Maximum likelihood estimator and UMVUE

[safina]

Homework Statement

Let X_{1}, ... , X_{n} be a random sample from f\left(x; \theta\right) = \theta x^{\theta - 1} I_{(0, 1)}\left(X\right), where \theta > 0.
a. Find the maximum-likelihood estimator of \theta/\left(1 + \theta\right).

b. Is there a function of \theta for which there exists an unbiased estimator whose variance coincides with the Cramer-Rao lower bound?

The Attempt at a Solution

a.) I understand that in getting the maximum likelihood estimator of \theta, we should be finding the value of \theta that will maximize the likelihood function.
We will do this by taking the derivative of the likelihood function with respect to \theta and equate this derivative to zero; or take the derivative of the logarithm of the likelihood function with respect to \theta and equate it to zero.
But I cannot figure out how to find the MLE of \theta/\left(1 + \theta\right).

b.) Please help me also to figure out what to do in solving this problem b.

 

[lanedance]

not 100% on these, but I think as \theta/\left(1 + \theta\right) is monotonically increasing for \theta > 0 it will be given using the maximum liklihood value for \theta

alternatively you could let a(\theta) = \theta/\left(1 + \theta\right) then solve for \theta(a) and substitute into your probabilty distribution and solve for the MLE for a

 

[lanedance]

how about this, if you have the Liklihood function consider it as
L(\theta) = L(\theta(a))

when you maximise, you find theat such that
\frac{d L(\theta)}{d \theta} = 0

considering this for a, you get
\frac{d}{da} L(\theta(a)) = \frac{d L(\theta)}{d \theta} \frac{d \theta}{d a}

which as the 2nd term is non-zero, gives the same result using the MLE for theta