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...

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)

It is known that Xi 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

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