Question

Show that the sample mean x, is sufficient for estimating the parameter λof the Poisson distribution.

Answer 1

We will be using Neymann factorization Theorem for proving this. Let us first write down the joint density of independent Poisson random variables with the same parameter.

$f_{\lambda}(X_1,\ \ldots, X_n) = e^{-n\lambda}\times\frac{\lambda^{\sum X_i}}{\prod_i X_i!}$

This can be written as $f_{\lambda}(X_1,\ \ldots, X_n) = h(X_1,\ \ldots, X_n) \cdot g(\lambda, T(X))$

Here, T(X) = $\overline{X}$ by choosing $h(X_1,\ \ldots, X_n) = \frac{1}{\prod_i X_i!}$ and $g(\lambda, T(X)) = e^{-n\lambda}\times\lambda^{n\overline{X}}$

Now, Neymann factorization states that if we can decompose the density in the above way then T(X) which is the sample mean is sufficient for estimating the parameter $\lambda$ . Hence proved.