Manuel Montenegro

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.

Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data

In previous studies we have stated that the well-known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables, when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one-dimensional Euclidean space. In this paper we show that the one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). This extension is immediately developed by combining some tools in the literature, namely, bootstrap techniques on Banach spaces, a metric between fuzzy sets based on the support function, and an embedding of the space of fuzzy random variables into a Banach space which is based on the support function.

Two-sample hypothesis tests of means of a fuzzy random variable

Abstract In this paper we will consider some two-sample hypothesis tests for means concerning a fuzzy random variable in two populations. For this purpose, we will make use of a generalized metric for fuzzy numbers, and we will develop an exact study for the case of normal fuzzy random variables and an asymptotic study for the case of simple general fuzzy random variables.

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.

Regression and correlation analyses of a linear relation between random intervals

In this paper we develop regression and corrclation analyses of a certain general linear relation between two random elements whose values are non-empty compact intervals. To this purpose, we firstly extend the least-squares method to deal with the involved random elements on the basis of a generalized metric defined on the space of the considered intervals. As a complementary study, a coefficient quantifying the strength of the linear relation between the two random elements is also presented, and a discussion of the extreme values for this measure is presented. A real-life example illustrates these studies.

Multi-sample test-based clustering for fuzzy random variables

A clustering method to group independent fuzzy random variables observed on a sample by focusing on their expected values is developed. The procedure is iterative and based on the p-value of a multi-sample bootstrap test. Thus, it simultaneously takes into account fuzziness and stochastic variability. Moreover, an objective stopping criterion leading to statistically equal groups different from each other is provided. Some simulations to show the performance of this inferential approach are included. The results are illustrated by means of a case study.

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.

Descriptive analysis of responses to items in questionnaires. Why not using a fuzzy rating scale

In evaluating aspects like quality perception, satisfaction or attitude which are intrinsically imprecise, the fuzzy rating scale has been introduced as a psychometric tool that allows evaluators to give flexible and quite accurate, albeit non numerical, ratings. The fuzzy rating scale integrates the skills associated with the visual analogue scale, because of the total freedom in assessing ratings, with the ability of fuzzy linguistic variables to capture the natural imprecision in evaluating such aspects.Thanks to a recent methodology, the descriptive analysis of the responses to a fuzzy rating scale-based questionnaire can be now carried out. This paper aims to illustrate such an analysis through a real-life example, as well as to show that statistical conclusions can often be rather different from the conclusions one could get from either Likert scale-based responses or their fuzzy linguistic encoding. This difference encourages the use of the fuzzy rating scale when statistical conclusions are important, similarly to the use of exact real-valued data instead of grouping them.

Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications

The fuzzy rating scale was introduced as a tool to measure intrinsically ill-defined/ imprecisely-valued attributes in a free way. Thus, users do not have to choose a value from a class of prefixed ones (like it happens when a fuzzy semantic representation of a linguistic term set is considered), but just to draw the fuzzy number that better represents their valuation or measurement. The freedom inherent to the fuzzy rating scale process allows users to collect data with a high level of richness, accuracy, expressiveness, diversity and subjectivity, what is especially valuable for statistical purposes.

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.