Neyman–Pearson Lemma for Fuzzy Hypotheses Testing with Vague Data

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which both hypotheses and observations are imprecise. This paper tries to develop a new approach for testing fuzzy hypothesis when the available data are fuzzy, too. First, some definitions are provided, such as: fuzzy sample space, fuzzy-valued random sample, and fuzzy-valued random variable. Then, the problem of fuzzy hypothesis testing with vague data is formulated. Finally, we state and prove a generalized Neyman–Pearson Lemma for such problem. The proposed approach is illustrated by some numerical examples.

[1]  María Rosa Casals,et al.  A note on the operativeness of Neyman-Pearson tests with fuzzy information , 1989 .

[2]  R. Viertl,et al.  Testing hypotheses with fuzzy data: The fuzzy p-value , 2004 .

[3]  M. Gil,et al.  Constructive definitions of fuzzy random variables , 1997 .

[4]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[5]  Jean J. Saade,et al.  Extension of fuzzy hypothesis testing with hybrid data , 1994 .

[6]  J. Behboodian,et al.  Neyman-Pearson Lemma for fuzzy hypotheses testing , 1999 .

[7]  María Asunción Lubiano,et al.  Two-sample hypothesis tests of means of a fuzzy random variable , 2001, Inf. Sci..

[8]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[9]  Tadashi Imaizumi,et al.  A fuzzy statistical test of fuzzy hypotheses , 1993 .

[10]  Seyed Mahmoud Taheri,et al.  Fuzzy Hypotheses Testing with Fuzzy Data: A Bayesian Approach , 2002, AFSS.

[11]  Nearly ideal binary communication in squeezed channels , 2001, quant-ph/0104057.

[12]  Bernhard F. Arnold.  Testing fuzzy hypotheses with crisp data , 1998, Fuzzy Sets Syst..

[13]  Seyed Mahmoud Taheri,et al.  A Bayesian approach to fuzzy hypotheses testing , 2001, Fuzzy Sets Syst..

[14]  D. Ralescu.  Fuzzy random variables revisited , 1995 .

[15]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[16]  Bernhard F. Arnold,et al.  Testing fuzzy linear hypotheses in linear regression models , 2003 .

[17]  Przemyslaw Grzegorzewski,et al.  Testing statistical hypotheses with vague data , 2000, Fuzzy Sets Syst..

[18]  Pedro Gil,et al.  On the use of Zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information , 1986 .

[19]  Bernhard F. Arnold,et al.  An approach to fuzzy hypothesis testing , 1996 .

[20]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[21]  M. Casals,et al.  Bayesian testing of fuzzy parametric hypotheses from fuzzy information , 1993 .

[22]  Jean J. Saade,et al.  Fuzzy hypothesis testing with hybrid data , 1990 .

[23]  P. Billingsley,et al.  Probability and Measure , 1980 .

[24]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[25]  Madan M. Gupta,et al.  Approximate reasoning in expert systems , 1985 .

[26]  Iickho Song,et al.  A fuzzy decision problem based on the generalized Neyman-Pearson criterion , 1992 .

[27]  Baoding Liu.  Uncertainty Theory: An Introduction to its Axiomatic Foundations , 2004 .