An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

A new goodness-of-fit test for certain bivariate distributions applicable to traffic accidents

T. Cacoullos and H. Papageorgiou [On some bivariate probability models applicable to traffic accidents and fatalities, Int. Stat. Rev. 48 (1980) 345–356] studied a special class of bivariate discrete distributions appropriate for modeling traffic accidents, and fatalities resulting therefrom. The corresponding random variable may be written as $Z={(N,Y)}^{\prime }$, with $Y={\sum }_{j=1}^N{X}_j$, where ${\left\{X_j\right\}}_{j=1}^N$, are independent copies of a (discrete) random variable $X$, and $N$ is independent of ${\left\{X_j\right\}}_{j=1}^N$, and follows a Poisson law. If $X$ follows a Poisson law (resp. Binomial law), the resulting distribution is termed Poisson–Poisson (resp. Poisson–Binomial). L2-type goodness-of-fit statistics are constructed for the ‘general distribution’ of this kind, where $X$ may be an arbitrary discrete nonnegative random variable. The test statistics utilize a simple characterization involving the corresponding probability generating function, and are shown to be consistent. The proposed procedures are shown to perform satisfactorily in simulated data, while their application to accident data leads to positive conclusions regarding the modeling ability of this class of bivariate distributions.     

An efficient correction to the density-based empirical likelihood ratio goodness-of-fit test for the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model positively skewed data. An important issue is to develop a powerful goodness-of-fit test for the IG distribution. We propose and examine novel test statistics for testing the IG goodness of fit based on the density-based empirical likelihood (EL) ratio concept. To construct the test statistics, we use a new approach that employs a method of the minimization of the discrimination information loss estimator to minimize Kullback–Leibler type information. The proposed tests are shown to be consistent against wide classes of alternatives. We show that the density-based EL ratio tests are more powerful than the corresponding classical goodness-of-fit tests. The practical efficiency of the tests is illustrated by using real data examples.     

On empirical likelihood test for predictability

Predicting asset prices is a critical issue in statistics and finance. In this article, by incorporating the recent advances in nonparametric approaches, we propose the empirical likelihood test for the predictability for the direction of price changes. Under some regularity conditions, the test statistic has an asymptotic χ² distribution under the null hypothesis that the direction of price change cannot be predicted. This test procedure is easy to implement and presents better finite sample performances than other popular causality tests, as reported in some Monte Carlo experiments.

1. Hightlights

2. We propose a non parametric likelihood test for predictability.

3. The test involves no user-chosen parameter or estimation of covariance matrix.

4. The test is simple to implement and has standard asymptotics.

5. The test has significantly better sizes than several popular tests with satisfactory power.

     

A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison

The Rayleigh distribution has been used to model right skewed data. Rayleigh [On the resultant of a large number of vibrations of the some pitch and of arbitrary phase. Philos Mag. 1880;10:73–78] derived it from the amplitude of sound resulting from many important sources. In this paper, a new goodness-of-fit test for the Rayleigh distribution is proposed. This test is based on the empirical likelihood ratio methodology proposed by Vexler and Gurevich [Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput Stat Data Anal. 2010;54:531–545]. Consistency of the proposed test is derived. It is shown that the distribution of the proposed test does not depend on scale parameter. Critical values of the test statistic are computed, through a simulation study. A Monte Carlo study for the power of the proposed test is carried out under various alternatives. The performance of the test is compared with some well-known competing tests. Finally, an illustrative example is presented and analysed.     

Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples

A test based on the studentized empirical characteristic function calculated in a single point is derived. An empirical power comparison is made between this test and tests like the Epps–Pulley, Shapiro–Wilks, Anderson–Darling and other tests for normality. It is shown to outperform the more complicated Epps-Pulley test based on the empirical characteristic function and a Cramér-von Mises type expression in a simulation study. The test performs especially good in large samples and the derived test statistic has an asymptotic normal distribution which is easy to apply.     

An exact density-based empirical likelihood ratio test for paired data

The Wilcoxon rank-sum test and its variants are historically well-known to be very powerful nonparametric decision rules for testing no location difference between two groups given paired data versus a shift alternative. In this title, we propose a new alternative empirical likelihood (EL) ratio approach for testing the equality of marginal distributions given that sampling is from a continuous bivariate population. We show that in various shift alternative scenarios the proposed exact test is superior to the classic nonparametric procedures, which may break down completely or are frequently inferior to the density-based EL ratio test. This is particularly true in the cases where there is a nonconstant shift under the alternative or the data distributions are skewed. An extensive Monte Carlo study shows that the proposed test has excellent operating characteristics. We apply the density-based EL ratio test to analyze real data from two medical studies.     

Gini index based goodness-of-fit test for the logistic distribution

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.     

An empirical goodness-of-fit test for multivariate distributions

An empirical test is presented as a tool for assessing whether a specified multivariate probability model is suitable to describe the underlying distribution of a set of observations. This test is based on the premise that, given any probability distribution, the Mahalanobis distances corresponding to data generated from that distribution will likewise follow a distinct distribution that can be estimated well by means of a large sample. We demonstrate the effectiveness of the test for detecting departures from several multivariate distributions. We then apply the test to a real multivariate data set to confirm that it is consistent with a multivariate beta model.     

Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

Monitoring test for stability of copula parameter in time series

In this paper, we consider a monitoring procedure to detect changes of the copula parameter of strong mixing processes. We propose two monitoring procedures based on the cumulative sums of scores evaluated at consistent copula parameter estimates and their fluctuations. We investigate the asymptotic properties of our monitoring procedures under both the null of no change in the copula parameter and its alternative. We also illustrate a simulation study and a real data analysis.     

An alternative algorithm of the empirical likelihood estimation for the parameter of a linear regression model

ABSTRACT

Empirical likelihood (EL) is a nonparametric method based on observations. EL method is defined as a constrained optimization problem. The solution of this constrained optimization problem is carried on using duality approach. In this study, we propose an alternative algorithm to solve this constrained optimization problem. The new algorithm is based on a newton-type algorithm for Lagrange multipliers for the constrained optimization problem. We provide a simulation study and a real data example to compare the performance of the proposed algorithm with the classical algorithm. Simulation and the real data results show that the performance of the proposed algorithm is comparable with the performance of the existing algorithm in terms of efficiencies and cpu-times.     

Balanced augmented empirical likelihood for regression models

This paper studies the problem of convex hull constraint in conventional empirical likelihood. Specifically, in the framework of regression, a balanced augmented empirical likelihood (BAEL) procedure through adding two synthetic data points is proposed. It can be used to resolve the under-coverage issue, especially in small-sample or high-dimension setting. Furthermore, some asymptotic properties for proposed BAEL ratio statistic are established under mild conditions. The proposed approach performs robust to different random errors by choosing a robust loss function. Extensive simulation studies and a real example are carried out to support our results.     

Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.     

A test of fit for a continuous distribution based on the empirical convex conditional mean function

Gupta and Kirmani (2008 Gupta, R.C., Kirmani, S.N.U.A. (2008). Characterization based on convex conditional mean function. J. Stat. Plann Inference. 138:964–970.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the convex conditional mean function (CCMF) characterizes the distribution function completely. In this paper, we introduce a consistent estimator of CCMF and call it empirical convex conditional mean function (ECCMF). Then we construct a simple consistent test of fit based on the integrated squared difference between ECCMF and CCMF. The theoretical and asymptotic properties of the estimator ECCMF and the proposed test statistic are studied. The performance of the constructed test is investigated under different distributions using simulations.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

Empirical likelihood method for the multivariate accelerated failure time models

In applications, multivariate failure time data appears when each study subject may potentially experience several types of failures or recurrences of a certain phenomenon, or failure times may be clustered. Three types of marginal accelerated failure time models dealing with multiple events data, recurrent events data and clustered events data are considered. We propose a unified empirical likelihood inferential procedure for the three types of models based on rank estimation method. The resulting log-empirical likelihood ratios are shown to possess chi-squared limiting distributions. The properties can be applied to do tests and construct confidence regions without the need to solve the rank estimating equations nor to estimate the limiting variance-covariance matrices. The related computation is easy to implement. The proposed method is illustrated by extensive simulation studies and a real example.