Likelihood principle and stopping rules

[1/0%]

LaTeX

Show LaTeX

Problem 4.2.2 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 4.2 "First Examples"

(a)(b)(c)

For the situation of Example 2.1, Lindley and Phillips (1976) give a detailed account of the effect of stopping rules, which we can illustrate as follows. Let $X$ be the number of successes in $n$ Bernoulli trials with success probability $p$ .

(a)

Suppose that the number of Bernoulli trials performed is a prespecifed number $n$ , so that we have the binomial sampling model, $\mathbb {P}\left(X=x\right)=\binom {n}{x} p^{x}(1-p)^{n-x}$ , $x=0,1, \ldots , n$ . Calculate the Bayes risk of the Bayes estimator (1.1.12) and the UMVU estimator of $p$ .

(b)

Suppose that the number of Bernoulli trials performed is a random variable $N$ . The value $N=n$ was obtained when a prespecified number, $x$ , of successes was observed so that we have the negative binomial sample model, $\mathbb {P}\left(N=n\right)=\binom {n-1}{x-1} p^{x}(1- p)^{n-x}$ , $n=x$ . Calculate the Bayes risk of the Bayes estimator and the UMVU estimator of $p$ .

(c)

Calculate the mean squared errors of all three estimators under each model. If it is unknown which sampling mechanism generated the data, which estimator do you prefer overall?

References

[LeCa98] Lehmann E.L., Casella G. Theory of Point Estimation. 2nd ed., Springer, 1998.