Generalized Modified Inverse Weibull Distribution: Its Properties and Applications
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Generalized Modified Inverse Weibull Distribution: Its Properties and Applications Hadi Saboori and Ghobad Barmalzan University of Zabol, Zabol, Iran
Seyyed Masih Ayat University of Zabol, Zabol, Iran Abstract In this paper, we introduce a new useful continuous distribution called generalized modified inverse Weibull distribution. This distribution is a fourparameter extension of the modified inverse Weibull which generalizes some well-known distributions. Various statistical and probabilistic properties are derived such as rth moment, moment generating function, Renyi and Shannon entropies and hazard rate function. We also discuss estimation of the parameters by maximum likelihood and provide the information matrix. The likelihood ratio order (which implies the hazard rate and usual stochastic orders) between smallest order statistics from two independent heterogeneous samples of this new family are discussed. Finally, a real numerical example is also considered for illustrative purposes. AMS (2000) subject classification. Primary: 62E15; Secondary: 62F10. Keywords and phrases. Modified inverse Weibull distribution, Generalized modified inverse Weibull distribution, Hazard rate function, Maximum likelihood estimation, Order statistics, Stochastic comparisons.
1 Introduction The inverse Weibull distribution has been used to model degradation of mechanical components such as pistons, crankshafts of diesel engines, as well as breakdown of insulating fluid to mention a few areas. The usefulness and applications of inverse Weibull (IW) distribution in various areas including reliability and branching processes can be seen in Keller and Kamath (1982) and in references therein. There are several generalizations for this distribution including those of Khan and Robert (2012) on the exponentiated generalized inverse Weibull distribution, Elbatal and Muhammed (2014) on the transmuted modified inverse Weibull distribution and Aryal and Tsokos (2011) on the modified inverse Weibull distribution.
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In this paper, we generalize the modified inverse Weibull distribution and call the new class, the Generalized Modified Inverse Weibull (GMIW) distribution. The major reasons for introducing this family are as follows. • The quality of procedures utilized in statistical analysis mainly depends on the assumed probability model or distribution. Therefore, considerable effort has gone in the development of large classes of standard probability distributions along with relevant statistical methodologies. In fact, there are many continuous univariate distributions in statistical literature, but their applications have not produced useful results in the environmental, financial, biomedical sciences, engineering and economic areas. Therefore, the extension of existing distribution is essential for applications. • The proposed four-parameter family of distributions contains many flexible lifetime distributions as special sub-models. These family of models include the Inverse Weibull distribution ‘(see Khan et al., 2008)’, Flexible
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