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Learn how to use the likelihood ratio test statistic to test hypotheses about the parameters of a population distribution. See examples for normal and exponential distributions and the relation to MLEs.
Learn how to use the generalized likelihood ratio test (GLRT) to test hypotheses about parametric models. See examples, definitions, theorems, and approximations for large sample sizes.
9-3.4 Likelihood Ratio Test (extra!) • Neyman-Pearson Lemma: Likelihood-ratio test is the most powerful test of a specified value α when testing two simple hypotheses.# • simple hypotheses # H 0: θ=θ 0 and H 1: θ=θ 1
Likelihood Ratio Tests. Likelihood ratio tests (LRTs) have been used to compare two nested models. The form of the test is suggested by its name, LRT. _ = –2 log = / (^ ) ) , _ (^ 1 ) ) the ratio of two likelihood functions; the simpler model ( s ) has fewer parameters than the general ( g ) model.
Definition 8.2.1 The likelihood ratio test statistic for testing H0 : θ ∈ Θ0 versus H1 : θ ∈ Θc. 0 is. L(θ|x) supΘ0. λ(x) = . supΘ L(θ|x) A likelihood ratio test (LRT) is any test that has a rejection region of the form {x : λ(x) ≤ c}, where c is any number satisfying 0 ≤ c ≤ 1.
We want to test H 0: θ = θ 0 against H 1: θ 6= θ 0 using the log-likelihood function. We denote l (θ) the loglikelihood and bθ n the consistent root of the likelihood equation. Intuitively, the farther bθ n is from θ 0, the stronger the evidence against the null hypothesis. How far is fifar enoughfl? AD February 2008 3 / 30
For a composite hypothesis we define the likelihood ratio test statistic as λ(x) = −2log sup θ∈Θ 0 L( θ;x) sup θ∈Θ L(θ;x) = −2log L(ˆˆ;x) L(θˆ;x), where θˆ= argmax θ∈Θ and ˆˆ θ = argmax θ∈Θ 0. As Θ 0 ⊆ Θ, we would generally have that θˆ∈ Θ A, sup θ∈Θ L(x;θ) = sup θ∈Θ A L(x;θ) = L(x; ˆθ).
