Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

A New Conjoint Analysis Procedure with Application to Marketing Research

Abstract

In this article we propose some extensions and applications of the nonparametric combination of dependent rankings (see Pesarin, F., Lago, A. (2000). Nonparametric combination of department rankings with applications to the quality assessment of industrial products. Metron LVIII (1–2):39–52.) This methodology is applied to Conjoint Analysis in order to aggregate (ex ante) preferences from a group of individuals. Furthermore, a new global association test (GAT) is introduced in order to test for the association of the global ranking with all attributes of interest. The GAT procedure allows the experimenter to have clear indications on significant attributes by considering the intensity of the optimal weights given by the procedure itself. This may help the experimenter in interpreting the usual analysis involving the normal plot for detecting active effects.     

Nonparametric Bayes factors based on empirical likelihood ratios

Bayes methodology provides posterior distribution functions based on parametric likelihoods adjusted for prior distributions. A distribution-free alternative to the parametric likelihood is use of empirical likelihood (EL) techniques, well known in the context of nonparametric testing of statistical hypotheses. Empirical likelihoods have been shown to exhibit many of the properties of conventional parametric likelihoods. In this paper, we propose and examine Bayes factors (BF) methods that are derived via the EL ratio approach. Following Kass and Wasserman (1995), we consider Bayes factors type decision rules in the context of standard statistical testing techniques. We show that the asymptotic properties of the proposed procedure are similar to the classical BF's asymptotic operating characteristics. Although we focus on hypothesis testing, the proposed approach also yields confidence interval estimators of unknown parameters. Monte Carlo simulations were conducted to evaluate the theoretical results as well as to demonstrate the power of the proposed test.     

Weighted local linear composite quantile estimation for the case of general error distributions

It is known that for nonparametric regression, local linear composite quantile regression (local linear CQR) is a more competitive technique than classical local linear regression since it can significantly improve estimation efficiency under a class of non-normal and symmetric error distributions. However, this method only applies to symmetric errors because, without symmetric condition, the estimation bias is non-negligible and therefore the resulting estimator is inconsistent. In this paper, we propose a weighted local linear CQR method for general error conditions. This method applies to both symmetric and asymmetric random errors. Because of the use of weights, the estimation bias is eliminated asymptotically and the asymptotic normality is established. Furthermore, by minimizing asymptotic variance, the optimal weights are computed and consequently the optimal estimate (the most efficient estimate) is obtained. By comparing relative efficiency theoretically or numerically, we can ensure that the new estimation outperforms the local linear CQR estimation. Finite sample behaviors conducted by simulation studies further illustrate the theoretical findings.     

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

Semiparametric ARCH Models

This article introduces a semiparametric autoregressive conditional heteroscedasticity (ARCH) model that has conditional first and second moments given by autoregressive moving average and ARCH parametric formulations but a conditional density that is assumed only to be sufficiently smooth to be approximated by a nonparametric density estimator. For several particular conditional densities, the relative efficiency of the quasi-maximum likelihood estimator is compared with maximum likelihood under correct specification. These potential efficiency gains for a fully adaptive procedure are compared in a Monte Carlo experiment with the observed gains from using the proposed semiparametric procedure, and it is found that the estimator captures a substantial proportion of the potential. The estimator is applied to daily stock returns from small firms that are found to exhibit conditional skewness and kurtosis and to the British pound to dollar exchange rate.     

Nonparametric Entropy-Based Tests of Independence Between Stochastic Processes

This article develops nonparametric tests of independence between two stochastic processes satisfying β-mixing conditions. The testing strategy boils down to gauging the closeness between the joint and the product of the marginal stationary densities. For that purpose, we take advantage of a generalized entropic measure so as to build a whole family of nonparametric tests of independence. We derive asymptotic normality and local power using the functional delta method for kernels. As a corollary, we also develop a class of entropy-based tests for serial independence. The latter are nuisance parameter free, and hence also qualify for dynamic misspecification analyses. We then investigate the finite-sample properties of our serial independence tests through Monte Carlo simulations. They perform quite well, entailing more power against some nonlinear AR alternatives than two popular nonparametric serial-independence tests.     

Modifications of the Empirical Likelihood Interval Estimation with Improved Coverage Probabilities

The empirical likelihood (EL) technique has been well addressed in both the theoretical and applied literature in the context of powerful nonparametric statistical methods for testing and interval estimations. A nonparametric version of Wilks theorem (Wilks, 1938 Wilks , S. S. ( 1938 ). The large-sample distribution of the likelihood ratio for testing composite hypotheses . Annals of Mathematical Statistics 9 : 60 – 62 .[Crossref] , [Google Scholar]) can usually provide an asymptotic evaluation of the Type I error of EL ratio-type tests. In this article, we examine the performance of this asymptotic result when the EL is based on finite samples that are from various distributions. In the context of the Type I error control, we show that the classical EL procedure and the Student's t-test have asymptotically a similar structure. Thus, we conclude that modifications of t-type tests can be adopted to improve the EL ratio test. We propose the application of the Chen (1995 Chen , L. ( 1995 ). Testing the mean of skewed distributions . Journal of the American Statistical Association 90 : 767 – 772 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) t-test modification to the EL ratio test. We display that the Chen approach leads to a location change of observed data whereas the classical Bartlett method is known to be a scale correction of the data distribution. Finally, we modify the EL ratio test via both the Chen and Bartlett corrections. We support our argument with theoretical proofs as well as a Monte Carlo study. A real data example studies the proposed approach in practice.     

Comparison of nonparametric analysis of variance methods: A vote for van der Waerden

ABSTRACT

For two-way layouts in a between-subjects analysis of variance design, the parametric F-test is compared with seven nonparametric methods: rank transform (RT), inverse normal transform (INT), aligned rank transform (ART), a combination of ART and INT, Puri & Sen's L statistic, Van der Waerden, and Akritas and Brunners ANOVA-type statistics (ATS). The type I error rates and the power are computed for 16 normal and nonnormal distributions, with and without homogeneity of variances, for balanced and unbalanced designs as well as for several models including the null and the full model. The aim of this study is to identify a method that is applicable without too much testing for all the attributes of the plot. The Van der Waerden test shows the overall best performance though there are some situations in which it is disappointing. The Puri & Sen's and the ATS tests show generally very low power. These two and the other methods cannot keep the type I error rate under control in too many situations. Especially in the case of lognormal distributions, the use of any of the rank-based procedures can be dangerous for cell sizes above 10. As already shown by many other authors, nonnormal distributions do not violate the parametric F-test, but unequal variances do, and heterogeneity of variances leads to an inflated error rate more or less also for the nonparametric methods. Finally, it should be noted that some procedures show rising error rates with increasing cell sizes, the ART, especially for discrete variables, and the RT, Puri & Sen, and the ATS in the cases of heteroscedasticity.