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29MiscLikelihood

# Copyright c 2012 iowa state university statistics 511

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Unformatted text preview: interval for θi is ˆ ˆ θi ± z1−α/2 Σii , where z1−α/2 is the 1 − α/2 quantile of the N (0, 1) ˆ ˆ distribution and Σii is element (i, i) of Σ. Copyright c 2012 (Iowa State University) Statistics 511 14 / 30 An approximate p-value for testing H0 : θ = θ 0 is ˆ ˆ −1 ˆ Pr[χ2 ≥ (θ − θ 0 ) Σ (θ − θ 0 )], k where χ2 is a central χ2 random variable with k k degrees of freedom. Copyright c 2012 (Iowa State University) Statistics 511 15 / 30 MULTIVARIATE DELTA METHOD Suppose g is a function from IRk to IRm , i.e., g1 (θ ) g (θ ) 2 k for θ ∈ IR , g(θ ) = . . . gm (θ ) for some functions g1 , . . . , gm . Suppose g is differentiable with derivative matrix ∂ g (θ) 1 · · · ∂ gm (1θ) ∂θ ∂θ .1 .. . . . D≡ . . . . ∂ g1 (θ ) ∂ gm (θ ) · · · ∂θk ∂θk Copyright c 2012 (Iowa State University) Statistics 511 16 / 30 ˆ Now suppose θ has mean θ and variance Σ. Then Taylor’s Theorem implies ˆ ˆ g(θ ) ≈ g(θ ) + D (θ − θ ) which implies ˆ ˆ E[g(θ )] ≈ g(θ ) + D E(θ − θ ) = g(θ ) and ˆ ˆ Var[g(θ )] ≈ Var[g(θ ) + D (θ − θ )] = D ΣD. Copyright c 2012 (Iowa State University) Statistics 511 17 / 30 ˆ· If θ ∼ N (θ , Σ), it follows that ˆ· g(θ ) ∼ N (g(θ ), D ΣD). In practice, we often need to estimate D by ˆ ˆ replacing θ in D with θ to obtain D. Similarly, we often need to replace Σ with an ˆ estimate Σ. Copyright c 2012 (Iowa State University) Statistics 511 18 / 30 LIKELIHOOD RATIO BASED INFERENCE Suppose we wish to test the null hypothesis that a reduced model provides an adequate ﬁt to a dataset relative to a more general full model that includes the reduced model as a special case. Copyright c 2012 (Iowa State University) Statistics 511 20 / 30 Deﬁne Λ as Reduced Model Maximized Likelihood . Full Model Maximized Likelihood Λ is...
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