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GS-distributions: a new family of distributions for continuous unimodal variables. (English) Zbl 1445.62031
Summary: The choice of the best-suited statistical distribution for modeling data is not a trivial issue. Unless a sound theoretical background exists for selecting a particular distribution, one will usually resort to testing various candidates and select a distribution based on its fit to the observed data. While this is a legitimate strategy, it is more objective and efficient to define a sufficiently general family that can be used for this purpose. This approach has a long tradition in statistics, and resulted in various families of distributions, most notably Pearson’s. Given such a family, modeling a data set requires estimating the appropriate parameters within this family and assessing the resulting fit. As a contribution to this methodology, the Generalized S-distribution is introduced here as a new family of distributions that can serve as statistical models for unimodal continuous distributions. The article begins with a description of the rationale for defining this family. It then discusses its basic properties and introduces a numerical procedure for determining appropriate parameters using maximum likelihood estimation. Finally, the paper illustrates the distribution and methods with several examples.

MSC:
62E15 Exact distribution theory in statistics
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
62F10 Point estimation
Software:
AS 99; Mathematica
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[1] Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions, 9th Printing. Dover Publications, New York. · Zbl 0543.33001
[2] Balthis, W.L.; Voit, E.O.; Meaburn, G.M., Setting prediction limits for Mercury concentrations in fish having high bioaccumulation potential, Environmetrics, 7, 429-439, (1996)
[3] Gupta, R.D.; Kundu, D., Generalized exponential distributions, Austral. New Zealand J. statist., 41, 173-188, (1999) · Zbl 1007.62503
[4] Hernández-Bermejo, B.; Sorribas, A., Analytical quantile solution for the S-distribution, random number generation and statistical data modelling, Biometrical J., 43, 1017-1025, (2001) · Zbl 1083.62503
[5] Hill, I.D.; Hill, R.; Holder, R.L., Algorithm AS 99: Fitting Johnson curves by moments, Appl. statist., 25, 180-189, (1976)
[6] Jones, M.C., The complementary beta distribution, J. statist. plan. inference, 104, 329-337, (2002) · Zbl 1004.60009
[7] Jones, M.C., Families of distributions arising from distributions of order statistics, Test, 13, 1-43, (2004) · Zbl 1110.62012
[8] Johnson, N.L., Systems of frequency curves generated by methods of translation, Biometrika, 36, 149-176, (1949) · Zbl 0033.07204
[9] Kamps, U., A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations, Metrika, 38, 215-225, (1991) · Zbl 0735.62011
[10] Lu, M., Tilley, B.C., Li, S., 1998. Issues on permutation tests: applications in analysis of CT lesion volume in the NINDS T-PA stroke trial. 1998 Proceedings of the Biopharmaceutical Section, American Statistical Association, Alexandria, VA, pp. 27-32.
[11] March, J.; Trujillano, J.; Tort, M.; Sorribas, A., Estimating conditional distributions using a method based on S-distributions: reference percentile curves for body mass index in Spanish children, Growth dev. aging, 67, 59-72, (2003)
[12] Morgenthaler, S.; Tukey, J.W., Fitting quantiles: doubling, HR, HQ, and HHH distributions, J. comp. graph. statist., 9, 180-195, (2000)
[13] Nagahara, Y., The PDF and CF of Pearson type IV distributions and the ML estimation of the parameters, Statist. probab. lett., 43, 251-264, (1999) · Zbl 0930.62013
[14] Parzen, E., Nonparametric statistical data modelling (with comments), J. amer. statist. assoc., 74, 105-131, (1979)
[15] Pearson, E.S., Hartley, H.O. (Eds.), 1972. Biometrika Tables for Statisticians, vol. 2. Cambridge University Press, Cambridge. · Zbl 0255.62003
[16] Podladchikova, O.; Lefebvre, B.; Krasnoselskikh, V.; Podladchikov, V., Classification of probability densities on the basis of Pearson’s curves with application to coronal heating simulations, Nonlinear process. geophys., 10, 323-333, (2003)
[17] Savageau, M.A., Growth equations: a general equation and a survey of special cases, Math. biosci., 48, 267-278, (1980) · Zbl 0419.92001
[18] Savageau, M.A., A suprasystem of probability distributions, Biometrical J., 24, 323-330, (1982) · Zbl 0493.62026
[19] Schwacke, L.H., 2000. SDIST user’s manual. Internal Report, Department of Biometry and Epidemiology, Medical University of South Carolina.
[20] Simpson, K.N., Itzler, R., 1996. Exploratory economic analysis of data from randomized clinical trials of antiretroviral therapy in HIV disease populations and recommendations for future study design and analysis protocols. UNC Report to Glaxo Wellcome.
[21] Sorribas, A.; March, J.; Voit, E.O., Estimating age-related trends in cross-sectional studies using S-distributions, Statist. med., 19, 697-713, (2000)
[22] Sorribas, A.; March, J.; Trujillano, J., A new parametric method based on S-distributions for computing receiver operating characteristic curves for continuous diagnostic tests, Statist. med., 21, 1215-1235, (2002)
[23] Tsoularis, A., Analysis of logistic growth models, Math. biosci., 179, 21-55, (2002) · Zbl 0993.92028
[24] Turner, M.E.; Pruitt, K.M., A common basis for survival, growth and autocatalysis, Math. biosci., 39, 113-123, (1978)
[25] Voit, E.O., S-system analysis of endemic infections, Comput. math. appl., 20, 161-173, (1990) · Zbl 0728.92019
[26] Voit, E.O., The S-distribution: a tool for approximation and classification of univariate, unimodal probability distributions, Biometrical J., 7, 855-878, (1992) · Zbl 0775.62027
[27] Voit, E.O., A maximum likelihood estimator for shape parameters of S-distributions, Biometrical J., 42, 471-479, (2000) · Zbl 0959.62023
[28] Voit, E.O.; Schwacke, L.H., Scalability properties of the S-distribution, Biometrical J., 40, 665-684, (1998) · Zbl 0914.62008
[29] Voit, E.O.; Yu, S., The S-distribution: approximation of discrete distributions, Biometrical J., 36, 205-219, (1994) · Zbl 0960.62508
[30] Wolfram, S., The Mathematica book, (1999), Wolfram Media/Cambridge University Press Cambridge · Zbl 0924.65002
[31] Yu, S.; Voit, E.O., A simple, flexible failure model, Biometrical J., 37, 595-609, (1995) · Zbl 0837.62097
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