Ronald H. Randles

A Two-Sample Adaptive Distribution-Free Test

Abstract An adaptive distribution-free test is proposed for the two-sample location problem. First, the data are used to assess the tailweight and skewness of the underlying distributions. This leads to the selection and then application, with the same data, of one of several common rank tests for shift, such as the Mann-Whitney-Wilcoxon test. The preliminary selection is made in a way that insures the testing procedure is distribution-free. A Monte Carlo study shows that the adaptive test has excellent power over a wide class of distributions and is preferable to certain prominent nonadaptive tests.

Generalized Linear and Quadratic Discriminant Functions Using Robust Estimates

Abstract Two new methods of constructing robust linear and quadratic discriminant functions are introduced. The first is a generalization of Fishers procedure for finding a linear discriminant function. It places less weight on those observations that are far from the overlapping regions of the two populations. The second new method substitutes M-estimates of the means and the covariance matrices into the usual expressions for the linear and quadratic discriminant functions. Monte Carlo results indicate lower misclassification probabilities for these schemes compared to Fishers linear discriminant function in cases of heavy-tailed or contaminated distributions.

Adaptive distribution-free tests

Some adaptive distribution-free tests about location parameters of symmetric distributions are processed for the one-and two sample cases. The preliminary inferences in these adaptive schemes is to classify the underlying distributions as having light, medium, or heave rails with a statistic that is assentialy the range of the sample divided by the mean deviation from the sample median. If this inference suggests medium tails, a wilcoxon-type test is used; for heavy tails, a median-type test is used. A modified Wilcoxon test (alirnating middle sample itams) is processed for ligntatailed distributions and portions of its null distribution are trasulated. The favorable performance of these adaotive rets is demanstrated by a monta carla study.

Distribution-Free Partial Discriminant Analysis

Abstract A distribution-free rank procedure is proposed for use in partial discrimination problems involving two populations. It is shown that this procedure can be applied with virtually any discriminant function. Moreover, the discriminant function may be selected after observing the samples on which it is to be based. Using Monte Carlo methods the rank procedure is compared with a normal theory and a tolerance region procedure. The rank procedure was the only one that adequately controlled the probabilities of misclassification while maintaining relatively small probabilities of not classifying an observation.

Discriminant Analysis Based on Ranks

Abstract A model-free rank procedure is proposed for the two-population discrimination problem that enables the practitioner to better control the balance between the two probabilities of misclassification. The method is applied to the discriminant functions resulting from normal assumptions and also to an adaptive one which is a weighted average of the linear and quadratic discriminant functions, where the weights are determined from the data. A Monte Carlo study shows that the rank method can greatly improve the balance between the two misclassification probabilities while keeping their average comparatively small.

On the Selection of the Underlying Distribution and Adaptive Estimation

Abstract In our approach to adaptive inference, selection of the true underlying family of distributions is needed. A Bayesian selection rule, which is based on non-informative priors for the parameters, and a multiple decision solution, that maximizes the weighted sum of the probabilities of correct selection, are presented; both of these yield a decision rule of the same form. A modified maximum likelihood solution is also given and is used to construct an adaptive estimate of the location of a symmetric distribution. By a Monte Carlo study, it is found that this adaptive estimator compares most favorably to more standard estimators.

Certain Uncorrelated and Independent Rank Statistics

Abstract Some interesting examples of uncorrelated simple linear rank statistics are listed. Consideration is then given to conditions under which general rank statistics are uncorrelated or have joint distributions which enjoy certain symmetric properties. Two simple linear rank statistics that are uncorrelated are found, under suitable restrictions, to be asymptotically independent. This is followed by some illustrations in which two rank statistics are exactly independent. The importance of these results about independence in actual testing is then noted.

A Power Approximation for the Chi-Square Goodness-of-Fit Test: Simple Hypothesis Case

Abstract The limiting distribution of the chi-square goodness-of-fit statistic Tn under alternatives is noncentral chi-square if the alternative probabilities approach the null probabilities at an appropriate rate as n → ∞. It is shown that for fixed alternative probabilities, the limiting distribution of (Tn − μ n )/σ n is standard normal. Both of these asymptotic results can be used to approximate the power of the goodness-of-fit test. Numerical comparisons between these two approximations indicate that for large values of the true power, the normal approximation is best, but for moderate values of power, the chi-square approximation is best.

An Adaptive Procedure for Selecting the Population With Largest Location Parameter

An adaptive procedure for selecting the population with the largest (smallest) location parameter is given. A Monte Carlo sampling study is presented indicating that this procedure performs as well as the means procedure when the underlying distribution is medium tailed like the normal. It is shown to be superior to both the means procedure and the rank sum procedure when the underlying distribution is either very light tailed (e.g., the uniform distribution) or very heavy tailed (e.g., the Cauchy distribution).

An Adaptive M-Estimator and Its Application to a Selection Problem

A robust adaptive estimation procedure for location estimation problems is developed. Classification in this procedure is done on the basis of skewness and tailweight, using two statistics that are ratios of linear functions of sample order statistics. The associated estimators are of the general type known as M-estimators, Following the development of the adaptive location estimation procedure, an application to the k population selection problem is given. Monte Carlo results show the superiority of the adaptive procedure to the sample means procedure, the rank sum procedure, and the previously developed adaptive procedure of Randles, Ramberg, and Hogg.