simulation results show that the proposed empirical likelihood method has better finite-sample performance than other competitors.

Semiparametric empirical likelihood confidence intervals for AUC under a density ratio model

Inferences on the area under a receiver operating characteristic curve (AUC) are usually based on a fully parametric approach or a fully nonparametric approach. A semiparametric empirical likelihood method is proposed to construct confidence intervals for AUC by assuming a density ratio model for the diseased and non-diseased population densities. The limiting distribution of the semiparametric empirical log likelihood ratio statistic for AUC has a scaled chi-square distribution. The proposed semiparametric empirical likelihood approach is shown, via a simulation study, to be more robust than a fully parametric approach and is more accurate than a fully nonparametric approach. Some results on simulation and an analysis of two real examples are presented.

Semiparametric Inference for Data with A Continuous Outcome from A Two-Phase Probability-Dependent Sampling Scheme

Multi-phased designs and biased sampling designs are two of the well recognized approaches to enhance study efficiency. In this paper, we propose a new and cost-effective sampling design, the two-phase probability dependent sampling design (PDS), for studies with a continuous outcome. This design will enable investigators to make efficient use of resources by targeting more informative subjects for sampling. We develop a new semiparametric empirical likelihood inference method to take advantage of data obtained through a PDS design. Simulation study results indicate that the proposed sampling scheme, coupled with the proposed estimator, is more efficient and more powerful than the existing outcome dependent sampling design and the simple random sampling design with the same sample size. We illustrate the proposed method with a real data set from an environmental epidemiologic study.

Data Fusion Using Empirical Likelihood

The authors propose a robust semi-parametric empirical likelihood method to integrate all available information from multiple samples with a common center of measurements. Two different sets of estimating equations are used to improve the classical likelihood inference on the measurement center. The proposed method does not require the knowle- dge of the functional forms of the probability density functions of related populations. The advantages of the proposed method are demonstrated through extensive simulation studies by comparing the mean squared errors, coverage proba- bilities and average lengths of confidence intervals with those from the classical likelihood method. Simulation results suggest that our approach provides more informative and efficient inference than the conventional maximum likelihood estimator if certain structural relationship exists among the parameters of relevant samples.

Semi-parametric hybrid empirical likelihood inference for two-sample comparison with censored data

Two-sample comparison problems are often encountered in practical projects and have widely been studied in literature. Owing to practical demands, the research for this topic under special settings such as a semiparametric framework have also attracted great attentions. Zhou and Liang (Biometrika 92:271-282, 2005) proposed an empirical likelihood-based semi-parametric inference for the comparison of treatment effects in a two-sample problem with censored data. However, their approach is actually a pseudo-empirical likelihood and the method may not be fully efficient. In this study, we develop a new empirical likelihood-based inference under more general framework by using the hazard formulation of censored data for two sample semi-parametric hybrid models. We demonstrate that our empirical likelihood statistic converges to a standard chi-squared distribution under the null hypothesis. We further illustrate the use of the proposed test by testing the ROC curve with censored data, among others. Numerical performance of the proposed method is also examined.