Suppose that thetai i1n are independent unbiased estimators

Suppose that (theta)i, i=1,....n are independent unbiased estimators of theta with Var(theta)=(sigma)^2. Consider a combined estimator (theta)c=Sum(ai*(theta)i) where sum(ai)=1.

a)Show that (Theta)c is unbiased.

Solution

a.

Since i is unbiased of ,E(i)=

c=sum(ai*i),with sum(ai)=1

Then,E(c)=sum(ai*E(i))=sum(ai)*= [since sum(ai)=1

Therefore c is unbiased for .

b. Var(c)=sum(ai^2*i^2) i=1,2

define f(ai)=sum(ai^2*i^2)+*(sum(ai)-1) ,where is the lagrangian multiplier

parially differentiating wrt ai,we get,

ai=(-)/(2*i^2) ,i=1(1)2

but sum(ai)=1

so,=(-2)/sum(1/i^2)

therefore,

ai=(1/i^2)/sum(1/i^2) ,i=1,2 and second order derivative of f(ai) is positive at this value of ai,so variance is minimized.(proved)