53 ∙ let x h h 1 q be independent normal h h 2

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Unformatted text preview: 53 ∙ Let X h : h 1,..., q be independent Normal h , h 2 random variables. Then X h / h are independent Normal h / h ,1 random variables. The nonzero means are the keys. ∙ The variable ∑ i 1 q X i i 2 has a noncentral chi-square distribution with q degrees-of-freedom and noncentrality parameter ∑ h 1 q h h 2 54 ∙ The density of the noncentral chi-square distribution is not especially important to us. ∙ If W is a q 1 multivariate normal with mean and positive definite variance , recall that − 1/2 W Normal − 1/2 , I q . It follows that − 1/2 W ′ − 1/2 W W ′ − 1 W has a noncentral chi-square distribution with q degress of freedom and noncentrality parameter ′ − 1 55 ∙ We can apply the noncentral 2 distribution the testing against local alternatives. Under H n 1 , V − 1/2 n ̂ n − V − 1/2 n ̂ n − n V − 1/2 d → Normal , I p V − 1/2 Normal V − 1/2 , I p 56 ∙ Therefore, n ̂ n − ′ V − 1 ̂ n − has an asymptotic noncentral chi-square distribution with p degrees-of-freedom and noncentrality parameter ′ V − 1 ∙ It can be shown that the larger the noncentrality parameter, the more likely the null will be rejected in favor of the local alternative (asymptotically). 57 ∙ Further, if V 2 − V 1 is positive semidefinite for pd matrices V 1 and V 2 , V 1 − 1 − V 2 − 1 is psd , and so ′ V 1 − 1 − V 2 − 1 ′ V 1 − 1 − ′ V 2 − 1 ≥ or ′ V 1 − 1 ≥ ′ V 2 − 1 This means that tests based on estimators with a smaller asymptotic variance have higher asymptotic local power. 58 ∙ For a given estimator ̂ n where V Avar n ̂ n − n , we can estimate the local power by estimating the noncentrality parameter: ̂ n ′ V ̂ n − 1 and using the noncentral chi-square distribution for power calculations. 59 ∙ In many cases, we want to test general linear restrictions of the form H : R a where R is a q p matrix with q ≤ p and rank R q (which ensures there are really q linearly independent restrictions to test) and a is q 1; these are both known and selected by us. Local alternatives are of the form H n 1 : R n a / n and so, again, the alternative converges to the null at rate 1/ n . 60 ∙ As usual, the test statistic is defined so that it has a known distribution under the null. In this case, T n n R ̂ n − a ′ RV ̂ n R ′ − 1 n R ̂ n − a n R ̂ n − a ′ RV ̂ n R ′ − 1 R ̂ n − a R ̂ n − a ′ R V ̂ n / n R ′ − 1 R ̂ n − a R ̂ n − a ′ Avar R ̂ n − 1 R ̂ n − a where the final expression is a good way of remembering how we obtain a limiting chi-square distribution under H . (Remember, Avar ̂ n V ̂ n / n .) 61 ∙...
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53 ∙ Let X h h 1 q be independent Normal h h 2 random...

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