Chapter 8: Problem 9
Let \(Y_{1}
Chapter 8: Problem 9
Let \(Y_{1}
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Get started for freeLet the random variable \(X\) have the pdf \(f(x ; \theta)=(1 / \theta) e^{-x /
\theta}, 0
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid \(N\left(\theta_{1}, \theta_{2}\right)
.\) Show that the likelihood ratio principle for testing \(H_{0}:
\theta_{2}=\theta_{2}^{\prime}\) specified, and \(\theta_{1}\) unspecified,
against \(H_{1}: \theta_{2} \neq \theta_{2}^{\prime}, \theta_{1}\)
unspecified, leads to a test that rejects when
\(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \leq c_{1}\) or
\(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \geq c_{2}\)
where \(c_{1}
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 \(H_{1}: \theta=2\)
Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a Poisson distribution with parameter \(\theta .\) Let \(L(\theta)\) be the joint pdf of \(X_{1}, X_{2}, \ldots, X_{10} .\) The problem is to test \(H_{0}: \theta=\frac{1}{2}\) against \(H_{1}: \theta=1\) (a) Show that \(L\left(\frac{1}{2}\right) / L(1) \leq k\) is equivalent to \(y=\sum_{1}^{n} x_{i} \geq c\) (b) In order to make \(\alpha=0.05\), show that \(H_{0}\) is rejected if \(y>9\) and, if \(y=9\) reject \(H_{0}\) with probability \(\frac{1}{2}\) (using some auxiliary random experiment). (c) If the loss function is such that \(\mathcal{L}\left(\frac{1}{2}, \frac{1}{2}\right)=\mathcal{L}(1,1)=0\) and \(\mathcal{L}\left(\frac{1}{2}, 1\right)=1\) and \(\mathcal{L}\left(1, \frac{1}{2}\right)=2\) show that the minimax procedure is to reject \(H_{0}\) if \(y>6\) and, if \(y=6\), reject \(H_{0}\) with probability \(0.08\) (using some auxiliary random experiment).
Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a normal distribution \(N(\theta, 100) .\) Show that \(C=\left\\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \bar{x}=\sum_{1}^{n} x_{i} / n\right\\}\) is a best critical region for testing \(H_{0}: \theta=75\) against \(H_{1}: \theta=78\). Find \(n\) and \(c\) so that $$P_{H_{0}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{0}}(\bar{X} \geq c)=0.05$$ and $$P_{H_{1}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{1}}(\bar{X} \geq c)=0.90$$
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