Chapter 8: Problem 8
If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a beta distribution with parameters \(\alpha=\beta=\theta>0\), find a best critical region for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta=2\)
Chapter 8: Problem 8
If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a beta distribution with parameters \(\alpha=\beta=\theta>0\), find a best critical region for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta=2\)
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Get started for freeShow that the likelihood ratio principle leads to the same test when testing a simple hypothesis \(H_{0}\) against an alternative simple hypothesis \(H_{1}\), as that given by the Neyman-Pearson theorem. Note that there are only two points in \(\Omega\).
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid with pmf \(f(x ; p)=p^{x}(1-p)^{1-x}, x=0,1\), zero elsewhere. Show that \(C=\left\\{\left(x_{1}, \ldots, x_{n}\right): \sum_{1}^{n} x_{i} \leq c\right\\}\) is a best critical region for testing \(H_{0}: p=\frac{1}{2}\) against \(H_{1}: p=\frac{1}{3} .\) Use the Central Limit Theorem to find \(n\) and \(c\) so that approximately \(P_{H_{0}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.10\) and \(P_{H_{1}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.80\).
Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a normal distribution \(N\left(0, \sigma^{2}\right) .\) Find a best critical region of size \(\alpha=0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=2 .\) Is this a best critical region of size \(0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=4 ?\) Against \(H_{1}: \sigma^{2}=\sigma_{1}^{2}>1 ?\)
Let \(Y_{1}
Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate normal distribution with \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}, \rho=\frac{1}{2}\), where \(\mu_{1}, \mu_{2}\), and \(\sigma^{2}>0\) are unknown real numbers. Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0, \sigma^{2}\) unknown against all alternatives. The likelihood ratio \(\Lambda\) is a function of what statistic that has a well- known distribution?
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