Suppose X Binomial(n, p), ˆp is the MLE of p, and ˆ is the MLE of = p(1 p). (a)

Question

Suppose X Binomial(n, p), ˆp is the MLE of p, and ˆ is the MLE of = p(1 p). (a) Show that ˆ is asymptotically efficient when p 6= 1/2. You can assume Theorem 10.1.2 with () = only (since the statement for general lacks a necessary condition). Briefly explain why the asymptotic efficiency does not hold when p = 1/2. (b) Find the limiting distribution of ˆ when p = 1/2 (Use Theorem 5.5.26). (c) With the exact expression of Var ˆ, calculate limn n 2 Var ˆ.

book that used statistical inference by casella

Explanation / Answer

Let L = f(x; n,p) (limlit 1 to n) = C(n,x) px qn-x (limlit 1 to n) = C(n,1)C(n,2)C(n,3).....C(n,n) p1+2+3....+n qn+(n-1)+(n-2)+.....+1

Taking log on both side, we get

logL = log(C(n,1)C(n,2)C(n,3).....C(n,n)) + logp1+2+3....+n + logqn+(n-1)+(n-2)+.....+1

logL = A + (1+2+3+.....+n)logp + (1+2+3+.....+n) log (1-p)

Differentaiting both sides wrt p, we get

1/L L/p = 0+ (1+2+3+.....+n)/p + (1+2+3+.....+n)/(1-p)*(-1)

= (1+2+3+.....+n)( 1/p - 1/(1-p))

So, L/p = 0

(1-p -p)/p(1-p) = 0

1 - 2p = 0

p = 1/2

So, the maximum likelihood Estimator of p is = 1/2

Asymptotic efficiency does not hold, because variances are different for different values of p.

(b) The limiting distribution of p when p = 1/2 i s given by

P(x;, n, 1/2) = C(n,x) (1/2)x (1-1/2)n-x = C(n,x) (1/2)n

(c) Var(p) = Var (x/n) = 1/n2 Var(x) = 1/n2 npq = pq/n = (1/2) * (1/2) /n = 1/(4n)

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