N for sufciently large n n i1 where copyright

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Unformatted text preview: lim Pr[||θ − θ || > ε] = 0 for any ε > 0. n→∞ For sufficiently large n, ˆ· θ ∼ N (θ , I−1 (θ )), where Copyright c 2012 (Iowa State University) Statistics 511 7 / 30 ∂ (θ ) ∂ (θ ) ∂θ ∂θ ∂ 2 (θ ) = −E ∂ θ∂ θ ∂ 2 (θ ) = −E . ∂θi ∂θj i,j∈{1,...,k} I(θ ) = E Copyright c 2012 (Iowa State University) Statistics 511 8 / 30 I (θ ) is known as the Fisher Information matrix. I (θ ) can be approximated by the observed Fisher Information matrix, which is given by 2 ˆ ˆ(θ ) ≡ −∂ (θ ) I ∂ θ∂ θ ˆ θ =θ . I(θ ) and ˆ(θ ) may depend on unknown nuisance Iˆ parameters. In such cases, nuisance parameters are replaced by consistent estimators. Copyright c 2012 (Iowa State University) Statistics 511 9 / 30 A Simple Example i.i.d. Suppose y1 , . . . , yn ∼ N (µ, σ 2 ). If we are interested in inference for µ, we can take θ = µ and treat σ 2 as a nuisance parameter. It is straightforward to show that ¯· is the unique y solution to the likelihood equation. Furthermore, it is straightforward to show that I (θ) = ˆ(θ) = n/σ 2 . Iˆ Copyright c 2012 (Iowa State University) Statistics 511 10 / 30 A Simple Example (continued) Thus, we have ˆy θ = ¯· ∼ N (θ = µ, I −1 (θ) = σ 2 /n) and · ¯· ∼ N (µ, s2 /n) y for sufficiently large n, where 2 s= Copyright c 2012 (Iowa State University) n i=1 (yi − ¯· )2 y . n−1 Statistics 511 11 / 30 WALD TESTS AND CONFIDENCE INTERVALS Suppose for large n that · ˆ ˆ −1/2 (θ − θ ) ∼ N (0, I) Σ and · ˆ ˆ −1 ˆ (θ − θ ) Σ (θ − θ ) ∼ χ2 . k Copyright c 2012 (Iowa State University) Statistics 511 12 / 30 ˆ For example, suppose θ is the MLE of θ and ˆ(θ ) is Iˆ the observed information matrix. Then under regularity conditions, we have · ˆ [ˆ(θ )]1/2 (θ − θ ) ∼ N (0, I) Iˆ and · ˆ (θ − θ ) ˆ(θ )(θ − θ ) ∼ χ2 Iˆ ˆ k for sufficiently large n. Copyright c 2012 (Iowa State University) Statistics 511 13 / 30 An approximate 100(1 − α)% confidence...
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