Power Generalized Weibull Distribution Based on Record Values and Associated Inferences with Bladder Cancer Data Example

Abstract

The power generalized Weibull distribution has been proposed recently by [24] as an alternative to the gamma, Weibull and the exponentiated Weibull distributions. The power generalized Weibull family is suitable for modeling data that indicate nonmonotone hazard rates and can be used in survival analysis and reliability studies. Usefulness and flexibility of the family are illustrated by reanalyzing Efron’s data pertaining to a head-and-neck cancer clinical trial. These data involve censoring and indicate unimodal hazard rate. For this distribution, some recurrence relations are established for the single and product moments of upper record values. Further, using these relations, we have obtained means, variances and covariances of upper record values from samples of sizes up to 10 for various values of the parameters and present them in figures. Real data set is analyzed to illustrate the flexibility and importance of the model.

Keywords

Record values / Moments / Characterization / Power generalized Weibull distribution

References

[1.] Ahsanullah M. . Record Statistics, 1995 New York Nova Science Publishers

[2.] Arnold BC, Balakrishnan N, Nagaraja HN. . A First course in Order, 1992 New York John Wiley and Sons

[3.] Arnold BC, Balakrishnan N, Nagaraja HN. . Record, 1998 New York John Wiley and Sons,

[4.] Balakrishnan N, Ahsanullah M. Relations for single and product moments of record values from exponential distribution, Open journal of Statistics. J. Appl. Statist. Sci., 1993, 2: 73-87

[5.] Balakrishnan N, Ahsanullah M. Recurrence relations for single and product moments of record values from generalized Pareto distribution. Comm. Statist. Theory Methods, 1994, 23: 2841-2852,

[6.] Basak, P., Balakrishnan, N.: Maximum likelihood prediction of future record statistics. In: Mathematical and Statistical Methods in Reliability, 7, in Series on Quality, Reliability, and Engineering 578 Statistics, pp. 159-175. World Scientific Publishing, Singapore

[7.] Burr IW. Cumulative frequency functions. Ann. Math. Statistics, 1942, 13: 215-232,

[8.] Chandler KN. The distribution and frequency of record values. J. Roy. Statist. Soc. Ser B, 1952, 14: 220-228

[9.] Glick N. Breaking records and breaking boards. Arner. Math. Monthly, 1978, 85: 2-26,

[10.] Gupta RD, Kundu D. Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom. J., 2001, 43: 117-130,

[11.] Gradshteyn IS, Ryzhik IM. . Tables of Integrals, 2014 Series of Products Academic Press, New York

[12.] Grunzien Z, Szynal D. Characterization of uniform and exponential distributions via moments of the k \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-th record values with random indices Appl. Statist. Sci., 1997, 5: 259-266

[13.] Kumar D. Explicit expressions and statistical inference of generalized rayleigh distribution based on lower record values. Math. Methods of Statistics, 2015, 24: 225-241,

[14.] Kumar D, Jain N, Gupta S. The type I generalized half logistic distribution based on upper record values. J. Prob. Statistics, 2015, 2015: 01-11,

[15.] Kumar D. kth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$kth$$\end{document} lower record values from of Dagum distribution. Discussion Math. Prob. Statistics, 2016, 36: 25-41,

[16.] Kumar D, Dey S, Ormoz E, MirMostafaee SMTK. Inference for the unit-Gompertz model based on record values and inter-record times with an application. Rendiconti del Circolo Matematico di Palermo, 2020, 69: 1295-1319,

[17.] Lawless JF. . Statistical Models and Methods for Lifetime Data, 1982 2 New York Wiley

[18.] Lee and Wang. . Statistical methods for survival data analysis, 2003 John Wiley and Sons New York Annals of Statistics

[19.] Miroslav MR, Kundu D. Marshall-Olkin generalized exponential distribution. Metron, 2015, 73: 317-333,

[20.] Nash JC. On best practice optimization methods in R. J. Statistical Soft., 2014, 60: 1-14,

[21.] Nash JC. . Nonlinear parameter optimization using R tools, 2014 Chichester John Wiley and Sons,

[22.] Nash, J.C.: nlmrt: Functions for nonlinear least squares solutions. R package version 2014.5.4, URL http://CRAN.R-project.org/package=nlmrt (2014)

[23.] Nevzorov VB. Records. Theo. Prob. Appl., 1987, 32: 201-228,

[24.] Nikulin N, Haghighi F. . On the power generalized Weibull family: model for cancer censored data, 2009 LXVII Metron 75-86

[25.] Pawlas P, Szynal D. Relations for single and product moments of k \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-th record values from exponential and Gumbel distributions. J. Appl. Statist. Sci., 1998, 7: 53-61

[26.] Pawlas P, Szynal D. Recurrence relations for single and product moments of k - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k-$$\end{document}th record values from Pareto, generalized Pareto and Burr distributions. Comm. Statist. Theory Methods, 1999, 28: 1699-1709,

[27.] Pawlas P, Szynal D. Recurrence relations for single and product moments of k - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k-$$\end{document}th record values from Weibull distribution and a characterization. J. Appl. Stats. Sci., 2000, 10: 17-25

[28.] Resnick SI. . Extreme values, regular variation and point processes, 1973 New York Springer-Verlag

[29.] Singh S, Dey S, Kumar D. Statistical inference based on generalized Lindley record values. J. Appl. Statistics, 2020, 47(9): 1543-1561,

[30.] Shorrock RW. Record values and inter-record times. J. Appl. Probab., 1973, 10: 543-555,

[31.] Sultan KS. Record values from the modified Weibull distribution and applications. Int. Math. Forum, 2007, 41: 2045-2054,

[32.] Weibull W. A statistical distribution function of wide applicability. J. Appl. Mech.-TransASME, 1951, 18: 293-297,