UMP Test

Question 1

Uniformly Most Powerful (UMP) Test

Answer 1

A hypothesis test that is most powerful for all values in the alternative hypothesis space. Maximizes power uniformly over composite alternatives.

Question 2

UMP Test Conditions

Answer 2

1. Size condition: max P(X in R* | θ in H0) = α. 2. Power condition: for all θ in H1, P(X in R* | θ) ≥ P(X in R | θ) for any other test R of size α.

Question 3

UMP Test: Step 1

Answer 3

Reduce the problem to a simple vs. simple test by selecting a specific parameter value (e.g., λ1 > λ0) from the composite alternative.

Question 4

UMP Test: Step 2

Answer 4

Construct the most powerful test using Neyman-Pearson lemma for H0: λ = λ0 vs. H1: λ = λ1.

Question 5

Likelihood Ratio Test for Exponential

Answer 5

Reject H0 if (λ0/λ1)^n * exp(-(λ0 - λ1) * sum Xi) ≤ c. Rejection region depends on whether λ1 > λ0 or λ1 < λ0.

Question 6

Rejection Region for λ1 > λ0

Answer 6

Reject H0 if 2λ0 * sum Xi < χ²_{1-α, 2n}. This is a lower-tailed test using the chi-square distribution.

Question 7

Rejection Region for λ1 < λ0

Answer 7

Reject H0 if 2λ0 * sum Xi > χ²_{α, 2n}. This is an upper-tailed test using the chi-square distribution.

Question 8

Chi-Square Relation

Answer 8

If Xi ~ Exp(λ), then 2λ * sum Xi ~ χ²_{2n}. Used to derive the critical region of the UMP test.

Question 9

UMP for Two-Sided Alternative?

Answer 9

No UMP test exists for H1: λ ≠ λ0 because different values of λ1 require different rejection regions. No single region works uniformly.

Question 10

Key Insight for UMP Tests

Answer 10

A UMP test for one-sided alternatives can be derived using Neyman-Pearson if the rejection region does not depend on specific λ1, only its inequality to λ0.