Uniform most powerful test

[macangyed]

Hi all, I am new here :wave:

I got a question listed below. Thanks for your generous help.

Let X1;...;Xn be a random sample from the uniform distribution on (Theta; Theta + 1). Find the uniform most powerful test at level  alpha (Type I error) = 0:01 for testing H0 : Theta = 1 vs Ha : Theta > 1. Calculate the power of this test.

 

[Dason]

Hi! :welcome: We are glad that you posted here! This looks like a homework question though. Our homework help policy can be found here. We mainly just want to see what you have tried so far and that you have put some effort into the problem. I would also suggest checking out this thread for some guidelines on smart posting behavior that can help you get answers that are better much more quickly.

 

[macangyed]

Thanks Dason, I am sorry if I brought any trouble to you. I am very lost in this question since it is a irregular case.

So far I only think

Under H0: P(sup Xi>=Theta|Theta)=1, P(inf Xi<Theta+1|Theta)=0
=> P(X1>=Theta|Theta)*...*P(Xn>=Theta|Theta)=1 ， P(X1<Theta+1|Theta)*...*P(Xn<Theta+1|Theta)=1

Test: Rej H0 if either Y1<Theta or Y1>Theta+1

But how to calculate the power of the test?

 

[Dason]

Well what is the definition of the power of a test?

 

[macangyed]

Under Ha, the the probability that Xi fall in the complement of rejection region.

In this question, the equation is Power=P(sup Xi>Theta+1|Theta)+P(inf Xi<Theta|Theta). Then how? Using order statistic?