Q10 IAI May 2009

X1,X2,...,Xn is a random sample from a U(-theta,theta), theta>0

Q. Find MLE of theta

A. Likelihood fn = 1/(2xtheta)^n -theta =<X1 =< Xn =< theta

The MLE cannot be found using usual calculus method

The likelihood fn is maximum if -theta =<X1 or Xn =< theta

Thus, MLE is MAx(-X1,Xn)


Could someone clarify how the bold part works out?

ALso, MAx(-X1,Xn) means the maximum between these 2 or the maximum between X1,X2,...,Xn ?

PS: X1 =

 

Here we want to maximise 1/(2thetha)^n, so we need as small valu of thetha as possible , but there is a constraint that thetha cannot be smaller than Xn since Xn is a sample from U(-thetha,thetha), similarly thetha should be smaller than or equal to X1. Thus it is said max of -x1 and xn , its not max of x1,x2.....xn.

Lets take an example suppose the distribution is U(-1,1) but we dont know it, and we take a sample of 5 observations say-0.89 , -0.5 , 0 , 0.66 , 0.98 , so now we want to maximise fn = 1/(2thetha)^5 , so now we need smallest value of thetha so we take 0.98(xn) , here we rule out -0.89 (x1) because as we got one observation as 0.98 thetha has to be greater than or equal to 0.98 , so it is maximum of (-x1 and xn) in this case it is maximum of (0.89 , 0.98).
Hope this helps

 

Perfect
The example helped