Fuzzy decision in testing hypotheses by fuzzy data: Two case studies (original) (raw)
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In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate the approach.
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This paper deals with the problem of testing statistical hypotheses when both the hypotheses and data are fuzzy. To this end, we first introduce the concept of fuzzy p-value and then develop an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level. Numerical examples are provided to illustrate the approach for different cases.
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Journal of Uncertainty Analysis and Applications, 2016
In this paper, a new method is proposed for testing fuzzy hypotheses based on the following two generalized p-values: (1) the generalized p-value of null fuzzy hypothesis against alternative fuzzy hypothesis and (2) the generalized p-value of alternative fuzzy hypothesis against null fuzzy hypothesis. In the proposed method, each generalized p-value is formulated on the basis of Zadeh's probability measure of fuzzy events. The introduced p-value method has several advantages over the common p-value methods for testing fuzzy hypotheses. A few illustrative examples and also an agricultural example, based on a real-world data set, are given to clarify the proposed method.
Proceedings of the 9th International Joint Conference on Computational Intelligence, 2017
We extend the classical approach of hypothesis testing to the fuzzy environment. We propose a method based on fuzziness of data and on fuzziness of hypotheses at the same time. The fuzzy p-value with its α-cuts is provided and we show how to defuzzify it by the signed distance method. We illustrate our method by numerical applications where we treat a one and a two sided test. For the one-sided test, applying our method to the same data and performing tests on the same significance level, we compare the defuzzified p-values between different cases of null and alternative hypotheses.
European Journal of Operational Research, 1986
Departamento de Matemáticas Universidad de Ov i e do Oviedo, SPA 1 N This paper is devoted to the problem of testing statistical hypotheses about an experiment, when the available information from its sampling is 11 vague 11 • When the information supplied by the experimental sampling is exact, the problem of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem. In addition, decision procedures may be used in problems of testing hypotheses. In a similar manner, the problem of testing statistical hypothese about an experiment when the available sample information is vague, is approached in this paper as a particular fuzzy decision problem (as defined by H. Tanaka, T. Okuda and K. Asai). This approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information, whereas the present information about it can be expressed by means of fuzzy information. The preceding framework allows us to extend the notion of risk function and sorne nonfuzzy decision procedures to the fuzzy case, and particularize them to the problem of testing. Finally, several illustrative examples are presented.
Fuzzy statistical tests based on fuzzy confidence intervals
2011
A general method is proposed to construct fuzzy tests for testing statistical hypotheses about an unknown fuzzy parameter, when the data available are observations of a fuzzy random variable. The proposed method to construct such tests is essentially based on the one-to-one correspondence between the acceptance region of any level test and the level confidence interval for the same parameter. By using this equivalency in the case of fuzzy environment, we construct the fuzzy test showing the degree of acceptability of the null and alternative hypotheses of interest. A numerical example in the field of lifetime study is given to clarify the theoretical results. Keywords: Fuzzy confidence interval, Fuzzy parameter, Fuzzy random variable, Fuzzy test, Lifetime data 1 INTRODUCTION AND BACKGROUND In the classical theory of parametric statistical inference there is a one-to-one relationship between a subset of the parameter space for which the null hypothesis is accepted and the structure o...
Testing Statistical Hypotheses in Fuzzy Environment
1997
In traditional statistics all parameters of the mathematical model and possible observations should be well defined. Sometimes such assumption appears too rigid for the real-life problems, especially while dealing with linguistic data or imprecise requirements. To relax this rigidity fuzzy methods are incorporated into statistics. We review hitherto existing achievements in testing statistical hypotheses in fuzzy environment, point out their advantages or disadvantages and practical problems. We propose also a formalization of that decision problem and indicate the directions of further investigations in order to construct a more general theory.
Testing Statistical Hypotheses Based on Fuzzy Confidence Intervals
Austrian Journal of Statistics, 2016
A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the case when the available data are also imprecise. The proposed method is based on the relationship between the acceptance region of statistical tests at level β and confidence intervals for the parameter of interest at confidence level 1 − β. First, a fuzzy confidence interval is constructed for the fuzzy parameter of interest. Then, using such a fuzzy confidence interval, a fuzzy test function is constructed. The obtained fuzzy test, contrary to the classical approach, leads not to a binary decision (i.e. to reject or to accept the given null hypothesis) but to a fuzzy decision showing the degrees of acceptability of the null and alternative hypotheses. Numerical examples are given to demonstrate the theoretical results, and show the possible applications in testing hypotheses based on fuzzy observations. Zusammenfassung: Aufbauend auf der Beziehung zwischen Konfidenzintervallen für Parameter von stochastischen Modellen und statistischen Tests für Parameterhypothesen, wird eine Verallgemeinerung für den Fall unscharfer Daten und zugehörigen unscharfen Konfidenzintervallen vorgeschlagen. Die zugehörigen verallgemeinerten Tests liefern unscharfe Entscheidungen mit Graden von Annahmen bzw. Ablehungen von Hypothesen. Numerische Beispiele aus der Lebensdaueranalyse zeigen die Anwendbarkeit von solchen statistischen Tests für unscharfe Beobachtungen.
A Comparative Study of One-Sample t-Test Under Fuzzy Environments
This paper proposes a method for testing hypotheses over one sample t-test under fuzzy environments using trapezoidal fuzzy numbers (tfns.). In fact, trapezoidal fuzzy numbers have many advantages over triangular fuzzy numbers as they have more generalized form. Here, we have approached a new method where trapezoidal fuzzy numbers are defined in terms of alpha level of trapezoidal interval data and based on this approach, the test of hypothesis is performed. More over the proposed test is analysed under various types of trapezoidal fuzzy models such as Alpha Cut Interval, Membership Function, Ranking Function, Total Integral Value and Graded Mean Integration Representation. And two numerical examples have been illustrated. Finally a comparative view of all conclusions obtained from various test is given for a concrete comparative study.