On Some Aspects of Exponential Intervened Geometric Distribution
Keywords:
Characterizations, limit distribution of extremes, maximum likelihood, parametric bootstrap, stochastic orderingAbstract
In this paper, we study different aspects of Exponential Intervened Geometric (EIG) distribution. EIG distribution arises as the distribution of random minimum and is a generalization of extended exponential distribution. The shape properties of the probability density function and hazard rate function of EIG are studied, along with structural properties such as moments, moment generating
function, skewness and kurtosis, mean deviation about mean and median. Expression for various reliability measures corresponding to EIG distribution are derived along with stochastic ordering property. Expression for quantiles are obtained and random number generation is discussed. The distributions of order statistics are derived and limit distributions of sample extrema are obtained. Four
characterizations of EIG distribution are proved. The parameters of EIG are estimated through the method of maximum likelihood (ML) and a simulation study is conducted to show the performance of ML estimates. The existence and uniqueness of ML estimates are proved. The EIG model is fitted to a real data set and is showed that the model performs better as compared to ten competitive models. Also, the adequacy of the model for the data set is established using parametric bootstrap approach.
References
Adamidis K, Loukas S. A lifetime distribution with decreasing failure rate. Stat Probabil Lett. 1998;
: 35-42.
Arnold BC, Balakrishnan N, Nagaraja HN. A first course in order statistics. New York: John Wiley and Sons; 1992.
Balakrishnan N, Ristic MM. Multivariate families of gamma-generated distributions with finite or infinite support above or below the diagonal. J Multivarate Anal. 2016; 143: 194-207.
Dey S, Nassar M, Kumar D. Alpha power transformed inverse Lindley distribution: A distribution with an upside-down bathtub-shaped hazard function. J Comput Appl Math. 2019; 348: 130-145.
Dhanavanthan P. Compound intervened Poisson distribution. Biometrical J. 1998; 40: 641-646.
Dhanavanthan P. Estimation of the parameters of compound intervened Poisson distribution. Biometrical J. 2000; 42: 315-320.
Glanzel W, Hamedani GG. Characterizations of univariate continuous distributions. Stud Sci Math Hung. 2001; 37: 83-118.
Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products- Seventh Edition. New York: Academic Press; 2007.
Hamedani GG, Safavimanesh F. Characterizations and infinite divisibility of certain 2016 univariate continuous distributions. Int. Math. Forum. 2017; 12: 195-228.
Henningsen A, Toomet O. maxLik: A package for maximum likelihood estimation in R. Comput Stat. 2011; 26: 443-458.
Huang M, Fung KY. Intervened truncated Poisson distribution. Sankhya B (2008). 1989; 51: 302-310.
Jayakumar K, Sankaran KK. Exponential intervened Poisson distribution. Commun Stat Theory Methods. 2019; 50: 3063-3093.
Kotz S, Nadarajah S. Extreme value distributions: Theory and applications. London: Imperial College Press; 2000.
Kumar CS, Sreejakumari S. Extended intervened Geometric distribution. J Stat Distrib Appl. 2016; 2: 8-13.
Kus C. A new lifetime distribution. Comput Stat Data Anal. 2007; 51: 4497- 509.
Lemonte AJ. A new exponential-type distribution with constant, decreasing, increasing, upside-down
bathtub and bathtub-shaped failure rate function. Comput Stat Data Anal. 2013; 62: 149-170.
Lemonte AJ. The beta log-logistic distribution. Braz J Probab Stat. 2014; 28: 313-332.
Madhavi A, Kundu D. A new method for generating distributions with an application to exponential distribution. Commun Stat Theory Methods. 2017; 46 : 6543-6557.
Maguire BA, Pearson ES, Wynn AHA. The time intervals between industrial accidents. Biometrika.
; 39: 16-80.
Marshall AW, Olkin I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997; 84: 641-652.
Marshall AW, Olkin I. Life distributions: Structure of nonparametric, semiparametric, and parametric families. New York: Springer; 2007.
Nadarajah S, Jayakumar K, Ristic MM. A new family of lifetime models. J Stat Comput Simul. 2013; 83: 1389-1404.
Scollinik DPM. On the intervened generalized Poisson distribution. Commun Stat Theory. 2006; 35: 953-963.
Shanmugam R. An inferential procedure for the Poisson intervention parameter. Biometrics. 1992; 48: 559-565.
Shanmugam R. An intervened Poisson distribution and its medical application. Biometrics. 1985; 41: 1025-1029.
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