4.2. Asymptotic Null Distributions of GLRT; Wald & Rao Statistics

Theorem

Assume the Foutz’s assumption holds. Let

• \(W_n=(\hat\theta_n-\theta_0)^TI(\theta_0)(\hat\theta_n-\theta_0)\)(Wald);

• \(V_n=nT_n(\theta_0)^TI(\hat\theta_n)T_n(\theta_0)\)(Rao);

Then,

1. \(-2\log\lambda_n\stackrel{d}\rightarrow \chi_k^2\);

2. \(W_n\stackrel{d}\rightarrow \chi_k^2\);

3. \(V_n\stackrel{d}\rightarrow \chi_k^2\).

• 즉 GLRT와 Wald, Rao 모두 \(\chi^2\)로 점근수렴하기 때문에 large \(n\)일 경우 카이스퀘어 테스트를 할 수 있다다.

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