Question

Q4. Suppose we have a random sample X,, X,, ,X, from the following distribution from U(8,0+1) Let X),Xa).,X) be the order statistics. Show that (Xy,Xu)is sufficient statistics for of ? . Is family of distributions of (Xa),X( )) complete? Is this minimal sufficient?

Detailed and complete answer needed only. Thank you.

Answer 1

Note: The distribution of Range (R) can be found easily by the integrating the joint distribution of f(R,X₍₁₎) over the range of X_{(1) which is from (0,1).}

and to find f(R,X₍₁₎) we find the joint pdf of f(X_(n),X₍₁₎) then by variable change we convert it to a joint pdf of (R,,X₍₁₎)

Thanks