Question

Let X_1,…, X_n  be a random sample from the Bernoulli distribution, say P[X=1]=θ=1-P[X=0].

and

Cramer Rao Lower Bound of θ(1-θ)

=((1-2θ)^2 θ(1-θ))/n

Find the UMVUE of θ(1-θ) if such exists.

can you proof [part (b) ] using (Leehmann Scheffe Theorem step by step solution) to proof [∑X1-nXbar^2 ]/(n-1) is the umvue , I have the key solution below

x is complete and sufficient.

S^2=∑ [X1-Xbar ]^2/(n-1) is unbiased estimator of θ(1-θ) since the sample variance is an unbiased estimator of the population variance. Furthermore,

S^2= [*∑X1)^2-(∑Xbar)^2 ]/(n-1)= [∑X1)-nXbar^2 ]/(n-1) is a function of ∑Xi , hence, by Leehmann Scheffe Theorem S^2 is UMVUE of θ(1-θ)

Answer 1

It is known that X_i follows Bernoulli( theta)

It is of the form of one parameter exponential family

where,   

Therefore by one parameter exponential family,

is complete and sufficient for

Now, , because

We know that we can estimate the population variance by sample variance which is also an unbiased estimate of population variance

denotes the sample variance

  

Hence, S² is an unbiased estimate of

Lehman sceffe theorem says that, if T is a complete sufficient stat for some parameter p , and T' be an unbiased estimator of g(p) for all p, then E(T'|T) is the UMVUE of g(p)

Here,

p=, g(p) =

T' = S² which is an unbiased estimate of  

is complete and sufficient for

Therefore,

, because S² is a function of

therefore S² is the UMVUE by Lehman Sceffe