A Modified Chi‐squared Goodness‐of‐fit Test for the Three‐parameter Weibull Distribution and its Applications in Reliability

Abstract

The Weibull probability distribution plays an important role in a statistical analysis of lifetime or response data in reliability and survival studies. To extend applications of the distribution to a wider class of failure-rate situations, several modifications of the classical Weibull model were proposed. Mudholkar, Srivastava and Freimer [MUD 95] as well as Mudholkar, Srivastava and Kollia [MUD 96] introduced the exponentiated and generalized Weibull families, which include distributions with unimodal and bathtub failure rates and also possess a broad class of monotone hazard rates. Bagdonavicius and Nikulin [BAG 02] have proposed another generalization -the power generalized Weibull family, which also includes all possible failure rate functions.One of the main problems of statistical modeling is the selection of a proper probability distribution to be used in an analysis. Different criteria, both parametric and non-parametric, may be used for such a selection. Chi-squared tests are among them and it is worth considering these tests in more detail.Fisher [FIS 24] showed that the limit distribution of Pearson's sum essentially depends on a method of parameter estimation. He proved that using efficient estimators Chapter written by Vassilly VOINOV, Roza ALLOYAROVA and Natalie PYA.