91Ó°ÊÓ

Consider the single response variable \(Y\) with \(Y \sim\) binomial \((n, \pi)\) (a) Find the Wald statistic \((\widehat{\pi}-\pi)^{T} \mathfrak{I}(\widehat{\pi}-\pi)\) where \(\widehat{\pi}\) is the maximum likelihood estimator of \(\pi\) and \(\mathfrak{I}\) is the information. (b) Verify that the Wald statistic is the same as the score statistic \(U^{T} \mathfrak{I}^{-1} U\) in this case (see Example 5.2 .2 ). (c) Find the deviance $$2[l(\widehat{\pi} ; y)-l(\pi ; y)]$$ (d) For large samples, both the Wald/score statistic and the deviance approximately have the \(\chi^{2}(1)\) distribution. For \(n=10\) and \(y=3\) use both statistics to assess the adequacy of the models: (i) \(\pi=0.1\) (ii) \(\pi=0.3\) (iii) \(\pi=0.5\) Do the two statistics lead to the same conclusions?