CT3 IAI May 2007 Question 8

A random sample of size n is taken from distribution with pdf
f2=2X/(θ^2); 0<X<θ
=0 otherwise

a)Write down the likelihood function and hence by drawing the rough sketch of the likelihood function, obtain the maximum likelihood estimator (mle) for θ .
b)Examine if the mle is unbiased

Kindly explain didn't understand the given solution.

Thanks & Regards,
Anjum

 

This is somewhat similar to example 10.5 on page 14 chapter 10. The usual method of differentiation doesn't work here. MLE in this case is "maximum of sample values" which is less than θ. So we're always underestimating our parameter. Hence the estimator is biased.

 

Thank You

 

This is a special case when equating the first order derivative to zero gives result to theta-cap = 1/zero. Also, the second order derivative gives a positive result. That means likelihood is minimum when theta-cap tends to infinity. It is given that f(x) holds true for 0<x<theta. That means theta>x(i). So, the MLE(theta) is max(x(i)). This is what I understood from the worked out example in page 14 (Chap - Point Estimation) of study material.