Under the assumptions outlined in Theorem 10.11, show that Rao's efficient score statistic, which is given by \(Q=n^{-1} U_{n}^{2}\left(\theta_{0}ight) I^{-1}\left(\theta_{0}ight)\) has an asymptotic ChiSquared \(\left[1, \delta^{2} I\left(\theta_{0}ight)ight]\) distribution under the sequence of alternative hypothesis \(\left\{\theta_{1, n}ight\}_{n=1}^{\infty}\) where \(\theta_{1, n}=\theta_{0}+n^{-1 / 2} \delta\).

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Theorem 10.11. Suppose that X1,..., Xn is a set of independent and iden- tically distributed random variables from a distribution F(x|0) with density f(x0), where 0 has parameter space . Suppose that the conditions of The- orem 10.5 hold, then, under the null hypothesis Ho: 000, An X as no where X has CHISQUARED(1) distribution. =