Goodness-of-fit tests for the Weibull distribution based on the Laplace transform

Krit, Meryam

Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 135-151.

L’objectif de cet article est de développer de nouveaux tests d’adéquation à la loi de Weibull à deux paramètres basés sur la transformée de Laplace. Le principe de ces tests consiste à mesurer la proximité entre la transformée de Laplace théorique et sa version empirique. Trois méthodes d’estimation des paramètres de la loi de Weibull sont utilisées pour simplifier la construction des statistiques. L’article propose aussi une nouvelle version de la statistique de Cabaña et Quiroz utilisant les estimateurs de maximum de vraisemblance des paramètres. Ces tests ne sont pas asymptotiques, ils peuvent être utilisés pour des échantillons de petite taille. Une comparaison exhaustive des tests proposés est présentée. Parmi tous les tests d’adéquation utilisés, les meilleurs tests sont identifiés. Les résultats dépendent fortement de la forme du taux de hasard de la loi sous-jacente.

The aim of this paper is to develop new goodness-of-fit (GOF) tests for the two-parameter Weibull distribution based on the Laplace transform. The principle of the tests relies on the measure of the closeness between the theoretical Laplace transform and its empirical version. Three estimation methods are used to simplify the building of the statistics. The paper also introduces a new version of Cabaña and Quiroz statistic using the maximum likelihood estimators of the parameters. All these tests are not asymptotic and can be used for small samples size. A comprehensive comparison study is presented. Among all the proposed GOF tests, the best ones are identified. The results strongly depend on the shape of the underlying hazard rate.

Keywords: Reliability, Goodness-of-fit tests, Weibull distribution, Extreme Value distribution, Laplace transform
Mot clés : Fiabilité, Tests d’adéquation, Loi de Weibull, Loi des valeurs extrêmes, Transformée de Laplace

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