In the context of the proof of Theorem 10.12, prove that \(\Lambda_{n}=n\left(\hat{\theta}_{n}-ight.\) \(\left.\theta_{0}ight)^{2} I\left(\theta_{0}ight)+o_{p}\left(n^{-1}ight)\), as \(n ightarrow \infty\) where \(\theta_{1, n}=\theta_{0}+n^{-1 / 2} \delta\).

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Theorem 10.12. Suppose that X1.....X, is a set of independent and iden- tically distributed random variables from a distribution F(x0) with density f(x0), where 0 has parameter space . Suppose that the conditions of Theorem 10.5 hold, then, under the sequence of alternative hypotheses (01,n) where 01,n=00+n-1/28, An X as n where X has CHISQUARED [1, 621(00)] distribution.