Maximum likelihood estimator and UMVUE

In summary, the conversation discusses finding the maximum-likelihood estimator of \theta/\left(1 + \theta\right) and whether there exists an unbiased estimator whose variance coincides with the Cramer-Rao lower bound for a given probability distribution. The method of finding the maximum-likelihood estimator is by taking the derivative of the likelihood function with respect to \theta and equating it to zero. Alternatively, one can use a function a(\theta) = \theta/\left(1 + \theta\right) and solve for \theta(a) to find the MLE. The conversation also considers the possibility of using the MLE for \theta to find the MLE for a function of \theta.
  • #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.
 
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  • #2
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
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
 

What is a maximum likelihood estimator (MLE)?

A maximum likelihood estimator is a statistical method used to estimate the parameters of a probability distribution by finding the set of values that maximizes the likelihood of a given set of data. In other words, it is a way to find the most likely values for the unknown parameters of a probability distribution based on the observed data.

How is the maximum likelihood estimator calculated?

The maximum likelihood estimator is calculated by taking the derivative of the likelihood function with respect to each parameter, setting the derivatives equal to zero, and solving for the parameter values that maximize the likelihood function.

What is an unbiased minimum variance estimator (UMVUE)?

An unbiased minimum variance estimator is a type of estimator that is both unbiased (its expected value is equal to the true value of the parameter being estimated) and has the smallest possible variance among all unbiased estimators. In other words, it is the most efficient unbiased estimator for a given parameter.

How is the UMVUE different from the MLE?

The UMVUE and MLE are different in that the UMVUE is unbiased and has the smallest possible variance among all unbiased estimators, while the MLE may not be unbiased and may have a larger variance. However, in some cases, the UMVUE and MLE may be the same.

When should the UMVUE be used instead of the MLE?

The UMVUE should be used instead of the MLE when unbiasedness and minimum variance are the most important factors in selecting an estimator. This is often the case in situations where the cost of estimation errors is high, such as in medical research or financial forecasting.

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