Asymptotic properties for simplifed likelihood based tests for the Weibull distribution, 8th International Conference on Modeling in Industrial Maintenance and Reliability,

Goodness-of-fit (GOF) tests are a sophisticated tool which aims to determine whether a probability distribution is adapted to a given data set. In reliability analysis, the two-parameter Weibull distribution is a very widely used model. There is a huge literature on GOF tests for the exponential distribution, but GOF tests for the Weibull distribution have been much less studied. That is why we developed simplified likelihood based GOF tests for the Weibull distribution. The principle of these tests consists in nesting the Weibull distribution in larger families with three parameters and testing the value of the third parameter by using the likelihood ratio, Wald or score procedures. These tests are simplified by getting rid of the nuisance parameters using parameters estimators. This simplification allows to obtain exact tests that can be applied to small samples. However, general results on asymptotic properties of these tests are not available. The aim of this paper is to study the asymptotic properties of these GOF tests in a particular case when the Weibull distribution is nested in the Generalized Gamma distribution. The Delta method is used to prove the asymptotic convergence of the Wald and score test statistics to the chi-squared distribution and the second-order Delta method is applied to prove the asymptotic convergence of the likelihood ratio test statistic to a quadratic form of Normal random variables.