slides_13_inferasymptotic

# If w is a q 1 multivariate normal with mean and

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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

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We can apply the noncentral 2 distribution the testing against local alternatives. Under H n 1 , V 1/2 n ̂ n 0  V 1/2 n ̂ n n  V 1/2 d Normal 0 , I p V 1/2 Normal V 1/2 , I p 56
Therefore, n ̂ n 0 V 1 ̂ n 0 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

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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 0 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