Chapter 6: Problem 3
Given the pdf
$$f(x ; \theta)=\frac{1}{\pi\left[1+(x-\theta)^{2}\right]},
\quad-\infty
Chapter 6: Problem 3
Given the pdf
$$f(x ; \theta)=\frac{1}{\pi\left[1+(x-\theta)^{2}\right]},
\quad-\infty
All the tools & learning materials you need for study success - in one app.
Get started for freeLet \(X\) 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 \(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}\) be a random sample from a Cauchy
distribution with median \(\theta\), that is, with pdf.
$$f(x ; \theta)=\frac{1}{\pi} \frac{1}{1+(x-\theta)^{2}},
\quad-\infty
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the Poisson distribution with \(0<\theta \leq 2\). Show that the mle of \(\theta\) is \(\widehat{\theta}=\min \\{\bar{X}, 2\\}\).
Let \(S^{2}\) be the sample variance of a random sample of size \(n>1\) from \(N(\mu, \theta), 0<\theta<\infty\), where \(\mu\) is known. We know \(E\left(S^{2}\right)=\theta\) (a) What is the efficiency of \(S^{2} ?\) (b) Under these conditions, what is the mle \(\widehat{\theta}\) of \(\theta\) ? (c) What is the asymptotic distribution of \(\sqrt{n}(\widehat{\theta}-\theta) ?\)
Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are iid with pdf \(f(x ; \theta)=(1 /
\theta) e^{-x / \theta}, 0
What do you think about this solution?
We value your feedback to improve our textbook solutions.