I was working on the following problem: Consider two probability density functions on [ 0

Edith Mayer

Edith Mayer

Answered question

2022-05-07

I was working on the following problem:
Consider two probability density functions on [ 0 , 1 ] : f 0 ( x ) = 1 and f 1 ( x ) = 2 x. Among all tests of the null hypothesis H 0 : X f 0 ( x ) versus the alternative X f 1 ( x ), with significance level α = 0.1, how large can the power possibly be?
I think we need to begin by looking at an arbitrary test with significance level α = 0.1 but I am having a hard time doing this. I am not even sure this is the direction we want to head in so I was hoping to get some hints regarding this problem.

Answer & Explanation

tomatoland45wt8wm

tomatoland45wt8wm

Beginner2022-05-08Added 19 answers

Let { Γ 0 , Γ 1 } be a partition of [0,1] into (measurable) sets and the decision rule be that hypothesis H i is true if the observation X belongs to Γ i . Then, the false alarm probability is
α = Γ 1 f 0 ( x ) d x = 0.1
while the power of the test is
β = Γ 1 f 1 ( x ) d x .
Now,since f 0 ( x ) is the uniform density, we know that the total "length" of the set Γ 1 is 0.1 and so the question is
What should we choose Γ 1 to be so as to maximize Γ 1 f 1 ( x ) d x
Sketching the shape of the density f 1 ( x ) might reveal the answer to you in a flash, and then you can write down a more formal way of getting to it if you like.

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