Let F(x) be a known distribution function, the random variables η1,η2... be independent of the common distribution function F(x−ϑ), where ϑ 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 there exist a Lebesgue measure ε Borel set E ("confidence set") and a Borel-measurable function tn(x1,...,xn) ( n = 1,2, ...) such that for any ϑ we have
P(ϑ−tn(η1,...,ηn)∈E)>1−e−cn(n>n0(ε,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". probability and statsprobability