AISM 54, 585-594

## Likelihood ratio statistic for exponential mixtures

### Gabriela Ciuperca

Laboratoire de Probabilités, Combinatoire et Statistique, Domaine de Gerland, Bat. 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 $f_\gamma(x)$ 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.

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