Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.     

Jackknife empirical likelihood test for mean residual life functions

Mean residual life (MRL) function is an important function in survival analysis which describes the expected remaining life given survival to a certain age. In this article, we propose a non parametric method based on jackknife empirical likelihood through a U-statistic to test the equality of two mean residual functions. The asymptotic distribution of the test statistic has been derived. Simulations are conducted to illustrate the performance of the proposed test under different distributional assumptions and compare with some existing method. The proposed method is then applied to two real datasets.     

A moment-based empirical likelihood ratio test for exponentiality using the probability integral transformation

ABSTRACT

A simple and efficient goodness-of-fit test for exponentiality is developed by exploiting the characterization of the exponential distribution using the probability integral transformation. We adopted the empirical likelihood methodology in constructing the test statistic. The proposed test statistic has a chi-square limiting distribution. For small to moderate sample sizes Monte-Carlo simulations revealed that our proposed tests are much more superior under increasing failure rate (IFR) and bathtub decreasing-increasing failure rate (BFR) alternatives. Real data examples were used to demonstrate the robustness and applicability of our proposed tests in practice.     

EMPIRICAL LIKELIHOOD METHODS FOR THE GINI INDEX

The Gini index and its generalizations have been used extensively for measuring inequality and poverty in the social sciences. Recently, interval estimation based on nonparametric statistics has been proposed in the literature, for example the naive bootstrap method, the iterated bootstrap method and the bootstrap method via a pivotal statistic. In this paper, we propose empirical likelihood methods to construct confidence intervals for the Gini index or the difference of two Gini indices. Simulation studies show that the proposed empirical likelihood method performs slightly worse than the bootstrap method based on a pivotal statistic in terms of coverage accuracy, but it requires less computation. However, the bootstrap calibration of the empirical likelihood method performs better than the bootstrap method based on a pivotal statistic.     

Empirical likelihood test for equality of two distributions using distance of characteristic functions

In this paper, we investigate the problem of testing for the equality of two distributions. We employ a two-sample Jackknife Empirical Likelihood (JEL) approach to construct a test statistic whose limiting distribution is Chi-square distribution with degree of freedom 1, no matter what the data dimension (fixed) is. A variety of synthetic data experiments demonstrate that our JEL test statistic performs very well, with a very neat asymptotic distribution under the null hypothesis. Furthermore, we apply the test procedure to a real dataset to obtain competitive results.     

Cressie–Read Power-Divergence Statistics for Non-Gaussian Vector Stationary Processes

Abstract.  For a class of vector-valued non-Gaussian stationary processes, we develop the Cressie–Read power-divergence (CR) statistic approach which has been proposed for the i.i.d. case. The CR statistic includes empirical likelihood as a special case. Therefore, by adopting this CR statistic approach, the theory of estimation and testing based on empirical likelihood is greatly extended. We use an extended Whittle likelihood as score function and derive the asymptotic distribution of the CR statistic. We apply this result to estimation of autocorrelation and the AR coefficient, and get narrower confidence intervals than those obtained by existing methods. We also consider the power properties of the test based on asymptotic theory. Under a sequence of contiguous local alternatives, we derive the asymptotic distribution of the CR statistic. The problem of testing autocorrelation is discussed and we introduce some interesting properties of the local power.     

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.     

Conditional heteroscedasticity test for Poisson autoregressive model

In this article, the problem of interest is testing the conditional heteroscedasticity of Poisson autoregressive model. We construct a non parametric test statistic based on empirical likelihood method. The asymptotic distribution of the proposed statistic is derived and its finite-sample property is examined through Monte Carlo simulations. The simulation results show that the proposed method is good for practical use.     

Empirical likelihood ratio test for a mean change point model with a linear trend followed by an abrupt change

In this paper, a change point model with the mean being constant up to some unknown point, and increasing linearly to another unknown point, then dropping back to the original level is studied. A nonparametric method based on the empirical likelihood test is proposed to detect and estimate the locations of change points. Under some mild conditions, the asymptotic null distribution of an empirical likelihood ratio test statistic is shown to have the extreme distribution. The consistency of the test is also proved. Simulations of the powers of the test indicate that it performs well under different assumptions of the data distribution. The test is applied to the aircraft arrival time data set and the Stanford heart transplant data set.     

Linear regression analysis with inequality constraints on the regression parameters via empirical likelihood

An empirical likelihood ratio test is developed for testing for or against inequality constraints on regression parameters in linear regression analysis. The proposed approach imposes no parametric model nor identically distributing assumption on the random errors. The asymptotic distribution of the proposed test statistic under null hypothesis is shown to be of chi-bar-squared type. The asymptotic power under contiguous alternatives is also briefly discussed. Moreover, an adjusted empirical likelihood method is adopted to improve the small sample size behaviour of the proposed test. Several simulation studies are carried out to assess the finite sample performance of the proposed tests. The results reveal that the proposed tests could be valuable for improving inference efficiency. A real-life example is discussed to illustrate the theoretical results.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Second‐order Accurate Confidence Regions Based on Members of the Generalized Power Divergence Family

Recently, a technique based on pseudo‐observations has been proposed to tackle the so‐called convex hull problem for the empirical likelihood statistic. The resulting adjusted empirical likelihood also achieves the high‐order precision of the Bartlett correction. Nevertheless, the technique induces an upper bound on the resulting statistic that may lead, in certain circumstances, to worthless confidence regions equal to the whole parameter space. In this paper, we show that suitable pseudo‐observations can be deployed to make each element of the generalized power divergence family Bartlett‐correctable and released from the convex hull problem. Our approach is conceived to achieve this goal by means of two distinct sets of pseudo‐observations with different tasks. An important effect of our formulation is to provide a solution that permits to overcome the problem of the upper bound. The proposal, which effectiveness is confirmed by simulation results, gives back attractiveness to a broad class of statistics that potentially contains good alternatives to the empirical likelihood.     

A goodness-of-fit test for overdispersed binomial (or multinomial) models

Overdispersion or extra variation is a common phenomenon that occurs when binomial (multinomial) data exhibit larger variances than that permitted by the binomial (multinomial) model. This arises when the data are clustered or when the assumption of independence is violated. Goodness-of-fit (GOF) tests available in the overdispersion literature have focused on testing for the presence of overdispersion in the data and hence they are not applicable for choosing between the several competing overdispersion models. In this paper, we consider a GOF test proposed by Neerchal and Morel [1998. Large cluster results for two parametric multinomial extra variation models. J. Amer. Statist. Assoc. 93(443), 1078–1087], and study its distributional properties and performance characteristics. This statistic is a direct analogue of the usual Pearson chi-squared statistic, but is also applicable when the clusters are not necessarily of the same size. As this test statistic is for testing model adequacy against the alternative that the model is not adequate, it is applicable in testing two competing overdispersion models.     

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.     

Inference for random coefficient INAR(k) with the occasional level shift random noise based on dual empirical likelihood

The INAR(k) model has been widely used in various kinds of fields. However, there are little discussions about the INAR(k) model with the occasional level shift random noise. In this paper, the maximum likelihood estimation of parameter based on martingale difference sequence is given, the log empirical likelihood ratio test statistic is obtained and the test statistic converges to chi-square distribution, we prove that the confidence region of the parameter is convex. Furthermore, the numerical simulation of the proposed INAR(k) model is given, which illustrates the effectiveness of the model. Then, the proofs of asymptotic results are given in the Appendix.     

Correlation analysis with additive distortion measurement errors

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a additive fashion by an observed confounding variable. Two estimators, a direct-plug-in estimator and a residual-based estimator, are proposed. Their asymptotical results are obtained, and the residual-based estimator is shown asymptotically efficient. Moreover, we suggest an asymptotic normal approximation and an empirical likelihood-based statistic to construct the confidence interval. The empirical likelihood statistic is shown to be asymptotically chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators. These methods are applied to analyse the Boston housing price data for an illustration.     

A two sample ordered alternative test for means and variances

A two sample test of likelihood ratio type is proposed, assuming normal distribution theory, for testing the hypothesis that two samples come from identical normal populations versus the alternative that the populations are normal but vary in mean value and variance with one population having a smaller mean and smaller variance than the other. The small sample and large sample distribution of the proposed statistic are derived assuming normality. Some computations are presented which show the speed of convergence of small sample critical values to their asymptotic counterparts. Comparisons of local power of the proposed test are made with several potential competing tests. Asymptotics for the test statistic are derived when underlying distributions are not necessarily normal.     

Resampling Methods for Testing a Semiparametric Random Censorship Model

This paper presents a goodness-of-fit test for a semiparametric random censorship model proposed by Dikta (1998 ). The test statistic is derived from a model-based process which is asymptotically Gaussian. In addition to test consistency, the proposed test can detect local alternatives distinct n ^-1/2 from the null hypothesis. Due to the intractability of the asymptotic null distribution of the test statistic, we turn to two resampling approximations. We first use the well-known bootstrap method to approximate critical values of the test. We then introduce a so-called random symmetrization method for carrying out the test. Both methods perform very well with a sample of moderate size. A simulation study shows that the latter possesses better empirical powers and sizes for small samples.     

An improved test of equality of mean directions for the Langevin‐von Mises‐Fisher distribution

A multi‐sample test for equality of mean directions is developed for populations having Langevin‐von Mises‐Fisher distributions with a common unknown concentration. The proposed test statistic is a monotone transformation of the likelihood ratio. The high‐concentration asymptotic null distribution of the test statistic is derived. In contrast to previously suggested high‐concentration tests, the high‐concentration asymptotic approximation to the null distribution of the proposed test statistic is also valid for large sample sizes with any fixed nonzero concentration parameter. Simulations of size and power show that the proposed test outperforms competing tests. An example with three‐dimensional data from an anthropological study illustrates the practical application of the testing procedure.