LIKELIHOOD RATIO STATISTIC FOR EXPONENTIAL MIXTURES

GABRIELA CIUPERCA

Laboratoire de Probabilités, Combinatoire et Statistique, Domaine de Gerland, Bât. Recherche B,

Univ. Lyon 1, 50 Av. Tony-Garnier, 69366 Lyon cedex 07, France, e-mail: Gabriela.Ciuperca@univ-lyon1.fr

(Received July 17, 2000; revised February 5, 2001)

Abstract. Let f_{0}(x) be the exponential density and f_{γ}(x) the translation model. Let (X_{i})_{i=1,n} be i.i.d. random variables, with density g. We test that g is f_{0} against g is a simple mixture, using the LRT statistic. We prove that the LRT diverges to infinity with probability 1/2 and it is equal to 0 with probability 1/2. Therefore, the classical likelihood limiting theory does not hold.

Key words and phrases: Mixture models, likelihood test, exponential distribution.