## Testing fuzzy hypotheses based on vague observations: a $$p$$-value approach.(English)Zbl 1440.62104

Summary: 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.

### MSC:

 62F86 Parametric inference and fuzziness 62F03 Parametric hypothesis testing

ump; Maple
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### References:

 [1] Arnold BF (1998) Testing fuzzy hypotheses with crisp data. Fuzzy Sets Syst 94: 323–333 · Zbl 0940.62015 [2] Buckley JJ (2005) Fuzzy statistics: hypothesis testing. Soft Comput 9: 512–518 · Zbl 1079.62026 [3] Casals MR, Gil MA, Gil P (1986) On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information. Fuzzy Sets Syst 20: 175–190 · Zbl 0611.62018 [4] Colubi A (2009) Statistical inference about the means of fuzzy random variables: applications to the analysis of fuzzy- and real-valued data. Fuzzy Sets Syst 160: 344–356 · Zbl 1175.62021 [5] Couso I, Sanchez L (2008) Defuzzification of fuzzy p-values. In: Advances in soft computing, vol 48 (Soft methods for handling variability and imprecision). Springer, Heidelberg, pp 126–132 [6] Denoeux T, Masson MH, Hébert PA (2005) Nonparametric rank-based statistics and significance tests for fuzzy data. Fuzzy Sets Syst 153: 1–28 · Zbl 1062.62075 [7] Dubois D, Prade H (1988) Possibility theory. Plenum Press, New-York · Zbl 0645.68108 [8] Filzmoser P, Viertl R (2004) Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59: 21–29 · Zbl 1052.62009 [9] Geyer CJ, Meeden GD (2005) Fuzzy and randomized confidence intervals and p-values. Stat Sci 20: 358–366 · Zbl 1130.62319 [10] Grzegorzewski P (2000) Testing statistical hypotheses with vague data. Fuzzy Sets Syst 112: 501–510 · Zbl 0948.62010 [11] Grzegorzewski P (2001) Fuzzy tests–defuzzification and randomization. Fuzzy Sets Syst 118: 437–446 · Zbl 0996.62013 [12] Knight K (2000) Mathematical statistics. Chapman & Hall/CRC, Boca Raton · Zbl 0935.62002 [13] Lubiano MA, Gil MA (1999) Estimating the expected value of fuzzy random variables in random samplings from finite populations. Stat Pap 40: 277–295 · Zbl 0942.62010 [14] Maple 9.5, Waterloo Maple Inc., Waterloo, Canada [15] Neyman J, Pearson ES (1933) The theory of statistical hypotheses in relation to probabilities a priori. Proc Camb Phil Soc 29: 492–510 · Zbl 0008.02401 [16] Parchami A, Taheri SM, Mashinchi M (2010) Fuzzy p-value in testing fuzzy hypotheses with crisp data. Stat Pap 51: 209–226 · Zbl 1247.62105 [17] Taheri SM, Arefi M (2009) Testing fuzzy hypotheses based on fuzzy test statistic. Soft Comput 13: 617–625 · Zbl 1170.62016 [18] Taheri SM, Behboodian J (1999) Neyman–Pearson Lemma for fuzzy hypotheses testing. Metrika 49: 3–17 · Zbl 1093.62520 [19] Taheri SM, Behboodian J (2001) A Bayesian approach to fuzzy hypotheses testing. Fuzzy Sets Syst 123: 39–48 · Zbl 0983.62015 [20] Taheri SM (2003) Trends in fuzzy statistics. Austrian J Stat 32: 239–257 [21] Tanaka H, Okuda T, Asai K et al (1979) Fuzzy information and decision in a statistical model. In: Gupta MM (eds) Advances in fuzzy set theory and applications.. North-Holland, Amsterdam, pp 303–320 [22] Torabi H, Behboodian J, Taheri SM (2006) Neyman–Pearson lemma for fuzzy hypotheses testing with vague data. Metrika 64: 289–304 · Zbl 1103.62021 [23] Torabi H, Behboodian J (2007) Likelihood ratio test for fuzzy hypotheses testing. Stat Pap 48: 509–522 · Zbl 1125.62009 [24] Torabi H, Behboodian J (2005) Sequential probability ratio test for fuzzy hypotheses testing with vague data. Austrian J Stat 34: 25–38 [25] Viertl R (1991) On Bayes’ theorem for fuzzy data. Stat Pap 32: 115–122 · Zbl 0719.62011 [26] Viertl R (1996) Statistical methods for non-precise data. CRC Press, Boca Raton, Florida · Zbl 1047.93534 [27] Viertl R (2006) Univariate statistical analysis with fuzzy data. Comput Stat Data Anal 51: 133–147 · Zbl 1157.62368 [28] Wang X, Kerre EE (2001) Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets Syst 118: 387–405 · Zbl 0971.03055 [29] Watanabe N, Imaizumi T (1993) A fuzzy statistical test of fuzzy hypotheses. Fuzzy Sets Syst 53: 167–178 · Zbl 0795.62025 [30] Yuan Y (1991) Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst 43: 139–157 · Zbl 0747.90003 [31] Zadeh LA (1965) Fuzzy sets. Inf Control 8: 338–359 · Zbl 0139.24606
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