# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.