%0 Journal Article
%T The Generalized Inverse Generalized Weibull Distribution and Its Properties
%A Kanchan Jain
%A Neetu Singla
%A Suresh Kumar Sharma
%J Journal of Probability
%D 2014
%R 10.1155/2014/736101
%X The Inverse Weibull distribution has been applied to a wide range of situations including applications in medicine, reliability, and ecology. It can also be used to describe the degradation phenomenon of mechanical components. We introduce Inverse Generalized Weibull and Generalized Inverse Generalized Weibull (GIGW) distributions. GIGW distribution is a generalization of several distributions in literature. The mathematical properties of this distribution have been studied and the mixture model of two Generalized Inverse Generalized Weibull distributions is investigated. Estimates of parameters using method of maximum likelihood have been computed through simulations for complete and censored data. 1. Introduction The Generalized Weibull (GW) distribution possessing bathtub failure rate was introduced by Mudholkar and Srivastava [1]. Mudholkar et al. [2] and Mudholkar and Hutson [3] applied GW distribution for analysis of data relating to bus motor failure, head and neck cancer, and flood. Inverse distributions, namely, Inverse Gamma, Inverse Generalized Gamma, Inverse Weibull, and Inverse Rayleigh, have been studied in literature [4¨C8]. Keller et al. [9] studied shapes of density and failure rate function for the basic inverse model and Drapella [6] worked on Inverse Weibull (IW) distribution. Drapella [6] and Mudholkar and Kolia [10] suggested the names complementary Weibull and reciprocal Weibull. These distributions have applications in reliability engineering and medical sciences and are used for modelling infant mortality, wear-out periods, degradation of mechanical components [11], times of breakdown of an insulating fluid subject to the action of constant tension [5], and load-strength relationship for a component [12]. Aleem and Pasha [13] studied some distributional properties of IW. In this paper, we first introduce a three-parameter continuous probability distribution on the positive real line, known as Inverse Generalized Weibull (IGW) distribution. It is the distribution of reciprocal of a variable distributed according to the Generalized Weibull distribution. It can also be called Complementary or Reciprocal Generalized Weibull distribution. Using IGW distribution, a four-parameter distribution named as Generalized Inverse Generalized Weibull (GIGW) distribution is introduced and its properties are studied. The mixture of two GIGW distributions has been investigated. Empirical estimates of parameters have been found using maximum likelihood method for complete and censored data. An application to real data set has been provided to
%U http://www.hindawi.com/journals/jprob/2014/736101/