Is my approach correct to check the sufficiency of the static

Question

Let we have $n$ random variables from Poisson distribution with parameters $\lambda$. It is required to check the sufficiency of the following estimators

a). $(X_1,\sum_{i=2}^{n}X_i)$ b). $(X_1,\bar{X})$

My approach

The joint distribution function of the samples is:

$f(X|\theta)=\frac{e^{-n\lambda }\lambda^{\sum_{i=1}^{n} X_i}}{\prod X_i}$

For the first option, we can write the likelihood as:

$f(X|\theta)=\frac{e^{-n\lambda }\lambda^{X_1+\sum_{i=2}^{n} X_i}}{\prod X_i}$

The above likelihood equation can be expressed as the function of $h(x_1,x_2,...,x_n)$ and $g((x_1,\sum_{i=2}^{n}),\lambda)$. Hence, the first one is sufficient. But for the second one, i am not sure how can i express the second Statistic in the above form. I know it will be sufficient but not sure how to express it in terms of factorization theorem. Thanks.

Answer 1

You can write $$ \sum_{k=1}^n X_k = n\bar X.$$ The fact that you don't even need $X_1$ is irrelevant: if $\bar X$ is sufficient, then $(\bar X,X_1)$ certainly is.