Let X1 X2 Xn be a random sample from a distribution with pd

Let X_1, X_2, ..., X_n be a random sample from a distribution with pdf f(x | theta) = thetax^(theta-1), 0 < x < 1, theta > 0.

The method of moments estimator of tau(theta)=1/theta is tau(thetahat)=(1-xbar)/xbar.

B) Show that E[ (d/dtheta ln(f(x | theta)))^2]=Var(V_i) where V_i=ln(X_i).

C) Find the Cramer-Rao lower bound for an unbiased estimator of tau(theta)=1/theta

image: http://postimg.org/image/5z0zmwej9/295af4c2/

Solution

So to find be, I need the joint pdf first and use the factorization theorem

f(x|?,?)=g(T(x)|?,?)h(x), where T(x) is the sufficient statistic

So obviously the left part with the x3 will resolve out to h(x), but the right portion is where I\'m having trouble

?nie??(x??)22?2x=e??2?2?ni(xi??)2xi

This might be more of an arithmetic question, but I\'m having trouble isolating out the x\'s so I can get a T(x). Let me know if I\'m going wrong elsewhere.