The taxis in a town are marked with reference numbers $1,2,...,p$ where $p$ is unknown. I am observing $n \leq p$ of them with pairwise different reference numbers $x_1,x_2,...,x_n$. It is to be assumed that every combination $(x_1,x_2,...,x_3)$ has the same probability to occure. Let $g:=\max x_i$. How can I show that there is a unique function $f(g)$ being an unbiased estimator of $p$? Any hints?
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$\begingroup$ Hint: find $E[g\mid p]$. That should make finding an unbiased function of $g$ fairly obvious. Uniqueness may be harder. This is essentially the German tank problem $\endgroup$– HenryJul 11, 2017 at 10:33
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The meat of this exercise is proving that g is a complete and sufficient statistic. Once you have proven this, the rest is theory.
Completeness: This is nicely worked out here
Sufficientness: This is easy, for example using the factorisation theorem
Existence: Take any unbiased estimator $S$ of $p$. This is not necessarily a function of $g$ yet, so let $$f(g) = E_p[S|g]$$ Since $g$ is sufficient, $f(g)$ does not depend on $p$ so it is again an estimator. By the law of total expectation, $f(g)$ is again unbiased: $$E_p[f(g)] = E_p[E_p[S|g]] = E_p[S] = p$$
(a.s.) Uniqueness: This is where completness comes in. By the theorem of Lehmann-Scheffé, any unbiased estimator $h(g)$ (that is a function of $g$) must be the MVUE. In particular, $f(g)$ is the MVUE. The MVUE is a.s. unique, so $f(g)$ is a.s. equal to any other unbiased estimator $h(g)$ (as it is also the MVUE)
