Let n X , X ,..., X 1 2 be a random sample of size n > 2 from a distribution with p.d.f
f (x;θ ) =θ xθ-1 , 0 < x < 1, θ > 0
Find Cramer-Rao lower bound(CRLB)
The joint probability distribution is given by,
Taking logarithm in both sides we get,
Differentiating with respect of we get,
To get maximum likelihood value of we equate right side to zero.
Solving we get,
We know MLE follows invariant property.
So, required MLE of 1/ is given by
Cramer-Rao lower bound-
Cramer-Rao lower bound is given by
where
and
Also, we get,
So, CRLB is given by .
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