Madan L. Puri

Differentials of fuzzy functions

Abstract In this paper the Radstrom embedding theorem (Proc. Amer. Math. Soc. 3 (1952), 165) is generalized and is used to define the concept of the differential of a fuzzy function.

Nonparametric methods in general linear models

Distribution theory of rank statistics: Distribution Theory of Linear Rank-Order Statistics Distribution Theory of Signed Rank Order Statistics Distribution Theory of Multivariate Linear Rank-Order Statistics Nonparametric inference in linear models: Distribution-Free Rank-Order Tests for Some Linear Hypotheses Rank-Order Estimation Theory in Some Linear Models Asymptotically Distribution-Free Aligned Rank- Order Tests for Some General Linear Hypotheses Rank-Order Tests for Miscellaneous Problems in Linear Models Appendix.

CONVERGENCE THEOREM FOR FUZZY MARTINGALES

Abstract We study fuzzy set-valued measures in a Banach space and their relationships to fuzzy random variables. Our main result is a convergence theorem for fuzzy martingales. Our tool is a Radon-Nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures.

19 Nonparametric methods in design and analysis of experiments

Publisher Summary This chapter focuses on pure rank statistics in factorial designs. Pure rank statistics are invariant under any strict monotone transformation of the data and are robust against outliers. In addition, they are applicable to ordinal data such as scores in psychological tests, grading scales to describe the degree of the damage of plants or trees in ecological or environmental studies. The classical models of analysis of variance are generalized (ANOVA) in such a way that not only the assumption of normality of the error terms is relaxed but also the structure of the designs is introduced in a broader framework. In addition, the concept of treatment effects is redefined within this framework. To identify the testing problem underlying the different rank procedures, the relations between the hypotheses in the general model and in the standard linear model are investigated in the chapter. The chapter concludes with a discussion of random-factor model along with the rank procedures for heteroscedastic mixed models.

A possibility measure is not a fuzzy measure

Abstract In this note we show that a possibility measure is not a particular type of fuzzy measure, except in trivial cases.

The multivariate nonparametric Behrens–Fisher problem

Abstract In this paper, we consider the multivariate case of the so-called nonparametric Behrens–Fisher problem where two samples with independent multivariate observations are given and the equality of the marginal distribution functions under the hypothesis in the two groups is not assumed. Moreover, we do not require the continuity of the marginal distribution functions so that data with ties and, particularly, multivariate-ordered categorical data are covered by this model. A multivariate relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within each component and we derive the asymptotic distribution of this estimator. Moreover, the unknown asymptotic covariance matrix of the centered vector of the estimated relative treatment effects is estimated and its L 2 -consistency is proved. To test the hypothesis of no treatment effect, we consider the rank version of the Wald-type statistic (as used in Puri and Sen, Nonparametric Methods in Multivariate Analysis, Wiley, New York, 1971) and the rank version of the ANOVA-type statistic which was suggested by Brunner et al. [J. Amer. Statist. Assoc. 92 (1997) 1494–1502] for univariate nonparametric models. Simulations show that the ANOVA-type statistic appears to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads to more or less liberal decisions. Regarding the power, none of the two statistics is uniformly superior to the other.

Time series analysis via rank order theory: signed-rank tests for ARMA models

An asymptotic distribution theory is developed for a general class of signed-rank serial statistics, and is then used to derive asymptotically locally optimal tests (in the maximin sense) for testing an ARMA model against other ARMA models. Special cases yield Fisher-Yates, van der Waerden, and Wilcoxon type tests. The asymptotic relative efficiencies of the proposed procedures with respect to each other, and with respect to their normal theory counterparts, are provided.(This abstract was borrowed from another version of this item.)

Nonparametric Methods for Stratified Two-Sample Designs with Application to Multiclinic Trials

Abstract Motivated by some problems arising from multiclinic trials, we consider stratified two-sample designs. Nonparametric effects are defined and nonparametric hypotheses are formulated in a design where treatment, centers (strata), and interactions are assumed to be fixed factors. The interpretation of the nonparametric effects and hypotheses is analyzed in two classes of semiparametric models: the linear models and models with Lehmann alternatives. The case where centers and interactions are assumed to be random factors, the so-called mixed model, is also considered. Nonparametric effects and hypotheses are defined for general models, and their properties are analyzed in corresponding linear models and in models with Lehmann alternatives. The nonparametric effects are estimated by linear rank statistics where the ranks over all centers are used. The mixed model for repeated (baseline and endpoint) observations is briefly considered, and rank procedures are also proposed for this model. All procedure...

Asymptotically distribution-free aligned rank order tests for composite hypotheses for general multivariate linear models

For general multivariate linear models, a composite hypothesis does not usually induce invariance of the joint distribution under appropriate groups of transformations, so that genuinely distribution-free tests do not usually exist. For this purpose, some aligned rank order statistics are incorporated in the proposal and study of a class of asymptotically distribution-free tests. Tests for the parallelism of several multiple regression surfaces are also considered. Finally the optimal properties of these tests are discussed.

Central limit theorems for generalized set-valued random variables

Abstract We give central limit theorems for generalized set-valued random variables whose level sets are compact both in R d or in a Banach space under milder conditions than those obtained recently by the latter two authors.