Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

Empirical likelihood methods for discretely observed Gaussian moving averages

This paper is concerned with parametric estimation, model specification and autocorrelation diagnosis for stationary moving averages driven by a Wiener process. By incorporating the analysis of the spectral densities of the discretely observed trajectory, empirical likelihood methods based on moment conditions are developed to the dependent sequences in this paper for estimation and test. Theoretical properties of the empirical likelihood estimator for parameters are provided. Empirical likelihood ratio tests for model specification of the moving averages are proposed by means of the bootstrap strategy. Simulation and empirical case studies are carried out to confirm the effectiveness of the proposed estimation and test.     

Validation tests for semi-parametric models

Tests are proposed for validation of the hypothesis that a partial linear regression model adequately describes the structure of a given data set. The test statistics are formulated following the approach of Fourier-type conditional expectations first suggested by Bierens [Consistent model specification tests. J Econometr. 1982;20:105–134]. The proposed procedures are computationally convenient, and under fairly mild conditions lead to consistent tests. Corresponding bootstrap versions are compared with alternative procedures for a wide selection of different estimators of the underlying partial linear model.     

Semiparametric Negative Binomial Regression Models

Negative-binomial (NB) regression models have been widely used for analysis of count data displaying substantial overdispersion (extra-Poisson variation). However, no formal lack-of-fit tests for a postulated parametric model for a covariate effect have been proposed. Therefore, a flexible parametric procedure is used to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. Within the proposed modeling framework, a log-likelihood ratio test is constructed to evaluate the adequacy of a postulated parametric form of the covariate effect. Simulation experiments are conducted to study the power performance of the proposed test.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

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.     

Prediction bands for the EDF of a future sample

We develop both nonparametric and parametric methods for obtaining prediction bands for the empirical distribution function (EDF) of a future sample. These methods yield simultaneous prediction intervals for all order statistics of the future sample, and they also correspond to tests for the two-sample problem. The nonparametric prediction bands correspond to the two-sample Kolmogorov-Smirnov test and related nonparametric tests, but the parametric prediction bands correspond to entirely new parametric two-sample tests. The parametric prediction bands tend to outperform the nonparametric bands when the parametric assumptions hold, but they may have true coverage probabilities well below their nominal levels when the parametric assumptions fail. A new computational algorithm is used to obtain critical values in the nonparametric case.     

A Lack-of-Fit Test for Heteroscedastic Regression Models via Cosine-Series Smoothers

In this paper, a test is derived to assess the validity of heteroscedastic nonlinear regression models by a non‐parametric cosine regression method. For order selection, the paper proposes a data‐driven method that uses the parametric null model optimal order. This method yields a test that is asymptotically normally distributed under the null hypothesis and is consistent against any fixed alternative. Simulation studies that test the lack of fit of a generalized linear model are conducted to compare the performance of the proposed test with that of an existing non‐parametric kernel test. A dataset of esterase levels is used to demonstrate the proposed method in practice.     

Estimating the Conditional Error Distribution in Non‐parametric Regression

Abstract. We consider a general non‐parametric regression model, where the distribution of the error, given the covariate, is modelled by a conditional distribution function. For the estimation, a kernel approach as well as the (kernel based) empirical likelihood method are discussed. The latter method allows for incorporation of additional information on the error distribution into the estimation. We show weak convergence of the corresponding empirical processes to Gaussian processes and compare both approaches in asymptotic theory and by means of a simulation study.     

Non parametric multiple comparisons of non nested rival models

The practice for testing homogeneity of several rival models is of interest. In this article, we consider a non parametric multiple test for non nested distributions in the context of the model selection. Based on the linear sign rank test, and the known union–intersection principle, we let the magnitude of the data to give a better performance to the test statistic. We consider the sample and the non nested rival models as blocks and treatments, respectively, and introduce the extended Friedman test version to compare with the results of the test based on the linear sign rank test. A real dataset based on the waiting time to earthquake is considered to illustrate the results.     

Testing lack of fit of regression models under heteroscedasticity

A test is proposed for assessing the lack of fit of heteroscedastic nonlinear regression models that is based on comparison of nonparametric kernel and parametric fits. A data-driven method is proposed for bandwidth selection using the asymptotically optimal bandwidth of the parametric null model which leads to a test that has a limiting normal distribution under the null hypothesis and is consistent against any fixed alternative. The resulting test is applied to the problem of testing the lack of fit of a generalized linear model.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Restricted Estimation in Partially Linear Errors-in-Variables Models

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

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.     

An alternative procedure for performing a power analysis of Mantel's test

This study proposes a simple way to perform a power analysis of Mantel's test applied to squared Euclidean distance matrices. The general statistical aspects of the simple Mantel's test are reviewed. The Monte Carlo method is used to generate bivariate Gaussian variables in order to create squared Euclidean distance matrices. The power of the parametric correlation t-test applied to raw data is also evaluated and compared with that of Mantel's test. The standard procedure for calculating punctual power levels is used for validation. The proposed procedure allows one to draw the power curve by running the test only once, dispensing with the time demanding standard procedure of Monte Carlo simulations. Unlike the standard procedure, it does not depend on a knowledge of the distribution of the raw data. The simulated power function has all the properties of the power analysis theory and is in agreement with the results of the standard procedure.     

A dynamic system for Gompertz model

Two-parameter Gompertz distribution has been introduced as a lifetime model for reliability inference recently. In this paper, the Gompertz distribution is proposed for the baseline lifetimes of components in a composite system. In this composite system, failure of a component induces increased load on the surviving components and thus increases component hazard rate via a power-trend process. Point estimates of the composite system parameters are obtained by the method of maximum likelihood. Interval estimates of the baseline survival function are obtained by using the maximum-likelihood estimator via a bootstrap percentile method. Two parametric bootstrap procedures are proposed to test whether the hazard rate function changes with the number of failed components. Intensive simulations are carried out to evaluate the performance of the proposed estimation procedure.     

Asymptotic properties of kruskal-wallis test and friedman test in the analysis of variance models with random effects

In this paper, asymptotic properties of the Kruskal-Wallis test in the one-way analysis of variance model and that of the Friedman test in the two-way classification model are investigated under alternatives when the treatment effects are random. It is shown that the asymptotic distribution of each statistic is the same as a mixture of central chi-squared variables. Asymptotic comparisons of the tests with respect to their parametric competitors are also performed     

Statistical analysis of right-censored failure-time data with partially specified hazard rates

This paper discusses the analysis of right-censored failure-time data in which the failure rate may have different forms in different time intervals. Such data occur naturally, for example, in demography studies and leukemia research, and a number of methods for the analysis have been proposed in the literature. However, most methods are purely parametric or nonparametric. Matthews and Farewell (1982), for example, discussed this problem and proposed a method for testing a constant failure rate against a failure rate involving a change point. To estimate an absolute limit on the attainable human life span, Zelterman (1992) discussed a hazard function that has different parametric forms over different time intervals. We consider a different situation in which the hazard function may follow a parametric form before a change point and is completely unknown after the change point. To test the existence of the change point, a modified maximal-censored-likelihood-ratio test is proposed and its asymptotic properties are studied. A bootstrap method is described for finding critical values of the proposed test. Simulation results indicate that the test performs well.     

Smooth tests for the gamma distribution

The gamma distribution is often used to model data with right skewness. Smooth tests of goodness of fit are proposed for this distribution. Their powers are compared with powers of the Anderson–Darling test and tests based on the empirical Laplace transform, the empirical moment generating function and the independence of the mean and coefficient of variation that characterizes the gamma distribution.