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Inferential Statisticshard
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Consider the likelihood ratio test for H0:θ=θ0H_0: \theta = \theta_0H0​:θ=θ0​ versus H1:θ≠θ0H_1: \theta \neq \theta_0H1​:θ=θ0​. If the log-likelihood function is ℓ(θ)\ell(\theta)ℓ(θ), the statistic λ=−2(ℓ(θ0)−ℓ(θ^))\lambda = -2(\ell(\theta_0) - \ell(\hat{\theta}))λ=−2(ℓ(θ0​)−ℓ(θ^)) is used. Under the null hypothesis, what is the asymptotic distribution of λ\lambdaλ as n→∞n \to \inftyn→∞?