If X1 and X2 are independent random variables having a binom

If X₁ and X₂ are independent random variables having a binomial distributions with the parameters theta and n₁ and theta and n₂, is (X₁ + 2X₂)/(n₁ + 2n₂) a sufficient estimator of theta?

Solution

X1~Bin(n1,theta)

X2~Bin(n2,theta)

(X1,X1+2X2) follows a multinomial distribution and

P(X1|S=X1+2X2=s) depends on theta. So, X1+2X2 is not a sufficient statistic for theta.