Weighted likelihood test for a change in one component of a parametric mixture

Dominique Abgrall (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

In this article, we consider a closed sample of independent random variables that follow a finite mixture distribution with parametric components. The sample might contain at most one change in the parameters of the first component. If there is a change, the parameters which describe the distribution of the first component are different before and after the change-point while the other parameters of the mixture remain the same. To test whether there is a change or not, we introduce two alternative hypothesis tests. They are based on weighted likelihood ratios that can be computed with standard inference algorithms. With a technique from Davis et al. (1995), we derive the limit distribution of their statistics under the null hypothesis in the form of quadratic forms of a multidimensional Brownian motion, with the help of a dedicated functional limit theorem. We show that validity conditions of the main result hold for univariate finite Gaussian mixtures within the framework of Hathaway (1985). Numerical applications on simulated data illustrate the advantage of the alternative tests compared to a standard benchmark test. An application to Property and Casualty insurance real data is provided for the alternative tests.

Mots clés: applications to insurance data; weighted likelihood quotient test; weak limit theorems for dependent sequences; parametric mixture; change-point detection

Domaines: Statistiques [math.ST]

Dominique Abgrall. Weighted likelihood test for a change in one component of a parametric mixture. 2021. ⟨hal-03211913v2⟩