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. 
Statistics
Doingexercises Doingexercises
23
Report
  • Mathe Mathe
    0

    I feel the bounty is a little to low for this.

  • Daniel90 Daniel90
    0

    I second that.

  • Doingexercises Doingexercises
    0

    Bounty has been increased

  • Mathe Mathe
    0

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

  • Doingexercises Doingexercises
    0

    @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.

  • Mathe Mathe
    0

    Without that hypothesis the problem was too difficult!

Answer

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.
View the answer

2 Attachments

Mathe Mathe
3.2K
  • Doingexercises Doingexercises
    0

    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)?

  • Mathe Mathe
    0

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

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to 50% commission on every question your affiliated users ask or answer.