Suppose X has the multivariate normal distribution in Rk with unknown mean vector h and known positive definite covariance matrixC−1. Consider testing h = 0 versus |C1/2h| ≥ b for some b > 0, where |·| denotes the Euclidean norm.

(i) Show the test that rejects when |C1/2X|

2 > ck,1−α is maximin, where ck,1−α

denotes the 1 − α quantile of the Chi-squared distribution with k degrees of freedom.

(ii) Show that the maximin power of the above test is given P{χ2 k (b2) > ck,1−α}, where χ2 k (b2) denotes a random variable that has the noncentral Chi-squared distribution with k degrees of freedom and noncentrality parameter b2.