14.27 Let (zeta) denote a generic measure of association. For (K) independent multinomial samples of sizes (left{n_{k}
Question:
14.27 Let \(\zeta\) denote a generic measure of association. For \(K\) independent multinomial samples of sizes \(\left\{n_{k}\right\}\), suppose that \(\sqrt{n_{k}}\left(\hat{\zeta}_{k}-\zeta_{k}\right)\) \(\xrightarrow{d} N\left(0, \sigma_{k}^{2}\right)\) as \(n_{k} \rightarrow \infty\). A summary measure is
\[\bar{\zeta}=\frac{\sum_{k}\left(n_{k} / \hat{\sigma}_{k}^{2}\right) \hat{\zeta}_{k}}{\sum_{k}\left(n_{k} / \hat{\sigma}_{k}^{2}\right)}\]
a. Show that \(\sum_{k} z_{k}^{2}=V+\left[\bar{\zeta}^{2} / \hat{\sigma}^{2}(\bar{\zeta})\right]\), where
\[V=\sum_{k} \frac{n_{k}\left(\hat{\zeta}_{k}-\bar{\zeta}\right)^{2}}{\hat{\sigma}_{k}^{2}}, \quad z_{k}=\frac{n_{k}^{1 / 2} \hat{\zeta}_{k}}{\hat{\sigma}_{k}}, \quad \hat{\sigma}^{2}(\bar{\zeta})=\left(\sum_{k} \frac{n_{k}}{\hat{\sigma}_{k}^{2}}\right)^{-1} .\]
b. Suppose that \(n \rightarrow \infty\) with \(n_{k} / n \rightarrow \rho_{k}>0, k=1, \ldots, K\). State the asymptotic chi-squared distribution for each component in the partitioning in part (a). Indicate the hypothesis that each tests.
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Step by Step Answer:
Solution a Proof that zk V bar bar We can prove the equation using the following steps 1 Expand the squared term in zk zk nk12 k k nk k k 2 Separate t...View the full answer
