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A new way of quantifying the symmetry of a random variable: estimation and hypothesis testing. (English) Zbl 1348.62161

Summary: New measures of skewness for real-valued random variables are proposed. The measures are based on a functional representation of real-valued random variables. Specifically, the expected value of the transformed random variable can be used to characterize the distribution of the original variable. Firstly, estimators of the proposed skewness measures are analyzed. Secondly, asymptotic tests for symmetry are developed. The tests are consistent for both discrete and continuous distributions. Bootstrap versions improving the empirical results for moderated and small samples are provided. Some simulations illustrate the performance of the tests in comparison to other methods. The results show that our procedures are competitive and have some practical advantages.

MSC:

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
62G86 Nonparametric inference and fuzziness
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[1] Abdous, B.; Ghoudi, K.; Rémillard, B., Nonparametric weighted symmetry tests, Canadian journal of statistics, 31, 4, 357-381, (2003) · Zbl 1130.62328
[2] Baringhaus, L.; Henze, N., A characterization of and new consistent tests for symmetry, Communication in statistics—theory and methods, 21, 6, 1555-1566, (1992) · Zbl 0800.62244
[3] Bertoluzza, C.; Corral, N.; Salas, A., On a new class of distances between fuzzy numbers, Mathware & soft computing, 2, 71-84, (1995) · Zbl 0887.04003
[4] Butler, C.C., A test for symmetry using the sample distribution function, The annals of mathematical statistics, 40, 6, 2209-2210, (1969) · Zbl 0214.46002
[5] Cabaña, A.; Cabaña, E.M., Tests of symmetry based on transformed empirical processes, Canadian journal of statistics, 28, 4, 829-839, (2000) · Zbl 0966.62028
[6] Cao, R.; Lugosi, G., Goodness-of-fit tests based on the kernel density estimator, Scandinavian journal of statistics, 32, 4, 599-616, (2005) · Zbl 1091.62031
[7] Cheng, W.-H.; Balakrishnan, N., A modified sign test for symmetry, Communications in statistics—simulation and computation, 33, 3, 703-709, (2004) · Zbl 1101.62329
[8] Cohen, J.P.; Menjoge, S.S., One-sample run tests of symmetry, Journal of statistical planning and inference, 18, 1, 93-100, (1988) · Zbl 0629.62052
[9] Colubi, A.; Domínguez-Menchero, J.S.; López-Díaz, M.; Ralescu, D.A., A \(D_E [0,1] \operatorname{-} \operatorname{representation}\) of random upper semicontinuous functions, Proceedings of the American mathematical society, 130, 3237-3242, (2002) · Zbl 1005.28003
[10] Colubi, A.; González-Rodríguez, G., Triangular fuzzification of random variables and power of distribution tests: empirical discussion, Computational statistics & data analysis, 51, 9, 4742-4750, (2007) · Zbl 1162.62342
[11] Dykstra, R.; Kochar, S.; Robertson, T., Likelihood ratio tests for symmetry against one-sided alternatives, Annals of the institute of statistical mathematics, 47, 4, 719-730, (1995) · Zbl 0843.62066
[12] Einmahl, J.H.J.; McKeague, I.W., Empirical likelihood based hypothesis testing, Bernoulli, 9, 2, 267-290, (2003) · Zbl 1015.62048
[13] Giné, E.; Zinn, J., Bootstrapping general empirical measures, Annals of probability, 18, 851-869, (1990) · Zbl 0706.62017
[14] González-Rodríguez, G.; Colubi, A.; Gil, M.A., A fuzzy representation of random variables: an operational tool in exploratory analysis and hypothesis testing, Computational statistics & data analysis, 51, 1, 163-176, (2006) · Zbl 1157.62303
[15] González-Rodríguez, G.; Colubi, A.; Gil, M.A., Fuzzy data treated as functional data. A one-way anova test approach, Computational statistics & data analysis, 56, 943-955, (2012) · Zbl 1243.62104
[16] Gupta, M.K., An asymptotically nonparametric test of symmetry, Annals of mathematical statistics, 38, 849-866, (1967) · Zbl 0157.48102
[17] Hettmansperger, T.P., Statistical inference based on ranks. wiley series in probability and mathematical statistics: probability and mathematical statistics, (1984), John Wiley & Sons Inc. New York
[18] Laha, R.G.; Rohatgi, V.K., Probability theory, (1979), Wiley New York · Zbl 0409.60001
[19] Lubiano, M.A., Colubi, A., Gil, M.A., González-Rodríguez, G., Coppi, R., 2006. Distribution estimation of a random variable based on a fuzzy representation. In: Proceedings of the 11th International Conference on Information Processing and Management of Uncertainty (IPMU’2006, Paris), pp. 718-723.
[20] McWilliams, T.P., A distribution-free test for symmetry based on a runs statistic, Journal of the American statistical association, 85, 412, 1130-1133, (1990)
[21] Mizushima, T., Estimation of symmetry parameter and tests for symmetry, Mathematica japonica, 52, 3, 359-376, (2000) · Zbl 0965.62032
[22] Mizushima, T.; Nagao, H., A test for symmetry based on density estimates, Journal of the Japan statistical society, 28, 2, 205-225, (1998) · Zbl 0924.62033
[23] Modarres, R.; Gastwirth, J.L., A modified runs test for symmetry, Statistics & probability letters, 31, 2, 107-112, (1996) · Zbl 0880.62050
[24] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Annals of probability, 11, 1373-1379, (1985) · Zbl 0583.60011
[25] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422, (1986) · Zbl 0592.60004
[26] Rothman, E.D.; Woodroofe, M., A cramér-von Mises type statistic for testing symmetry, Annals of mathematical statistics, 43, 2035-2038, (1972) · Zbl 0276.62045
[27] Trutschnig, W.; González-Rodríguez, G.; Colubi, A.; Gil, M.Á, A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread, Information sciences, 179, 23, 3964-3972, (2009) · Zbl 1181.62016
[28] Vorlickova, D., Asymptotic properties of rank tests of symmetry under discrete distributions, Annals of mathematical statistics, 43, 2013-2018, (1972) · Zbl 0263.62030
[29] Wilcoxon, F., Individual comparisons by ranking methods, Biometrics, 1, 80-83, (1945)
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