Generalized Likelihood Ratio Test Flashcards
What does GLRT stand for?
Generalized Likelihood Ratio Test
When should you use GLRT?
When UMP test does not exist or is hard to compute, or when intuitive tests are intractable.
What is the general form of the null and alternative hypotheses in GLRT?
H0: θ ∈ Θ0 vs H1: θ ∈ Θ \ Θ0
What is the unrestricted MLE?
The value of θ that maximizes the likelihood over the entire parameter space Θ.
What is the restricted MLE?
The value of θ that maximizes the likelihood assuming H0 is true (i.e., over Θ0).
What is the formula for the Generalized Likelihood Ratio (GLR)?
λ(X) = L(θ̂₀) / L(θ̂)
What is the decision rule for the GLRT?
Reject H0 if λ(X) ≤ c, where c is chosen based on significance level α.
What is the likelihood function?
L(θ) = ∏ f(Xi; θ) for i = 1 to n.
How is the restricted MLE found for a simple null hypothesis?
It is the fixed value of θ given in H0.
What distribution does -2 log(λ(X)) follow asymptotically under H0?
A chi-squared distribution with degrees of freedom equal to the difference in dimensionality between Θ and Θ₀.
What is the GLRT decision rule trying to compare?
The likelihood under the null hypothesis versus the maximum likelihood overall.
Why might the GLR be hard to simplify in some cases?
Because the ratio may be algebraically complex, especially with complicated distributions like the Pareto.
