A sufficient statistic for a geometric parameter

Question

You observe independent geometric counts $X_1, \dots, X_n$ on $1, 2, 3, \dots$ (trials up to and including the first success), each with unknown success probability $p$ .

Using the factorization idea, find a sufficient statistic for $p$ .

Accepted Answer

The factorization test. Write the joint mass function as "a piece that involves p and touches the data only through some summary" times "a piece with no p." Whatever summary appears is sufficient. Do it for the geometric. Each mass is p(1 - p)^{xi - 1}, so multiplying n of them, f(x; p) = \prod{i=1}^n p (1 - p)^{xi - 1} = p^n (1 - p)^{\sumi xi - n}. Every appearance of the data sits inside \sumi xi. Take g(T, p) = p^n (1 - p)^{T - n} with T = \sumi xi, and h(x) = 1. The factorization holds, so T = \sumi Xi is sufficient for p. Why it makes sense. The success probability is driven by how many …

A sufficient statistic for a geometric parameter

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