# MLE: Sampling Distribution, mean absolute deviation, consistency and bias

• Nov 4th 2012, 03:24 AM
Mick
MLE: Sampling Distribution, mean absolute deviation, consistency and bias
This thread follows on from the other one I posted which involved finding the maximum likelihood estimator.

Xi, i=1,...,n independent random variables with probability density functions fi(xi;theta)=(2/i*theta)*(xi/theta) for 0<xi<i*theta, 0 otherwise, theta a positive number.

Tn=max{X1/1, X2/2,...,Xn/n} is the maximum likelihood estimator of theta.

So I need to:

- Show that the sampling distribution of Tn is g(t)=(2n/theta)(t/theta)^(2n-1) for 0<t<theta, 0 otherwise.
I don't even know where to start with this one, any help would be appreciated!

-Find the Mean Absolute Deviation of Tn
The mean absolute deviation of Tn is given by MAD(Tn)=E[|Tn-theta|]. I understand the principle of mean absolute deviation just not sure how to approach it in this instance

-Show whether Tn is a consistent estimator for theta
I have to show that P[|Tn-theta|>epsilon] tends to 0 as n tends to infinity for any epsilon. Or that the bias and variance both tend to 0 as n tends to infinity. Not sure what the best approach is but if I establish |Tn-theta| above for the MAD I guess I can show it by definition?

- Show whether Tn is unbiased.
So I calculate b(theta)=E[Tn-theta] seems straightforward(ish!)

Thanks in advance, its the sampling distribution and mean absolute deviation that I probably have the least clue about.