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Chapter 14-10. Likelihood Ratio Test and Proof of Wilks’ Phenomenon

Recommended Post: 【Statistics】 Lecture 14. Statistical Tests

1. Summary

2. Proof

3. Examples

1. Summary

⑴ Likelihood Ratio Test: Given the null hypothesis H₀: θ = θ₀, and the alternative hypothesis H₁: θ ≠ θ₀, the rejection region for rejecting the null hypothesis H₀ can be set as follows (where ℒ is the likelihood function)

⑵ Generalized Likelihood Ratio Test: In the likelihood ratio test, the null hypothesis H0 can only be defined as a simple hypothesis like θ = θ₀, which is a limitation. Given the null hypothesis H₀: θ ∈ Θ₀, and the alternative hypothesis H₁: θ ∉ Θ₀, the rejection region for rejecting the null hypothesis H₀ can be set as follows (where ℒ is the likelihood function)

⑶ Wilks’ Phenomenon: When n is sufficiently large, for θ = (θ₁, ···, θ_k), null hypothesis H₀: θ ∈ Θ₀, and alternative hypothesis H₁: θ ∉ Θ₀, -2 log λ(X₁, ···, X_n) follows a chi-squared distribution with ν degrees of freedom

⑷ Simplified Wilks’ Phenomenon: When n is sufficiently large, for the null hypothesis H₀: θ = θ₀, and alternative hypothesis H₁: θ ≠ θ₀, θ ∈ ℝ, -2 log λ(X₁, ···, X_n) follows a chi-squared distribution with 1 degree of freedom

2. Proof

⑴ Proof for the simplified Wilks’ phenomenon

⑵ Using Taylor expansion

3. Examples

⑴ Example 1. Given X₁, ···, X_n ~ Poisson(λ), and null hypothesis H₀: λ = λ₀, H₁: λ ≠ λ₀, find the rejection region for significance level α.

⑵ Example 2. Given y₁, ···, y₅ following a multinomial distribution for θ = (p₁, ···, p₅), where L(θ) = p₁^y₁ ··· p₅^y₅, for the null hypothesis H₀: p₁ = p₂ = p₃, p₄ = p₅, and alternative hypothesis H₁, find the rejection region for significance level α.

⑶ Example 3. Chi-squared goodness-of-fit test: For S different bins (s = 1, ···, S), with frequencies Ns in each bin, null probabilities p_s⁰, and total sample size n, in addition to the chi-squared test, the likelihood ratio test can also be performed. However, as n → ∞, the chi-squared test statistic and the likelihood ratio test statistic are the same.

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