Chapter 6: Problem 3
Given the pdf
$$f(x ; \theta)=\frac{1}{\pi\left[1+(x-\theta)^{2}\right]},
\quad-\infty
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Let \(X\) be \(N(0, \theta), 0<\theta<\infty\) (a) Find the Fisher information \(I(\theta)\). (b) If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from this distribution, show that the mle of \(\theta\) is an efficient estimator of \(\theta\). (c) What is the asymptotic distribution of \(\sqrt{n}(\widehat{\theta}-\theta) ?\)
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the beta distribution with \(\alpha=\beta=\theta\) and \(\Omega=\\{\theta: \theta=1,2\\} .\) Show that the likelihood ratio test statistic \(\Lambda\) for testing \(H_{0}: \theta=1\) versus \(H_{1}: \theta=2\) is a function of the statistic \(W=\) \(\sum_{i=1}^{n} \log X_{i}+\sum_{i=1}^{n} \log \left(1-X_{i}\right)\)
Let \(X\) and \(Y\) be two independent random variables with respective pdfs
$$f\left(x ;
\theta_{i}\right)=\left\\{\begin{array}{ll}\left(\frac{1}{\theta_{i}}\right)
e^{-x / \theta_{i}} & 0
Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are iid with pdf \(f(x ; \theta)=(1 /
\theta) e^{-x / \theta}, 0
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N\left(\mu_{0}, \sigma^{2}=\theta\right)\) distribution, where \(0<\theta<\infty\) and \(\mu_{0}\) is known. Show that the likelihood ratio test of \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta \neq \theta_{0}\) can be based upon the statistic \(W=\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2} / \theta_{0}\) Determine the null distribution of \(W\) and give, explicitly, the rejection rule for a level \(\alpha\) test.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
