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(1) distribution under the null hypothesis \(H_{0}: \theta=\theta_{0}\), where

\[U_{n}\left(\theta_{0}ight)=\left.\sum_{i=1}^{n} \frac{\partial}{\partial \theta} \log \left[f\left(X_{i} ; \thetaight)ight]ight|_{\theta=\theta_{0}} .\]

Does this statistic require us to calculate the maximum likelihood estimator of \(\theta\) ?

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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. =