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Testing homogeneity in gamma mixture models. (English) Zbl 1034.62010

Let \(X=(X_1,\dots, X_n)\) be an i.i.d. sample from the PDF \(f\), and \(g_\lambda(x)= (\Gamma(\kappa))^{-1}\lambda^\kappa x^{\kappa-1} e^{-x\lambda}\), (\(\kappa\) is fixed and known). The hypothesis \(H_0: f=g_{\lambda_0}\) is tested against \(H_1: f=pg_\lambda+(1-p)g_{\lambda_0}\), where \(\lambda_0\) is known, and \(\lambda\) and \(p\) are unknown. The likelihood ratio test (LRT) with statistics \(\Lambda_n\) and the Gaussian process test with statistics \(M_n\) are considered. It is shown that under \(H_0\) \(2\Lambda_n=M_n^2+o_p(1)\) and \[ \lim_{n\to\infty}\Pr\{M_n^2-\log\log n+\log(16\pi^2/\kappa)\leq x\}=\exp(-e^{-x^2/2}). \] Simulation of the \(M_n\) distribution for small and moderate sample sizes is considered.

MSC:

62F05 Asymptotic properties of parametric tests
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