Hazard Rate Comparisons On Smallest Order Statistics From Heterogeneous Gamma Components

Order statistics from heterogeneous samples arise naturally in survival analysis,operations research,reliability theory,actuarial science and some other fields.There exist a large number of extensive studies on this topic in the literature.It should be remarked that most of the work was devoted to the cases of exponential,weibull and some specific models.Recently,with the increasing importance of the gamma distribu-tion in the fields of statistics,actuarial science and life testing,some researchers have attempted to study the comparison of maximum order statistics from heterogeneous gamma samples.However,few work on minimum order statistics has been done.In this thesis,we deal with the stochastic comparison of minimum order statistics from independent and heterogeneous gamma samples.We firstly explore how the scale parameter vector and the shape parameter vector affect the failure rate of the minimum order statistics.It is showed that greater difference among scale parameters vector or parameter vectors results in larger failure rate of minimum order statistics.The hazard rate order of minimum order statistics arising from two sets of independent gamma samples is established as well,wherein one set of samples are heterogeneous while the other is homogeneous.As a consequence,a lower bound for the hazard rate function of minimum order statistics from heterogeneous gamma samples is provided in terms of that from heterogeneous samples.