Exact Distributions Of Test Statistics Research Articles

Choosing the best Rukhin goodness-of-fit statistics

The testing for goodness-of-fit in multinomial sampling contexts is usually based on the asymptotic distribution of Pearson-type chi-squared statistics. However, approximations are not justified for those cases where sample size and number of cells permit the use of adequate algorithms to calculate the exact distribution of test statistics in a reasonable time. In particular, Rukhin statistics, containing χ 2 and Neyman's modified χ 2 statistics, are considered for testing uniformity. Their exact distributions are calculated for different sample sizes and number of cells. Several exact power comparisons are carried out to analyse the behaviour of selected statistics. As a result of the numerical study some recommendations are given. Conclusions may be extended to testing the goodness of fit to a given absolutely continuous cumulative distribution function.

Estimation of Moment Parameter in Elliptical Distributions

As a typical non-normal case, we consider a family of elliptically symmetric dis- tributions. Then, the moment parameter and its consistent estimator are presented. Also, the asymptotic expectation and the asymptotic variance of the consistent es- timator of the general moment parameter are given. Besides, the numerical results obtained by Monte Carlo simulation for some selected parameters are provided. The general moment parameter includes the important kurtosis parameter in the study of multivariate statistical analysis for elliptical populations. The kurtosis parameter, especially with relation to the estimation problem, has been considered by many authors. Mardia (1970, 1974) defined a measure of multi- variate sample kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Also, the testing normality was considered by using the asymptotic result. The related discussion of the kurtosis parameter under the elliptical distribution has been given by Anderson (1993), and Seo and Toyama (1996). Henze (1994) has discussed the asymptotic variance of the mul- tivariate sample kurtosis for general distributions. Here we deal with the estima- tion of the general moment parameters in elliptical distributions. In particular, we make a generalization of the results of Anderson (1993) and give an extension of asymptotic properties in Mardia (1970, 1974), Seo and Toyama (1996). In general, it is not easy to derive the exact distribution of test statistics or the percentiles for the testing problem under the elliptical populations, and so the asymptotic expansion of the statistics is considered. Especially that given up to the higher order includes not only the kurtosis parameter but the more gen- eral higher order moment parameters as well. Then we have to speculate about the estimation of the moment parameters as a practical problem. The present paper is organized in the following way. First, the probability density function, characteristic function and subclasses of the elliptical distribution are explained. Secondly, we prove the results concerning the moments and define the moment

Exact group-sequential designs for clinical trials with randomized play-the-winner allocation.

The use of both sequential designs and adaptive treatment allocation are effective in reducing the number of patients receiving an inferior treatment in a clinical trial. In large samples, when the asymptotic normality of test statistics can be utilized, a standard sequential design can be combined with adaptive allocation. In small samples the planned error rate constraints may not be satisfied if normality is assumed. We address this problem by constructing sequential stopping rules with specified properties by consideration of the exact distribution of test statistics under a particular adaptive allocation scheme, the randomized play-the-winner rule. Using this approach, compared to traditional equal allocation trials, trials with adaptive allocation are shown to require a larger total sample size to achieve a given power. More interestingly, the expected number patients allocated to the inferior treatment may also be larger for the adaptive allocation designs depending on the true success rates.

Generalized Hypergeometric Functions and Exact Distributions of Test Statistics

SYNOPTIC ABSTRACTIn this review article, an attempt has been made to give an up-to-date survey of the work done on the applications of generalized hypergeometric functions (especially G, H and R functions) to the exact distributions of test statistics. No attempt has been made to include details on the work connected with zonal polynomials, characteristic roots or asymptotic distributions.

A survey of algorithms for exact distributions of test statistics in r × c contingency tables with fixed margins

Testing of independence in an r × c tables is generally carried out by using a continuous limiting distribution of the test statistic, rather than the discrete distribution itself. This paper surveys algorithms for the computation of the latter. The central idea is the efficient enumeration of all tables with the same margins, or of a suitable subset thereof. The simplest case is Fisher's exact test for a 2 × 2 table and a one-side alternative hypothesis. Fisher's method is easily extended to r × c tables and to arbitrary statistics.

The Impact of Merger-Related Regulations Using Exact Distributions of Test Statistics

In Schipper and Thompson [1983] (henceforth ST), we estimated a pooled cross-section, time-series model of the return-generating process for the common stock in a sample of related firms. Some of the test statistics reported, based on linear constraints across estimated coefficients, are quadratic forms in the sample covariance matrix. Under the structure and assumptions of the model, these test statistics have asymptotic chi-squared distributions (as the number of time-series observations goes to infinity). In that paper, we did not discuss the exact distributions of these test statistics. In this paper, we show that under the assumption of normality of residuals, the test statistics reported in ST involving asymptotic chisquared distributions have convenient finite sample distributions which can be used to test some of the hypotheses examined there.1 In doing so we reaffirm that the average impact on shareholder wealth of the Williams Amendments was negative and statistically significant for the