The minimum inaccuracy estimates in \(\chi ^ 2\) tests for goodness of fit with fuzzy observations
type: Item
instance of: scholarly article
title: The minimum inaccuracy estimates in \(\chi ^ 2\) tests for goodness of fit with fuzzy observations
zbMATH Open document ID: 0665.62009
DOI: 10.1016/0378-3758(88)90055-9
author: Norberto Corral
author: M. Ángeles Gil
author: Pedro Gil
published in: Journal of Statistical Planning and Inference
publication date: 1988-01-01T00:00:00Z
review text: Experiments, where fuzzy information (fi) \(x_ k\) is obtained from X, are characterized. The information can be classified into classes and a fuzzy partition is determined. Under a hypothetical probability measure P the probability of the fi is calculable. An unknown probability vector is associated with the classes and fixes the behavior of the true probability of the fi system. Once a sample is selected the goodness of fit problem arises. The classical results of \(\chi^ 2\)-tests are reanalyzed in a fuzzy-set environment. The extension is straightforward and Pearson's statistic is used with a similar interpretation. An example fixes the similarity between the usual meaning of large values of the statistic in the classical frame and in the fuzzy one. When some parameters are unknown the corresponding results are stated by using a likelihood function. This agrees with the structure of a measure of inaccuracy (mi) of order \(\beta =2\) in information theory. The concept of minimum inaccuracy is used for deriving an estimator. It looks as an mi of order \(\beta =1\). Under some mild regularity conditions an estimator attaining the lower bound of the variance is the unique optimum estimate if some assumptions are fulfilled. Then estimators for some distributions are given. The mi-estimate for the \(\chi^ 2\)-statistic of goodness of fit is derived. An example illustrates its behavior.
Mathematics Subject Classification ID: 62B10
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 94D05
zbMATH DE Number: 4088702
zbMATH Keywords: fuzzy information
zbMATH Keywords: fuzzy partition
zbMATH Keywords: goodness of fit
zbMATH Keywords: Pearson's statistic
zbMATH Keywords: likelihood function
zbMATH Keywords: measure of inaccuracy
MaRDI profile type: Publication
full work available at URL: 0378-3758(88)90055-9
OpenAlex ID: W2062588968
cites work: On the use of Zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information
cites work: The fuzzy decision problem: An approach to the problem of testing statistical hypotheses with fuzzy information
cites work: The Use of Maximum Likelihood Estimates in $\chi^2$ Tests for Goodness of Fit
cites work: Q3734846
cites work: Q5842591
cites work: The fuzzy decision problem: An approach to the point estimation problem with fuzzy information
cites work: THE MINIMUM INACCURACY PRINCIPLE IN ESTIMATING POPULATION PARAMETERS FROM GROUPED DATA
cites work: Q3849332
cites work: Q4116383
cites work: An inaccuracy function of type \(\beta\)
cites work: Fuzzy sets
cites work: Probability measures of fuzzy events
cites work: Fuzzy sets as a basis for a theory of possibility
description: scientific article; zbMATH DE number 4088702