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Hakamipour, Nooshin; Rezaei, Sadegh; Nadarajah, Saralees

Compound geometric and Poisson models. (English) Zbl 1363.62010

Kybernetika 51, No. 6, 933-959 (2015).
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.

MSC:

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

Keywords:

exponential distribution; exponentiated exponential distribution; maximum likelihood estimation; Kolmogorov-Smirnov statistics

Software:

R; Maple
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\textit{N. Hakamipour} et al., Kybernetika 51, No. 6, 933--959 (2015; Zbl 1363.62010)
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References:

[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
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.