Best Tests Flashcards
What is the main goal of the ‘Best Tests’ lecture?
To define and identify the most powerful (best) hypothesis test of size α.
What are the hypotheses in the exponential test example?
H0: λ = λ0 vs. H1: λ = λ1, with λ1 > λ0.
What does it mean for a test to have size α?
It means the probability of a Type I error is α (rejecting H0 when H0 is true).
What is a ‘best test’ in hypothesis testing?
A test of size α that has the highest power for detecting the alternative hypothesis.
What is another term for a best test?
Most powerful test.
What lemma gives a procedure for finding the best test?
The Neyman-Pearson Lemma.
What does the Neyman-Pearson Lemma state?
Reject H0 if the likelihood ratio f(x; θ0)/f(x; θ1) ≤ c for some c.
What is a likelihood ratio?
The ratio of the likelihoods under H0 and H1: L(x; θ0)/L(x; θ1).
What is the rejection rule from the Neyman-Pearson Lemma in the exponential case?
Reject H0 if the sum of the Xi’s is ≤ some constant c.
How is the likelihood ratio simplified in the exponential case?
(λ0/λ1)^n * e^{-(λ0 - λ1)∑Xi}
What happens when you take the log of the exponential likelihood ratio?
It becomes linear in ∑Xi, simplifying the rejection rule.
What statistic emerges naturally from the likelihood ratio in the exponential case?
The sample mean (or the sum of the Xi’s).
How do we determine the critical value for the rejection rule?
Use the χ² distribution: c = χ²_{1-α, 2n} / (2nλ0)
Why does the sample mean appear in the best test for exponential distributions?
Because of its relationship to the χ² distribution via 2nλX̄ ~ χ²_{2n}.
What is the power of a test?
The probability of rejecting H0 when H1 is true.
Why is the test using the sample mean considered best?
It has the highest power while maintaining size α.
What does R* denote?
The rejection region of the best (most powerful) test.
What must hold for R* to be considered best?
P(X ∈ R* | H0) = α and P(X ∈ R* | H1) ≥ P(X ∈ R | H1) for all other tests R.
What does it mean for a rejection rule to come from Neyman-Pearson?
The rule and direction (e.g., reject for small means) emerge from the likelihood ratio.
What is the conclusion of the best test for exponential simple vs. simple?
Reject H0 if X̄ ≤ χ²_{1-α, 2n} / (2nλ0), which is the most powerful test.
