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Maximum likelihood estimator and UMVUE

  • Thread starter safina
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  • #1
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Homework Statement


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

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

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

Answers and Replies

  • #2
lanedance
Homework Helper
3,304
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not 100% on these, but I think as [itex]\theta/\left(1 + \theta\right)[/itex] is monotonically increasing for [itex]\theta > 0[/itex] it will be given using the maximum liklihood value for [itex]\theta [/itex]

alternatively you could let [itex]a(\theta) = \theta/\left(1 + \theta\right)[/itex] then solve for [itex]\theta(a) [/itex] and substitute into your probabilty distribution and solve for the MLE for a
 
  • #3
lanedance
Homework Helper
3,304
2
how about this, if you have the Liklihood function consider it as
[tex]L(\theta) = L(\theta(a)) [/tex]

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

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

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

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