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Statistical choice of non-separated one-parameter models. (English) Zbl 0731.62083

The purpose of this paper is to study the asymptotic choice between two models \(\{\) F(x\(| \alpha)\), \(\alpha\in A\subseteq {\mathbb{R}}\}\) and \(\{\) G(x\(| \beta \}\), \(\beta\in B\subseteq {\mathbb{R}}\}\), A and B being intervals but such that for \((\alpha_ 0,\beta_ 0)\), and only for this pair, we have \(F(x| \alpha_ 0)=G(x| \beta_ 0).\)
As examples consider the models with distribution functions \(F(x| \alpha)=x^{\alpha}\) \((0<\alpha <+\infty)\) and \(G(x| \beta)=1-(1- x)^{\beta}\) \((0<\beta <+\infty)\) on the support [0,1] which coincide for the uniform distribution \((\alpha_ 0=1\), \(\beta_ 0=1)\) or the models with densities \(f(x| \alpha)=\alpha x^{\alpha -1}e^{- x^{\alpha}}\) (Weibull distribution) \((0<\alpha <+\infty)\) and \(g(x| \beta)=(\Gamma (\beta))^{-1}x^{\beta -1}e^{-x}\) (gamma distribution) \((0<\beta <+\infty)\), on the support \({\mathbb{R}}_+\), which coincide for the exponential distribution \((\alpha_ 0=1\), \(\beta_ 0=1)\).

MSC:

62F99 Parametric inference
62F03 Parametric hypothesis testing
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References:

[1] M. G. KENDALL and A. STUART (1961):The Advanced Theory of Statistics, vol. II, C. Griffin and Cy. · Zbl 0416.62001
[2] J. TIAGO DE OLIVEIRA (1981): “Statistical choice of univariate extreme models,Statistical Distributions in Scientific Work, vol. 6, D. Reidell and Cy. · Zbl 0484.62036
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