Wilkes Theorem for Large Sample GRTS

Question 1

Generalized Likelihood Ratio (GLR)

Answer 1

A test statistic defined as the ratio of the likelihood under the null hypothesis to the likelihood under the alternative hypothesis: λ(X) = L(θ₀)/L(θ̂)

Question 2

GLRT Hypotheses

Answer 2

H₀: θ = θ₀ vs H₁: θ ≠ θ₀

Question 3

GLRT Test Statistic

Answer 3

The transformed test statistic is -2 ln(λ(X)), used to derive critical regions under Wilks’ Theorem

Question 4

Wilks’ Theorem

Answer 4

Under regularity conditions and large samples, -2 ln(λ(X)) converges in distribution to χ²₁ under H₀

Question 5

Critical Region Using Wilks’ Theorem

Answer 5

Reject H₀ if -2 ln(λ(X)) > χ²_{α,1}

Question 6

Regularity Conditions for Wilks’ Theorem

Answer 6

PDF must be differentiable, parameter must not define the support, and the log-likelihood must satisfy integration-differentiation conditions

Question 7

When Wilks’ Theorem Fails

Answer 7

Fails when the parameter appears in the support, e.g., Uniform(0, θ)

Question 8

Pareto Example MLE

Answer 8

θ̂ = n / Σ ln(1 + Xᵢ)

Question 9

Pareto Likelihood Function

Answer 9

L(γ) = γⁿ × ∏ (1 + xᵢ)^(−γ−1)

Question 10

R Function for χ² Critical Value

Answer 10

Use qchisq(1 − α, df = 1) in R to get the critical value for a GLRT

Question 11

Interpretation of -2 ln(λ(X))

Answer 11

Acts like a χ²₁ statistic for large n; used to test the null hypothesis

Question 12

Usage in Practice

Answer 12

Compute λ(X), transform with -2 ln, compare to critical value from χ²₁ to determine significance