Likelihood Ratio Statistic for Exponential Mixtures

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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.