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Cramér-Rao type inequality and a problem of mixture of distributions. (English) Zbl 0920.62027

Summary: The following problem is considered: Let \(X_1, X_2,\dots ,\) \(X_n,\dots \) be an infinite sequence of i.i.d. r.v.’s with distribution function \(\int _{-\infty }^\infty F(x{}\vartheta) dH(\vartheta)\), where \(F(x{}\vartheta)\) is a known distribution function on the real line, while the parameter \(\vartheta \) is also real \((-\infty <\vartheta <\infty)\) having a priori distribution \(H(\vartheta)\) which is to be determined by means of the infinite sample given above. The problem to have an explicit way of solution was raised by H. Robbins [Ann. Math. Stat. 35, 1-20, (1964; Zbl 0138.12304)], while the present investigation is closely connected with a Cramér-Rao type inequality given by the author [Contributions to Statistics, Jaroslav Hajek Mem. Vol., 253-262 (1979; Zbl 0419.62029); Lect. Notes Stat. 8, 284-289 (1981; Zbl 0476.62031)].

MSC:

62F10 Point estimation
62F15 Bayesian inference
62C10 Bayesian problems; characterization of Bayes procedures
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