# Sufficiency for a binomial population

The question at hand is: If X1 and X2 are independent random variables having binomial distributions with the parameters θ and n1 and θ and n2,

a ) show that (X1 +X2/n1+n2) is a sufficient estimator of θ. The textbook I am using shows how to find sufficient statistics as it relates to random variables from a Bernoulli population. Since the Bernoulli and binomial distributions are related I was wondering if I could use the Bernoulli to solve this? This is what the textbook does in their example:

(This image is from John E Freund's Mathematical Statistics with Applications 8th edition, Chapter 10 page 295)

Would I take the same approach?

b) In reference to part a, is (X1 +2X2)/(n1 +2n2) a sufficient estimator of θ? I am not understanding how to do this. Would I follow the same procedure as part a?

## 1 Answer

first hints:

a) What can you learn from the textbook derivation on the way to establish sufficiency? Try to reproduce the steps, starting from the Binomial likelihood instead of the Bernoulli likelihood. Once you obtain a factorisation, sufficiency follows. It is not necessary to prove that the conditional distribution of the sample given the sufficient statistic is independent from the parameter.

b) The question is unclear in that it mixes sufficiency and estimation: once you have produced a sufficient statistic, it is an estimator of anything you chose. To make it precisely an estimator of $\theta$, this sufficient statistic must enjoy some additional property like, e.g., unbiasedness. Now it may just be that the question is to determine whether or not$$\frac{X_1+2X_2}{n_1+2n_2}$$ is sufficient. A further hint is that, since$$\frac{X_1+X_2}{n_1+n_2}$$is sufficient, both sufficient statistics should be in one-to-one correspondence.

• For part a) so far I have understood that there are two ways. You can use the conditional density way (like in the picture I provided above) or you can use the factorization criterion. I understand the factorization theorem here and got the answer using it, but I don't understand the conditional density way. For part b) the question asks if (x1 + 2x2)/(n1 + 2n2) is a sufficient estimator of theta. Thats all the textbook has said. I am not sure how to approach this part with the factorization criterion method OR the conditional density method. – user198848 May 5 '18 at 18:07
• Yes part b) is just to determine whether or not (X1 + 2X2)/(N1 + 2N2) is sufficient. I am not sure whether the factorization theorem I used in part a) can be applied to this question. – user198848 May 5 '18 at 19:42
• Okay so if we let T1 = (X1 + X2)/(n1 + n2) and T2 = (X1 + 2X2/N1 + 2N2), we can conclude that T2 is not sufficient because there is no one to one mapping between T1 and T2? Is that interpretation correct ? – user198848 May 6 '18 at 8:37