A Bayesian approach to fuzzy hypotheses testing

doi:10.1016/S0165-0114(00)00134-2

Fuzzy Sets and Systems

Volume 123, Issue 1, 1 October 2001, Pages 39-48

https://doi.org/10.1016/S0165-0114(00)00134-2 Get rights and content

Abstract

In statistical decisions we may come across imprecise (fuzzy) concepts. One important case is a situation where we are interested in testing the hypotheses that are fuzzy rather than crisp. In this paper we consider and analyze the testing of fuzzy hypotheses on the basis of a Bayesian approach. We illustrate our method by some examples, and compare it with previous works on this topic.

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In addition, some authors discuss tests on parameters of specific distributions, that is, or without a specific distribution [19,20,26,28,30,34,36,38,39,58]. Moreover, there can also be found tests for regression coefficients [156], Hotelling-type tests [105], tests for the number of non-conforming objects in a lot [157] or tests for record-based Pareto analysis [35].
In statistical inference hypotheses related to different kinds of phenomena are formulated, and then data are collected and analyzed, which either confirm or falsify these hypotheses. Considering traditional statistics, in the underlying models hypotheses and sample data should be well defined. However, these models are often inadequate with regard to real-life problems, as theoretical specifications and observed information are frequently imprecise, vague, incomplete, qualitative, linguistic or noisy. To relax this rigidity, numerous researchers have proposed modifications and extensions of statistical inference approaches with the help of concepts of fuzzy statistics. In the meantime there are many papers on the topic of hypothesis testing in fuzzy environments, especially based on fuzzy hypotheses and/or by using fuzzy data. In order to structure this variety of contributions, proposals and applications, we give a comprehensive systematic review in this paper and offer a bibliography on fuzzy hypothesis testing. The paper seeks to consolidate the topic of fuzzy hypothesis testing with the purpose of supporting new researchers in this field and highlighting potential directions for future research.
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▷ Applications of the proposed methodology Applying Bayesian approach, [7] as also [38] firstly studied fuzzy hypothesis testing with crisp data under the basic assumption of availability of precise prior information (distribution) regarding the considered parameter. In particular, [7] considered one- and two-tailed hypotheses formulation and used the representation theorem to transform the original fuzzy null hypothesis testing problem to the problem of crisp null hypotheses testing against complementary alternative hypotheses on a family of γ-cuts.
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The approaches that fall into this category include [18,20,21,93,155]. All classical approaches use crisp probability values to represent the probability of events but recently the Bayesian approach has been generalised to accommodate fuzzy data [58,66,164,214]. The key idea of the fuzzy Bayesian approach is to use fuzzy numbers to represent the prior/conditional probabilities of the root/intermediate nodes [86,126,193,245,246,255].
Safety and reliability are rigorously assessed during the design of dependable systems. Probabilistic risk assessment (PRA) processes are comprehensive, structured and logical methods widely used for this purpose. PRA approaches include, but not limited to Fault Tree Analysis (FTA), Failure Mode and Effects Analysis (FMEA), and Event Tree Analysis (ETA). In conventional PRA, failure data about components is required for the purposes of quantitative analysis. In practice, it is not always possible to fully obtain this data due to unavailability of primary observations and consequent scarcity of statistical data about the failure of components. To handle such situations, fuzzy set theory has been successfully used in novel PRA approaches for safety and reliability evaluation under conditions of uncertainty. This paper presents a review of fuzzy set theory based methodologies applied to safety and reliability engineering, which include fuzzy FTA, fuzzy FMEA, fuzzy ETA, fuzzy Bayesian networks, fuzzy Markov chains, and fuzzy Petri nets. Firstly, we describe relevant fundamentals of fuzzy set theory and then we review applications of fuzzy set theory to system safety and reliability analysis. The review shows the context in which each technique may be more appropriate and highlights the overall potential usefulness of fuzzy set theory in addressing uncertainty in safety and reliability engineering.
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A Bayesian approach to fuzzy hypotheses testing

Abstract

Fuzzy Sets and Systems

European J. Oper. Res.

European J. Oper. Res.

Inform. Control

J. Math. Anal. Appl.

Statistical Decision Theory and Bayesian Analysis

Bayesian testing of fuzzy parametric hypotheses from fuzzy information, RAIRO

Oper. Res.