Using Liu estimator for detection of influential observations in linear measurement error models

Abstract

In this paper, we introduce Liu estimator for the vector of parameters in linear measurement error models and discuss its asymptotic properties. Based on the Liu estimator, diagnostic measures are developed to identify influential observations. Additionally, the analogs of Cook’s distance and likelihood distance are proposed to determine influential observations using case deletion approach. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics. Finally, the performance of the influence measures have been illustrated through simulation study and analyzing a real data set.     

Jackknife empirical likelihood method for testing the equality of two variances

In this paper, we propose a nonparametric method based on jackknife empirical likelihood ratio to test the equality of two variances. The asymptotic distribution of the test statistic has been shown to follow χ² distribution with the degree of freedom 1. Simulations have been conducted to show the type I error and the power compared to Levene's test and F test under different distribution settings. The proposed method has been applied to a real data set to illustrate the testing procedure.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

Estimation and Inference in Two-Step Econometric Models

A commonly used procedure in a wide class of empirical applications is to impute unobserved regressors, such as expectations, from an auxiliary econometric model. This two-step (T-S) procedure fails to account for the fact that imputed regressors are measured with sampling error, so hypothesis tests based on the estimated covariance matrix of the second-step estimator are biased, even in large samples. We present a simple yet general method of calculating asymptotically correct standard errors in T-S models. The procedure may be applied even when joint estimation methods, such as full information maximum likelihood, are inappropriate or computationally infeasible. We present two examples from recent empirical literature in which these corrections have a major impact on hypothesis testing.     

Empirical-likelihood-based Test for Partially Linear Single-index Models with Error-prone Linear Covariates

In this article, we consider whether the empirical likelihood ratio (ELR) test is applicable to testing for serial correlation in the partially linear single-index models (PLSIM) with error-prone linear covariates. It is shown that under the null hypothesis the proposed ELR statistic follows asymptotically a χ²-distribution with the scale constant and the degrees of freedom. A comparison between the ELR and the normal approximation method is also considered. Both simulated and real data examples are used to illustrate our proposed methodology.     

A Discussion of Permutation Tests Conditional to Observed Responses in Unreplicated 2 M Full Factorial Designs

ABSTRACT

In this article we present a new solution to test for effects in unreplicated two-level factorial designs. The proposed test statistic, in case the error components are normally distributed, follows an F random variable, though our attention is on its nonparametric permutation version. The proposed procedure does not require any transformation of data such as residualization and it is exact for each effect and distribution-free. Our main aim is to discuss a permutation solution conditional to the original vector of responses. We give two versions of the same nonparametric testing procedure in order to control both the individual error rate and the experiment-wise error rate. A power comparison with Loughin and Noble's test is provided in the case of a unreplicated 2⁴ full factorial design.     

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

ABSTRACT

In 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.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

Empirical likelihood inference for general transformation models with right censored data

In this work, we consider empirical likelihood inference for general transformation models with right censored data. The models are a class of flexible semiparametric survival models and include many popular survival models as their special cases. Based on the marginal likelihood function, we define an empirical likelihood ratio statistic. Under some regularity conditions, we show that the empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution. Through some simulation studies and a real data application, we show that our proposed procedure can work fairly well even for relatively small sample size and high censoring.     

Selecting the Optimal Transformation of a Continuous Covariate in Cox's Regression: Implications for Hypothesis Testing

ABSTRACT

Background: Many exposures in epidemiological studies have nonlinear effects and the problem is to choose an appropriate functional relationship between such exposures and the outcome. One common approach is to investigate several parametric transformations of the covariate of interest, and to select a posteriori the function that fits the data the best. However, such approach may result in an inflated Type I error. Methods: Through a simulation study, we generated data from Cox's models with different transformations of a single continuous covariate. We investigated the Type I error rate and the power of the likelihood ratio test (LRT) corresponding to three different procedures that considered the same set of parametric dose-response functions. The first unconditional approach did not involve any model selection, while the second conditional approach was based on a posteriori selection of the parametric function. The proposed third approach was similar to the second except that it used a corrected critical value for the LRT to ensure a correct Type I error. Results: The Type I error rate of the second approach was two times higher than the nominal size. For simple monotone dose-response, the corrected test had similar power as the unconditional approach, while for non monotone, dose-response, it had a higher power. A real-life application that focused on the effect of body mass index on the risk of coronary heart disease death, illustrated the advantage of the proposed approach. Conclusion: Our results confirm that a posteriori selecting the functional form of the dose-response induces a Type I error inflation. The corrected procedure, which can be applied in a wide range of situations, may provide a good trade-off between Type I error and power.     

Profile likelihood approaches for semiparametric copula and frailty models for clustered survival data

ABSTRACT

In clustered survival data, the dependence among individual survival times within a cluster has usually been described using copula models and frailty models. In this paper we propose a profile likelihood approach for semiparametric copula models with different cluster sizes. We also propose a likelihood ratio method based on profile likelihood for testing the absence of association parameter (i.e. test of independence) under the copula models, leading to the boundary problem of the parameter space. For this purpose, we show via simulation study that the proposed likelihood ratio method using an asymptotic chi-square mixture distribution performs well as sample size increases. We compare the behaviors of the two models using the profile likelihood approach under a semiparametric setting. The proposed method is demonstrated using two well-known data sets.     

Copula-based predictions in small area estimation

Unit-level regression models are commonly used in small area estimation (SAE) to obtain an empirical best linear unbiased prediction of small area characteristics. The underlying assumptions of these models, however, may be unrealistic in some applications. Previous work developed a copula-based SAE model where the empirical Kendall's tau was used to estimate the dependence between two units from the same area. In this article, we propose a likelihood framework to estimate the intra-class dependence of the multivariate exchangeable copula for the empirical best unbiased prediction (EBUP) of small area means. One appeal of the proposed approach lies in its accommodation of both parametric and semi-parametric estimation approaches. Under each estimation method, we further propose a bootstrap approach to obtain a nearly unbiased estimator of the mean squared prediction error of the EBUP of small area means. The performance of the proposed methods is evaluated through simulation studies and also by a real data application.     

Empirical likelihood and general relative error criterion with divergent dimension

In estimation of multiplicative or accelerated failure time models, the relative error criterion has been recognized as an alternative to the squared or absolute error criterion. The general relative error criterion introduced by Chen et al. [Least product relative error estimation. J Multivariate Anal. 2016;144:91–98] is a unified framework for efficient estimation, which includes the least absolute relative error estimation and least product relative error estimation as special cases. In this paper, by combining the empirical likelihood and general relative error criterion in multiplicative model, we develop a new empirical likelihood method for inference on the unknown parameters under high-dimensional setting. Limiting theory is established for the proposed empirical likelihood statistic. We conduct some simulation studies and real data analysis to evaluate the effectiveness of the proposed method.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.     

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.     

Smoothed empirical likelihood for GARCH models with heavy-tailed errors

ABSTRACT

This paper proposes an empirical likelihood (EL) method for estimating the GARCH(p, q) models with heavy-tailed errors. Using the kernel smoothing method, we derive a smoothed EL ratio statistic, which yields a smoothed EL estimator. Moreover, we derive a profile EL for the partial parameters in the presence of nuisance parameters. Simulations and empirical results are conducted to illustrate our proposed method.     

Statistical diagnosis for non-parametric regression models with random right censorship based on the empirical likelihood method

In this paper, we consider statistical diagnostic for non-parametric regression models with right-censored data based on empirical likelihood. First, the primary model is transformed to the non-parametric regression model. Then, based on empirical likelihood methodology, we define some diagnostic statistics. At last, some simulation studies show that our proposed procedure can work fairly well.     

Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

Empirical Characteristic Function Estimation and Its Applications

Abstract

This paper reviews the method of model-fitting via the empirical characteristic function. The advantage of using this procedure is that one can avoid difficulties inherent in calculating or maximizing the likelihood function. Thus it is a desirable estimation method when the maximum likelihood approach encounters difficulties but the characteristic function has a tractable expression. The basic idea of the empirical characteristic function method is to match the characteristic function derived from the model and the empirical characteristic function obtained from data. Ideas are illustrated by using the methodology to estimate a diffusion model that includes a self-exciting jump component. A Monte Carlo study shows that the finite sample performance of the proposed procedure offers an improvement over a GMM procedure. An application using over 72 years of DJIA daily returns reveals evidence of jump clustering.     

Penalized empirical likelihood inference for sparse additive hazards regression with a diverging number of covariates

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by applications in high-throughput genomic data analysis. In this paper, we propose a class of regularization methods, integrating both the penalized empirical likelihood and pseudoscore approaches, for variable selection and estimation in sparse and high-dimensional additive hazards regression models. When the number of covariates grows with the sample size, we establish asymptotic properties of the resulting estimator and the oracle property of the proposed method. It is shown that the proposed estimator is more efficient than that obtained from the non-concave penalized likelihood approach in the literature. Based on a penalized empirical likelihood ratio statistic, we further develop a nonparametric likelihood approach for testing the linear hypothesis of regression coefficients and constructing confidence regions consequently. Simulation studies are carried out to evaluate the performance of the proposed methodology and also two real data sets are analyzed.