This thread follows on from the other one I posted which involved finding the maximum likelihood estimator.
X_{i}, i=1,...,n independent random variables with probability density functions f_{i}(x_{i};theta)=(2/i*theta)*(x_{i}/theta) for 0<x_{i}<i*theta, 0 otherwise, theta a positive number.
T_{n}=max{X_{1}/1, X_{2}/2,...,X_{n}/n} is the maximum likelihood estimator of theta.
So I need to:
- Show that the sampling distribution of T_{n} 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 T_{n}
The mean absolute deviation of T_{n} is given by MAD(T_{n})=E[|T_{n}-theta|]. I understand the principle of mean absolute deviation just not sure how to approach it in this instance
-Show whether T_{n }is a consistent estimator for theta
I have to show that P[|T_{n}-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 |T_{n}-theta| above for the MAD I guess I can show it by definition?
- Show whether T_{n }is unbiased.
So I calculate b(theta)=E[T_{n}-theta] seems straightforward(ish!)
Thanks in advance, its the sampling distribution and mean absolute deviation that I probably have the least clue about.