Trends in Fuzzy Statistics

Authors

  • S. Mahmoud Taheri Isfahan University of Technology, Isfahan, Iran

DOI:

https://doi.org/10.17713/ajs.v32i3.459

Abstract

After introducing and developing fuzzy set theory, a lot of studies have been done to combine statistical methods and fuzzy set theory. Thisworks, called fuzzy statistics, have been developed in some branches.

In this article we review essential works on fuzzy estimation, fuzzy hypotheses testing, fuzzy regression, fuzzy Bayesian statistics, and some relevant fields.

References

B.F. Arnold. Statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size. Fuzzy Sets Syst., 75(2):365–372, 1995.

B.F. Arnold. An approach to fuzzy hypothesis testing. Metrika, 44:119–126, 1996.

B.F. Arnold. Testing fuzzy hypothesis with crisp data. Fuzzy Sets Syst., 94(2):323–333, 1998.

B.F. Arnold and O. Gerke. Testing fuzzy linear hypotheses in linear regression models. Metrika, 57:81–95, 2003.

B.F. Arnold and P. Stahlecker. Prediction in linear regression combining crisp data and fuzzy prior information. Statistics and Decisions, 16:19–33, 1998.

H. Bandemer and A. Gebhardt. Bayesian fuzzy kriging. Fuzzy Sets Syst., 112:405–418, 2000.

A. Bardossy. Note on fuzzy regression. Fuzzy Sets Syst., 37:65–75, 1990.

A. Bardossy, R. Hagaman, L. Duckstein, and I. Bogardi. Fuzzy least-squares regression: theory and applications. In J. Kacprzyk and M. Fedrizzi, editors, Fuzzy Regression

Analysis, pages 181–193. Physica-Verlag, Heidelberg, 1992.

J. Behboodian and A. Mohammadpour. Estimation of a fuzzy parameter and its application in hypothesis testing. In R. Borzouee, editor, Proc. of the Third Seminar on Fuzzy

Sets and Its Applications, pages 23–45. Univ. of Zahedan and Baluchestan, Zahedan, Iran, 2002.

J. Berg, W.M. Bergh, and U. Keymak. Probabilistic and statistical fuzzy set foundations of competitive exception learning. In Proc. of the 10-th IEEE Int. Conf. Fuzzy Systems, pages 1035–1038. Melbourne, 2001.

J.O. Berger, B. Boukai, and Y.Wang. Unified frequentist and Bayesian testing of a precise hypothesis. Stat. Sci., 12:133–160, 1997.

J.O. Berger and M. Delampady. Testing precise hypotheses. Stat. Sci., 2:317–352, 1987.

W.M. Bergh and J. Berg. Competitive exception learning using fuzzy frequency distributions. Neural Network World, 10:59–71, 2000.

S. Bodjanova. A generalized histogram. Fuzzy Sets Syst., 116:155–166, 2000.

J.J. Buckley. Decision making under risk: a comparison of Bayesian and fuzzy set method. Risk Analysis, 3:157–168, 1983a.

J.J. Buckley. Fuzzy decision making with data: applications to statistics. Fuzzy Sets Syst., 16:139–147, 1983b.

P.A. Burrough. GIS and geostatistics: essential partners for spatial analysis. Environmental and Ecological Statistics, 8:361–377, 2001.

K.Y. Cai. Parameter estimation of normal fuzzy variables. Fuzzy Sets Syst., 55:179–185, 1993.

K.Y. Cai, C.Y. Wen, and M.L. Zhang. Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context. Fuzzy Sets Syst., 42:145–172, 1991.

B. Cappelle and E.E. Kerre. On a possibilistic approach to reliability theory. In Proc. of the Second International Symposium on Uncertainty Modelling and Analysis, pages

–418. IEEE Computer Society Press, Univ. of Maryland, 1993.

B. Cappelle and E.E. Kerre. On the computability of possibilistic reliability. In B.M. Ayyoub and M.M. Gupta, editors, Uncertainty Analysis in Engineering and Sciences,

pages 325–337. Kluwer, Boston, 1997.

M.R. Casals. Bayesian testing of fuzzy parametric hypotheses from fuzzy information. RAIRO, Operations Research, 27:189–199, 1993.

M.R. Casals and M.A. Gil. A note on the operativeness of Neyman-Pearson tests with fuzzy information. Fuzzy Sets Syst., 30:215–220, 1989.

M.R. Casals, M.A. Gil, and P. Gil. On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information. Fuzzy Sets Syst., 20:175–190,

a.

M.R. Casals, M.A. Gil, and P. Gil. The fuzzy decision problem: An approach to the problem of testing statistical hypotheses with fuzzy information. Euro. J. Oper. Res.,

:71–382, 1986b.

A. Celmins. Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst., 22:260–269, 1987.

Y.O. Chang. Hybrid fuzzy least-squares regression analysis and its reliability measures. Fuzzy Sets Syst., 119:225–246, 2001.

Y.O. Chang and B.M. Ayyub. Hybrid least-squares regression analysis. In B.M. Ayyoub and M.M. Gupta, editors, Uncertainty Analysis in Engineering and Sciences, pages 179–191. Kluwer, Boston, 1997.

Y. Chen, B. Collier, P. Hu, and D. Quebedeaux. Objective evaluation of fabric softness. Textile Res. J., 70:443–448, 2000.

Y.Y. Chen. Statistical inference based on the possibility and belief measures. Trans. Amer. Math. Soc., 347:1855–1863, 1995.

Y.Y. Chen. Fuzzy analysis of statistical evidence. IEEE Trans. on Fuzzy Systems, 8: 796–799, 2000.

C.B. Cheng and E.S. Lee. Applying fuzzy adaptive network to fuzzy regression analysis. Comput. Math. Appl., 38:123–140, 1999.

C.B. Cheng and E.S. Lee. Fuzzy regression with radial basis function network. Fuzzy Sets Syst., 119:291–301, 2001.

N. Corral and M.A. Gil. The minimum inaccuracy fuzzy estimation: an extension of the maximum likelihood principle. Stochastica, 8:63–81, 1984.

N. Corral and M.A. Gil. A note on interval estimation with fuzzy data. Fuzzy Sets Syst., 28:209–215, 1988.

D.I. Dale. Probability, vague statements, and fuzzy sets. Philos. Science, 47:38–55, 1980.

M. Delgado, J.L. Verdegay, and M.A. Vila. Testing fuzzy-hypotheses, a Bayesian approach. In M.M. Gupta, editor, Approximate Reasoning in Expert Systems, pages 307–

North-Holland, Amsterdam, 1985.

P. Diamond. Least squares fitting of several fuzzy variables. In Proc. of Second IFSA Congress, pages 20–25. IFSA, Tokyo, 1987.

P. Diamond and R. Korner. Extended fuzzy linear models and least square estimates. Comput. Math. Appl., 33:15–32, 1997.

D. Dubois and H. Prade. Fuzzy sets and probability: misunderstandings, bridges, and gaps. In Proc. of the Second IEEE International Conference on Fuzzy Systems, pages 1059–1068. IEEE, Piscataway, 1993.

D. H. Dubois and Prade. Fuzzy Sets and Statistical Possibility Theory. Plenum Press, New York, 1988.

P. Dubois and H. Prade. Fuzzy sets and statistical data. Euro. J. Oper. Res., 25:345–356, 1986.

P. Dubois and H. Prade. Bayesian conditioning in possibility theory. Fuzzy Sets Syst., 92: 223–240, 1997.

J. Dunyak, I.W. Saad, and D. Wunsch. A theory of independent fuzzy probability for system reliability. IEEE Trans. on Fuzzy Systems, 7:286–294, 1999.

P. D‘Urso and T. Gastaldi. A least-squares approach to linear regression analysis. Comp. Stat. and Data Anal., 34:427–440, 2000.

T. Entani and H. Tanaka. Exponential possibility regression with interval outputs. In Proc. of the Fourth Asian Fuzzy Systems Symposium, pages 100–104. Tsukuba, Japan,

P. Filzmoser and R. Viertl. Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika (to appear), 2003.

D. Garcia, M.A. Lubiano, and C. Alonso. Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations. Inform. Sci., 138:

–184, 2001.

G.Z. Gertner and H. Zhu. Bayesian estimation in forest surveys when samples or prior information are fuzzy. Fuzzy Sets Syst., 77:277–290, 1997.

M.A. Gil. Fuzziness and loss of information in statistical problems. IEEE Trans. on Systems Man and Cybernet. SMC, 17:1012–1025, 1987.

M.A. Gil. A note on the connection between fuzzy numbers and random intervals. Stat. Prob. Lett., 13:311–319, 1992.

M.A. Gil, N. Corral, and P. Gil. The fuzzy decision problem: an approach to the point estimation problem with fuzzy information. Euro. J. Oper. Res., 22:26–34, 1985a.

M.A. Gil, N. Corral, and P. Gil. The minimum inaccuracy estimates in Â2 tests for goodness of fit with fuzzy observations. J. Stat. Plan. Inf., 19:95–115, 1985b.

M. Grabisch, H.T. Nguyen, and E. Walker. Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer, Dordrecht, 1995.

P. Grzegorzewski. Testing statistical hypotheses with vague data. Fuzzy Sets Syst., 112: 501–510, 2000.

P. Grzegorzewski. Testing fuzzy hypotheses with vague data. In C. Bertoluzzi, editor, Statistical Modeling, Analysis and Management of Fuzzy Data, pages 213–225. Physica-Verlag, Heidelberg, 2002.

P. Guo and H. Tanaka. Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst., 119:149–160, 2001.

B. Heshmaty and A. Kandel. Fuzzy linear regression and its application to forecasting in uncertain environment. Fuzzy Sets Syst., 15:159–191, 1985.

H. Hisdal. Are grades of membership probabilities? Fuzzy Sets Syst., 25:325–348, 1988.

D.H. Hong, J. Song, and H.Y. Do. Fuzzy least-squares linear regression analysis using shape preserving operations. Inform. Sci., 138:185–193, 2001.

C. Hwang and J. Yao. Independent fuzzy random variables and their application. Fuzzy Sets Syst., 82:335–350, 1996.

K. Jajuga. Linear fuzzy regression. Fuzzy Sets Syst., 20:343–353, 1986.

A. Kanagava, F. Tamaki, and H. Ohta. Control charts for process average and variability based on linguistic data. Inter. J. of Production Research, 2:913–922, 1993.

A. Kandel and W.J. Byatt. Fuzzy sets, fuzzy algebra, and fuzzy statistics. Proc. of The IEEE, 66(12):1619–1639, 1978.

P.R. Kersten. Fuzzy order statistics and their application to fuzzy clustering. IEEE Trans. on Fuzzy Systems, 7:708–712, 1999.

K.J. Kim, H. Moskowitz, and M. Koksalan. Fuzzy versus statistical linear regression. Euro. J. Oper. Res., 92:417–434, 1996.

G.J. Klir. Is there more to uncertainty than some probability theorists might have us believe? Int. J. General Systems, 15:347–378, 1989.

R. Korner. An asymptotic ®-test for the expectation of random fuzzy variables. J. Stat. Plan. Inf., 83:331–346, 2000.

B. Kosko. Fuzziness vs. probability. Inter. J. of General Systems, 17:211–240, 1990.

R. Kruse. Statistical estimation with linguistic data. Inform. Sci., 33:197–207, 1984.

R. Kruse and K.D. Meyer. Statistics with Vague Data, volume 33. Reidel, Dordrecht, 1987.

A.G. Lapiga and V.V. Polyakov. On statistical methods in fuzzy decision-making. Fuzzy Sets Syst., 47:303–311, 1992.

S. Lapointe and B. Bobee. Revision of possibility distributions: a Bayesian inference pattern. Fuzzy Sets Syst., 116:119–140, 2000.

M. Last, A. Schenker, and A. Kandel. Applying fuzzy hypothesis testing to medical data. In N. Zhong et al., editor, New Directions in Rough Sets, Data Mining, and

Granular-Soft Computing, Lecture Notes in Artificial Intelligence Series, Vol. 1711, pages 221–229. Springer, Berlin, 1999.

M. Laviolette and J.W. Seaman. Evaluating fuzzy representations of uncertainty. The Mathematical Scientist, 17:26–41, 1992.

M. Laviolette and J.W. Seaman. Unity and diversity of fuzziness - from a probability viewpoint. IEEE Trans. on Fuzzy Systems, 2:38–42, 1994.

M. Laviolette, J.W. Seaman, J.D. Barrett, and W.H. Woodall. A probabilistic and statistical view of fuzzy methods (with discussion). Technometrics, 37:249–296, 1995.

E.S. Lee. Fuzzy spatial statistics. In Selected Papers of Engineering Chemistry and Metallurgy, pages 151–157. Institute of Chemical Metallurgy, Chinese Academy of

Science, China, 1995.

E.S. Lee. Neuro-fuzzy estimation in spatial statistics. J. Math. Anal. Appl., 249:221–231, 2000.

M. Lopez-Diaz and M.A. Gil. Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications. J. Stat. Plan. Inf., 74:11–29, 1998.

M.A. Lubiano and M.A. Gil. Estimating the expected value of fuzzy random variables in random sampling from finite populations. Statistical Papers, 40:277–295, 1999.

M.A. Lubiano, M.A. Gil, and M. Lopez-Diaz. On the Rao-Blackwell Theorem for fuzzy random variables. Kybernetika, 35:167–175, 1999.

W. Luczynski and M. Matloka. Fuzzy regression models and their applications. J. Fuzzy Math., 3:583–589, 1995.

K.G. Manton, E. Stallard, and M.A. Woodbury. A multivariate event history based upon fuzzy states: estimation from longitudinal surveys with informative nonresponse. J. Official Statistics, 7:261–293, 1991.

K.G. Manton, M.A. Woodbury, and D.H. Tolley. Statistical Applications Using Fuzzy Sets. Wiley, New York, 1994.

R.A. McCain. Fuzzy confidence intervals. Fuzzy Sets Syst., 10:281–290, 1983.

M.L. Menendez. The minimum Á-divergence estimates with fuzzy observations: statistical applications. Fuzzy Sets Syst., 96:101–109, 1998.

M.L. Menendez, J.A. Pardo, and L. Pardo. Sufficient fuzzy information systems. Fuzzy Sets Syst., 32:97–105, 1989.

M.L. Menendez, J.A. Pardo, and L. Pardo. Some statistical applications of generalized jensen difference divergence measures for fuzzy information systems. Fuzzy Sets Syst.,

:169–180, 1992.

R. Mesiar and K. Piasecki. On a possible generalization of the Bayes method of inference. Fuzzy Sets Syst., 37:351–357, 1990.

M. Montenegro, M.R. Casals, M.A. Lubiano, and M.A. Gil. Two-sample hypothesis tests of a fuzzy random variable. Inform. Sci., 133:89–100, 2001.

S. Nahmias. Fuzzy variables. Fuzzy Sets Syst., 1:97–100, 1978.

W. Nather and R. Korner. Linear regression with random fuzzy numbers. In B.M. Ayyub and M.M. Gupta, editors, Uncertainty Analysis in Engineering and Sciences, pages

–211. Kluwer, Boston, 1998.

S.P. Niculescu and R. Viertl. A comparison between two fuzzy estimators for the mean. Fuzzy Sets Syst., 48:341–350, 1992.

T. Okuda. A statistical treatment of fuzzy observations: estimation problems. In Preprints of Second IFSA Congress, pages 51–55. IFSA, 1987.

M.G.A. Paris. Nearly ideal binary communication in squeezed channels. Physical Review A, 64:14304–14308, 2001.

W. Peizhuang and L. Xihui. Set-valued statistics. J. Engineering Math., 1, 1984.

G. Peters. Fuzzy linear regression with fuzzy intervals. Fuzzy Sets Syst., 63:45–55, 1994.

G. Peters. A linear forecasting model and its application to economic data. J. Forecast., 20:315–328, 2001.

K. Piasecki. On the Bayes formula for fuzzy probability measures. Fuzzy Sets Syst., 18: 183–185, 1986.

K. Piasecki. Note to Piasecki (1986). Fuzzy Sets Syst., 24:121–122, 1987.

D. Reden and W. Woodall. Properties of certain fuzzy linear regression methods. Fuzzy Sets Syst., 64:361–375, 1994.

I. Rojas, J. Ortega, F.J. Pelayo, and A. Prieto. Statistical analysis of the main parameters in the fuzzy inference process. Fuzzy Sets Syst., 102:157–173, 1999.

C. Romer and A. Kandel. Statistical tests for fuzzy data. Fuzzy Sets Syst., 72:1–26, 1995.

S. Roychowdhury andW. Pedrycz. Modeling temporal functions with granular regression and fuzzy rules. Fuzzy Sets Syst., 126:377–387, 2002.

J.J. Saade and H. Schwarzlander. Fuzzy hypothesis testing with hybrid data. Fuzzy Sets Syst., 35:197–212, 1990.

B. Sadeghpour Gildeh and D. Gien. A goodness of fit index to reliability analysis in fuzzy model. In A. Grmela, editor, Advances in Intelligent Systems, Fuzzy Systems,

Evolutionary Computation, pages 78–83. WSEAS Press, Greece, 2002a.

B. Sadeghpour Gildeh and D. Gien. dp;q-distance and Rao-Blackwell Theorem for fuzzy random variables. In Proc. of the 8th International Conference on Fuzzy Theory and

Technology. Durham, USA, 2002b.

M. Sakawa and H. Yano. Moltiobjective fuzzy linear regression analysis for fuzzy data. Fuzzy Sets Syst., 47:173–181, 1991.

A. Schenker, M. Last, and A. Kandel. Fuzzy hypotheses testing: verification-based data mining. Manuscript, 2000.

S. Schnatter. On propagation of fuzziness of data. Environmetrics, 2:241–252, 1991.

S. Schnatter. On statistical inference for fuzzy data with applications to descriptive statistics. Fuzzy Sets Syst., 50:143–165, 1992.

S. Schnatter. On fuzzy Bayesian inference. Fuzzy Sets Syst., 60:41–58, 1993.

M. Schnider and M. Craig. On the use of fuzzy sets in histogram equalization. Fuzzy Sets Syst., 45:271–278, 1992.

M. Schnider and A. Kandel. Expectations in fuzzy environments. J. Fuzzy Logic and Intelligent Systems, 3:76–93, 1993.

S. Schnider and A. Kandel. Properties of the fuzzy expected value and the fuzzy expected interval. Fuzzy Sets Syst., 26:373–385, 1988a.

S. Schnider and A. Kandel. Properties of the fuzzy expected value and the fuzzy expected interval in fuzzy environment. Fuzzy Sets Syst., 28:55–68, 1988b.

G. Sizhong. Fuzzy random set and fuzzy set-valued statistics. In Preprints of Second IFSA Congress, pages 20–25. IFSA, Tokyo, 1987.

J.C. Son, I. Song, and H.Y. Kim. A fuzzy decision problem based on the generalized Neyman-Pearson criterion. Fuzzy Sets Syst., 47:65–75, 1992.

W. Stallings. Fuzzy set theory versus Bayesian statistics. IEEE Trans. on Systems Man and Cybernet. SMC, 7:216–219, 1977.

S.M. Taheri. SPRT for non-precise hypotheses. Manuscript, 2003.

S.M. Taheri and J. Behboodian. Neyman-Pearson Lemma for fuzzy hypotheses testing. Metrika, 49:3–17, 1999.

S.M. Taheri and J. Behboodian. A Bayesian approach to fuzzy hypotheses testing. Fuzzy Sets Syst., 123:39–48, 2001.

S.M. Taheri and J. Behboodian. Fuzzy hypotheses testing with fuzzy data: a Bayesian approach. In R.P. Nikhil and M. Sugeno, editors, Advances in Soft Computing, AFSS-2002, pages 527–533. Springer, Berlin, 2002.

S. Takayanagi and N. Cliff. The fuzziness index for examining human statistical decisionmaking. In R.J. Marks, editor, Fuzzy Logic Technology and Applications, pages 509–514. IEEE Technical Activities Board, 1994.

H. Tanaka. Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst., 24:363–375, 1987.

H. Tanaka, I. Hayashi, and J. Watada. Possibilistic linear regression analysis based on possibility measure. In Preprints of Second IFSA Congress, pages 317–320. IFSA,

Tokyo, 1987.

H. Tanaka, H. Ishibuchi, and S. Yoshikawa. Exponential possibility regression analysis. Fuzzy Sets Syst., 69:305–318, 1995.

H. Tanaka and H. Lee. Exponential possibility regression analysis by identification method of possibilistic coefficients. Fuzzy Sets Syst., 106:155–165, 1999.

H. Tanaka, T. Okuda, and K. Asai. Fuzzy information and decision in statistical model. In M.M. Gupta et al., editor, Advances in Fuzzy Set Theory and Applications, pages

–320. North-Holland, Amsterdam, 1979.

H. Tanaka, S. Uejima, and K. Asai. Fuzzy linear regression model. IEEE Trans. Systems Man Cybernet., 10:2933–2938, 1980.

H. Tanaka, S. Uejima, and K. Asai. Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybernet., 12:903–907, 1982.

Y. Toyoura and J. Watada. Evaluation of fuzzy regression analysis. In Proc. of the Fourth Asian Fuzzy Systems Symposium, pages 1015–1020. Tsukuba, Japan, 2000.

Y. Uemura. A decision rule on fuzzy events. Japanese J. Fuzzy Theory and Systems, 3: 291–300, 1991.

Y. Uemura. A decision rule on fuzzy events under an observation. J. of Fuzzy Math., 1: 39–52, 1993a.

Y. Uemura. A simple decision rule on fuzzy events. Cybernet. Syst. Int. J., 24:509–521, 1993b.

R. Viertl. Is it necessary to develop a fuzzy Bayesian inference? In R. Viertl, editor, Probability and Bayesian Statistics. Plenum Press, New York, 1987.

R. Viertl. Estimation of the reliability function using fuzzy life time data. In P.K. Bose et al., editor, Quality for Progress and Development. Wiley Eastern, New Delhi, 1989.

R. Viertl. Statistical Methods for Non-Precise Data. CRC Press, Boca Raton, 1996.

R. Viertl. Statistics and integration of fuzzy functions. Environmetrics, 10:487–491, 1999.

R. Viertl. Statistics with one-dimensional fuzzy data. In C. Bertoluzzi et al., editor, Statistical Modeling, Analysis and Management of Fuzzy Data, pages 199–212. Physica-

Verlag, Heidelberg, 2002a.

R. Viertl. Statistical inference with non-precise data. In Encyclopedia of Life Support Systems. UNESCO, Paris, 2002b. On-line: www.eolss.unesco.org.

R. Viertl and D. Hareter. Bayes theorem for non-precise a-priori distribution and nonprecise data. Manuscript, 2002.

R. Viertl and H. Hule. On Bayes’ theorem for fuzzy data. Statistical Papers, 32:115–122, 1991.

P.Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.

J.H. Wang and T. Raz. On the construction of control charts using linguistic variables. Inter. J. Production Research, 28:477–487, 1990.

J.H.Wang and R.C. Tsaur. Resolution of fuzzy regression model. Euro. J. Oper. Re., 126: 637–650, 2000a.

J.H. Wang and R.C. Tsaur. Insight of a fuzzy regression model. Fuzzy Sets Syst., 112: 355–369, 2000b.

Z.Wang and S. Li. Fuzzy linear regression analysis of fuzzy valued variables. Fuzzy Sets Syst., 36:125–136, 1990.

N. Watanabe. Fuzzy random variables and statistical inference. Japanese J. of Fuzzy Theory and Systems, 8:833–846, 1996.

N. Watanabe and T. Imaizumi. A fuzzy statistical test of fuzzy hypotheses. Fuzzy Sets Syst., 53:167–178, 1993.

W.H. Woodall, K.L. Tsui, and G.R. Tucker. A review of statistical and fuzzy control based on categorical data. In Frontiers in Statistical Quality Control, pages 83–89.

Physica-Verlag, Heidelberg, 1997.

H. Wu. Fuzzy reliability analysis based on fuzzy numbers. Inform. Sci., 103:135–159, 1997.

L. Xihe. Stability of random membership frequency and fuzzy statistics. Fuzzy Sets Syst., 29:89–102, 1989.

R. Xu. S-Curve regression model in fuzzy environment. Fuzzy Sets Syst., 90:317–326, 1997.

R.R. Yager. Fuzzy prediction based on regression models. Inform. Sci., 26:45–63, 1982.

J. Yao and C. Hwang. Point estimation for n sizes of random sample with one vague data. Fuzzy Sets Syst., 80:205–215, 1996.

J. Yao and K. Wu. Sequential test for fuzzy hypothesis based on hybrid data and signed distance ranking. Manuscript, 2001.

J. Yen and L. Wang. Applications of statistical information criteria for optimal fuzzy model construction. IEEE Trans. on Fuzzy Systems, 6:362–372, 1998.

K.K. Yen, S. Ghoshray, and G. Roig. A linear regression model using triangular fuzzy number coefficients. Fuzzy Sets Syst., 106:167–177, 1999.

L.A. Zadeh. Probability and fuzziness are completibility rather than contradictory. Technometrics, 37:271–277, 1995.

L.A. Zadeh. Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. J. Stat. Plan. Inf., 105:233–264, 2002.

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Trends in Fuzzy Statistics. (2016). Austrian Journal of Statistics, 32(3), 239-257. https://doi.org/10.17713/ajs.v32i3.459