# Let $X$ be a single observation from the density $f(x) = (2θx + 1 − θ)I[0,1](x)$ with $−1≤ θ ≤ 1$. Find the most powerful test of size $α$ and its power

• Find a most powerful test of size $α$ of $H_0 : θ = 0$ versus $H_1 : θ = 1$.
• Find a uniformly most powerful test of size $α$ of $H_0 : θ = 0$ versus $H_1 : θ > 0$
• To test $H_0: θ ≤ 0$ versus $H_1: θ > 0$, the following procedure was used: Reject $H_0$ if $X$ exceeds $1/2$. Find the power and size of this test.
• I feel the bounty is a little to low for this.

• I second that.

• Bounty has been increased

• Is this for only one observation (n=1) or for a general random sample of n observations?

• @Rage, this is a good question, yes the original question states let X be a single observation from the density. I will update the question to make this clear.

• Without that hypothesis the problem was too difficult!

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.

2 Attachments

Mathe
2.9K
• Hi, just checking on page 2, the derivative (h'(x)) should the second term be -2θb(2θax+1-2θa) instead of -2θ b(2θax+1-2θb)?

• Yes, there was a typo in the solution. The final result does not change, though. Let me add a new version.