Sufficient statistics

Question

I'm learning about sufficient statistics and I understand basic stuff that it is the minimum information we need to represent a statistic (very vague definition, I know. I'm learning). I'm trying to solve problems in that. The problem is suppose a sample of size $n$ is taken from the geometric distribution with parameter $\pi$. How can I find a real valued sufficient statistic for this ?

Answer 1

The joint probability density function (pdf) of the sample is

$$\prod_{i=1}^n (1-\pi)^{x_i} \pi = \pi^n (1-\pi)^{\sum_{i=1}^n x_i}.$$

Let $T(x) = \sum_{i=1}^n x_i$. We see that the pdf can be expressed as a function that depends on the sample only through $T(x)$. Thus $T(x)$ is a sufficient statistic for $\pi$, by the Neyman-Fisher factorization theorem.

The Neyman-Fisher factorization theorem is a very useful tool for finding sufficient statistics. The Wikipedia page has several examples.

(This answer assumes one form of the geometric distribution. The form of $T(x)$ would be the same for the other.)