María Rosa Casals

Bootstrap approach to the multi-sample test of means with imprecise data

A bootstrap approach to the multi-sample test of means for imprecisely valued sample data is introduced. For this purpose imprecise data are modelled in terms of fuzzy values. Populations are identified with fuzzy-valued random elements, often referred to in the literature as fuzzy random variables. An example illustrates the use of the suggested method. Finally, the adequacy of the bootstrap approach to test the multi-sample hypothesis of means is discussed through a simulation comparative study.

On the use of Zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information

Abstract A statistical hypothesis is an assertion about the distribution of an experiment. We consider the study of the problem of testing a statistical hypothesis (that is, the problem of concluding whether or not the hypothesis is correct) on the basis of data from the experiment, when its outcomes do not provide exact but rather fuzzy information. For establishing optimality criteria of testing we will use the definition of probability of a fuzzy event, given by Zadeh, in order to extend both Neyman-Pearson and Bayes theories, to the fuzzy framework. Then, we will analyze several properties for the new criteria. Particularly, the goodness of optimal procedures in both the fuzzy and the nonfuzzy situation, will be compared for each criterion. Finally, we will apply the extended criteria for testing simple hypotheses. This application leads us to prefer Bayesian procedures to Neyman-Pearson procedures in the fuzzy context.

A distance-based statistical analysis of fuzzy number-valued data

Abstract Real-life data associated with experimental outcomes are not always real-valued. In particular, opinions, perceptions, ratings, etc., are often assumed to be vague in nature, especially when they come from human valuations. Fuzzy numbers have extensively been considered to provide us with a convenient tool to express these vague data. In analyzing fuzzy data from a statistical perspective one finds two key obstacles, namely, the nonlinearity associated with the usual arithmetic with fuzzy data and the lack of suitable models and limit results for the distribution of fuzzy-valued statistics. These obstacles can be frequently bypassed by using an appropriate metric between fuzzy data, the notion of random fuzzy set and a bootstrapped central limit theorem for general space-valued random elements. This paper aims to review these ideas and a methodology for the statistical analysis of fuzzy number data which has been developed along the last years.

A note on the operativeness of Neyman-Pearson tests with fuzzy information

Abstract In a previous paper we have studied the extension of the Neyman-Pearson optimality criterion of testing statistical hypotheses when the available experimental information involves fuzzy imprecision. This extension usually becomes unmanageable for small samples. Nevertheless. we will verify that when the sample size is large enough the application of the Central Limit Theorem determines an operative extension.

Introduction to ANOVA with Fuzzy Random Variables

In this paper we develop an introductory study on the Analysis of Variance when we deal with fuzzy random variables. Two different approaches are presented to solve the oneway ANOVA hypothesis testing. Finally, some remarks on future directions are included.

The Median of a Random Interval

In dealing with real-valued random variables, the median of the distribution is the ‘central tendency’ summary measure associated with its ‘middle position’. When available random elements are interval-valued, the lack of a universal ranking of values makes it impossible to formalize the extension of the concept of median as a middle-position summary measure. Nevertheless, the use of a generalized L 1 Hausdorff-type metric for interval data enables to formalize the median of a random interval as the central-tendency interval(s) minimizing the mean distance with respect to the random set values, by following the alternate equivalent way to introduce the median in the real-valued case. The expression for the median(s) is obtained, and main properties are analyzed. A short discussion is made on the main different features in contrast to the real-valued case.

Testing ‘Two-Sided’ Hypothesis about the Mean of an Interval-Valued Random Set

Interval-valued observations arise in several real-life situations, and it is convenient to develop statistical methods to deal with them. In the literature on Statistical Inference with single-valued observations one can find different studies on drawing conclusions about the population mean on the basis of the information supplied by the available observations. In this paper we present a bootstrap method of testing a ‘two-sided’ hypothesis about the (interval-valued) mean value of an interval-valued random set based on an extension of the t statistic for single-valued data. The method is illustrated by means of a real-life example.

Central tendency for symmetric random fuzzy numbers

Abstract Random fuzzy numbers are becoming a valuable tool to model and handle fuzzy-valued data generated through a random process. Recent studies have been devoted to introduce measures of the central tendency of random fuzzy numbers showing a more robust behaviour than the so-called Aumann-type mean value. This paper aims to deepen in the (rather comparative) analysis of these centrality measures and the Aumann-type mean by examining the situation of symmetric random fuzzy numbers. Similarities and differences with the real-valued case are pointed out and theoretical conclusions are accompanied with some illustrative examples.

Test of One-Sided Hypotheses on the Expected Value of a Fuzzy Random Variable

In this paper we present a procedure to test one-sided haypotheses about the population expected value of a fuzzy random variable. This procedure is based on a parameterized ranking function making the hypotheses being equivalent to classical ones for the population mean of a real-valued random variable.

Measuring the Dissimilarity Between the Distributions of Two Random Fuzzy Numbers

In a previous paper the fuzzy characterizing function of a random fuzzy number was introduced as an extension of the moment generating function of a real-valued random variable. Properties of the fuzzy characterizing function have been examined, among them, the crucial one proving that it unequivocally determines the distribution of a random fuzzy number in a neighborhood of 0. This property suggests to consider the empirical fuzzy characterizing function as a tool to measure the dissimilarity between the distributions of two random fuzzy numbers, and its expected descriptive potentiality is illustrated by means of a real-life example.