Hypothesis Testing

Flashcards reviewing key vocabulary and concepts related to hypothesis testing.

Hypothesis

A statement about a population parameter.

Hypothesis Test

The goal is to decide, based on a sample from the population, which of two complementary hypotheses is true.

Null Hypothesis (H0) and Alternative Hypothesis (Ha)

The two complementary hypotheses in a hypothesis testing problem.

Hypothesis Testing Procedure

A rule that specifies for which sample values the decision is made to accept H0 as true and for which sample values H0 is rejected and Ha is accepted as true.

Rejection Region / Critical Region

The subset of the sample space for which H0 will be rejected.

Acceptance Region

The complement of the rejection region.

Steps in Hypothesis Test

Setting the level of significance, choosing the appropriate test statistic, and establishing critical regions.

Likelihood Ratio Tests

A method of finding tests related to maximum likelihood estimation.

Likelihood Ratio Test (LRT)

A test that has a rejection region of the form {x: λ(x) ≤ c} where c is a number satisfying 0 ≤ c ≤ 1.

Bayesian Tests

Combining sample information with prior information using Bayes' Theorem to obtain the posterior distribution.

Union-Intersection Test

May be useful when the null hypothesis is conveniently expressed as an intersection.

Intersection-Union Test

May be useful if the null hypothesis is conveniently expressed as a union.

Type I Error

If θ ∈ Θ0, but the hypothesis test incorrectly decides to reject H0, then this has been made.

Type II Error

If θ ∈ Θ0c, but the test decides to accept H0, then this has been made.

Power Function

The function of θ defined by β(θ) = Pθ(X ∈ R), where R is the rejection region.

Size α Test

A test with power function β(θ) is this if sup(β(θ)) = α.

Level α Test

A test with power function β(θ) is this if sup(β(θ)) ≤ α.

Uniformly Most Powerful (UMP) Test

A test in class C with power function β(θ) is this if β(θ) ≥ β'(θ) for every θ ∈ Θ0c and every β'(θ) that is a power function of a test in class C.

P-value

A test statistic satisfying 0 ≤ p(x) ≤ 1 for every sample point x.

Valid P-value

If, for every θ ∈ Θ0 and every 0 ≤ α ≤ 1, Pθ(p(X) ≤ α) ≤ α