Compound geometric and Poisson models. (English) Zbl 1363.62010

Summary: Many lifetime distributions are motivated only by mathematical interest. Here, eight new families of distributions are introduced. These distributions are motivated as models for the stress of a system consisting of components working in parallel/series and each component has a fixed number of sub-components working in parallel/series. Mathematical properties and estimation procedures are derived for one of the families of distributions. A real data application shows superior performance of a three-parameter distribution (performance assessed with respect to Kolmogorov-Smirnov statistics, AIC values, BIC values, CAIC values, AICc values, HQC values, probability-probability plots, quantile-quantile plots and density plots) versus thirty one other distributions, each having at least three parameters.


62E10 Characterization and structure theory of statistical distributions
62N05 Reliability and life testing


R; Maple
Full Text: DOI Link


[1] Akaike, H.: A new look at the statistical model identification. IEEE Trans. Automat. Control 19 (1974), 716-723. · Zbl 0314.62039
[2] Bozdogan, H.: Model selection and Akaike’s Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika 52 (1987), 345-370. · Zbl 0627.62005
[3] Burnham, K. P., D., Anderson, R.: Multimodel inference: Understanding AIC and BIC in model selection. Sociolog. Methods Res. 33 (2004), 261-304.
[4] Ferguson, T. S.: A Course in Large Sample Theory. Chapman and Hall, London 1996. · Zbl 0871.62002
[5] Gupta, R. C., Gupta, P. L., Gupta, R. D.: Modeling failure time data by Lehman alternatives. Commun. Statist. - Theory and Methods 27 (1998), 887-904. · Zbl 0900.62534
[6] Gupta, R. D., Kundu, D.: Generalized exponential distributions. Australian and New Zealand J. Statist. 41 (1999), 173-188. · Zbl 1054.62013
[7] Hannan, E. J., Quinn, B. G.: The determination of the order of an autoregression. J. Royal Statist. Soc. B 41 (1979), 190-195. · Zbl 0408.62076
[8] Hurvich, C. M., Tsai, C.-L.: Regression and time series model selection in small samples. Biometrika 76 (1989), 297-307. · Zbl 0669.62085
[9] Kakde, C. S., Shirke, D. T.: On exponentiated lognormal distribution. Int. J. Agricult. Statist. Sci. 2 (2006), 319-326.
[10] Kolmogorov, A.: Sulla determinazione empirica di una legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari 4 (1933), 83-91. · Zbl 0006.17402
[11] Kolowrocki, K.: Reliability of Large Systems. Elsevier, New York 2004. · Zbl 0925.93028
[12] Leadbetter, M. R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer Verlag, New York 1987. · Zbl 0518.60021
[13] Lehmann, L. E., Casella, G.: Theory of Point Estimation. Second edition. Springer Verlag, New York 1998. · Zbl 0916.62017
[14] Lemonte, A. J., Cordeiro, G. M.: The exponentiated generalized inverse Gaussian distribution. Statist. Probab. Lett. 81 (2011), 506-517. · Zbl 1207.62028
[15] Marshall, A. W., Olkin, I.: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84 (1997), 641-652. · Zbl 0888.62012
[16] Mudholkar, G. S., Srivastava, D. K.: Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliability 42 (1993), 299-302. · Zbl 0800.62609
[17] Mudholkar, G. S., Srivastava, D. K., Friemer, M.: The exponential Weibull family: Analysis of the bus-motor-failure data. Technometrics 37 (1995), 436-445. · Zbl 0900.62531
[18] Mudholkar, G. S., Srivastava, D. K., Kollia, G. D.: A generalization of the Weibull distribution with application to the analysis of survival data. J. Amer. Statist. Assoc. 91 (1996), 1575-1583. · Zbl 0881.62017
[19] Nadarajah, S.: The exponentiated Gumbel distribution with climate application. Environmetrics 17 (2005), 13-23. · Zbl 0881.62017
[20] Nadarajah, S.: The exponentiated exponential distribution: A survey. Adv. Statist. Anal. 95 (2011), 219-251. · Zbl 1274.62113
[21] Nadarajah, S., Gupta, A. K.: The exponentiated gamma distribution with application to drought data. Calcutta Statist. Assoc. Bull. 59 (2007), 29-54. · Zbl 1155.33305
[22] Nadarajah, S., Kotz, S.: The exponentiated type distributions. Acta Applic. Math. 92 (2006), 97-111. · Zbl 1128.62015
[23] Nichols, M. D., Padgett, W. J.: A bootstrap control chart for Weibull percentiles. Qual. Reliab. Engrg. Int. 22 (2006), 141-151.
[24] Qian, L.: The Fisher information matrix for a three-parameter exponentiated Weibull distribution under type II censoring. Statist. Meth. 9 (2012), 320-329. · Zbl 1365.62367
[25] Team, R Development Core: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria 2014.
[26] Ristic, M., Nadarajah, S.: A new lifetime distribution. J. Statist. Comput. Simul. 84 (2014), 135-150.
[27] Schwarz, G. E.: Estimating the dimension of a model. Ann. Statist. 6 (1978), 461-464. · Zbl 0379.62005
[28] Shams, T. M.: The Kumaraswamy-generalized exponentiated Pareto distribution. European J. Appl. Sci. 5 (2013), 92-99.
[29] Smirnov, N.: Table for estimating the goodness of fit of empirical distributions. Ann. Math. Statist. 19 (1948), 279-281. · Zbl 0031.37001
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