The random variables xi x2bull bull bull xn are independent

Question

The random variables XI, X2.• • • X_n are independent and identically dis­tributed binomially with parameters in and p. They thus have the same probability function

f (x) = (7)P^r (1 - p)' ; _x = 0_{, 1,            m}

0 ;  otherwise.

Let X = [ X„

(a)   Show that the log-likelihood 1 (p; X) is given by

1 (p; X) = E log^m + E x, logp + (Tim - E x,) log (1 - ;

(b)   Show that the maximum likelihood estimator of p is

₁5.₌ m-1X = (nn) l

(c)   Show that the Cramer-Rao lower bound for the variance of unbiased estimators of r (p) is

{T^' (p)}² p(¹ - p) / (mn);

(d)  By using the above, show that the Cram6r-Rao lower bound for the variance of unbiased estimators of the common variance of the X„ namely mp - p) is

m(1- 2p)²p(1 - p) /n,

(e)   Give the asymptotic distribution of 1/f. Show that 1/p has mean equal to infinity. Is there any conflict between this fact and the asymptotic distribution you have found?

(f)   Show that the best apprmdmate size-a test of H₀ : p = 1/2 vs HA : p = 3/4 is to reject H₀ when

EX_; k,

where

E 2- N Q. _j=k (m_.n)

2"I^

Why is not possible to find an exact size-a test? Is this test Uniformly Most Powerful against HA : p > 1/2?