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estimation of shift parameter

Source: miklos schweitzer 1995 q12

October 6, 2021
probability and statsprobability

Problem Statement

Let F(x) be a known distribution function, the random variables η1,η2...\eta_1 , \eta_2 ... be independent of the common distribution function F(x−ϑ)F( x - \vartheta), where ϑ\vartheta is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of ε>0\varepsilon> 0 there exist a Lebesgue measure ε\varepsilon Borel set E ("confidence set") and a Borel-measurable function tn(x1,...,xn)t_n( x_1 ,. .., x_n ) ( n = 1,2, ...) such that for any ϑ\vartheta we have P(ϑ−tn(η1,...,ηn)∈E)>1−e−cn(n>n0(ε,F))P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} \qquad( n > n_0 ( \varepsilon, F ) ) Prove that a) if F is not absolutely continuous, then the shift parameter is "well estimated", b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".
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