Problem 66E

The likelihood function L(y1, y2, . . . , yn | θ) takes on different values depending on the arguments (y1, y2 , . . . , yn ). A method for deriving a minimal sufficient statistic developed by Lehmann and Scheff´e uses the ratio of the likelihoods evaluated at two points, (x1, x2, . . . , xn) and (y1, y2, . . . , yn ):

Many times it is possible to find a function g(x1, x2, . . . , xn ) such that this ratio is free of the unknown parameter θ if and only if g(x1, x2, . . . , xn ) = g(y1, y2, . . . , yn ). If such a function g can be found, then g(Y1, Y2, . . . , Yn ) is a minimal sufficient statistic for θ .

a Let Y1, Y2, . . . , Yn be a random sample from a Bernoulli distribution (see Example 9.6 and Exercise 9.65) with p unknown.

i Show that

ii Argue that for this ratio to be independent of p, we must have

iii Using the method of Lehmann and Scheff´e, what is a minimal sufficient statistic for p? How does this sufficient statistic compare to the sufficient statistic derived in Example 9.6 by using the factorization criterion?

b Consider the Weibull density discussed in Example 9.7.

i Show that

Reference

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let Y1, Y2, . . . , Ynbe independent Bernoulli random variables with

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