Chapter 7: Problem 6
Given that \(f(x ; \theta)=\exp [\theta K(x)+S(x)+q(\theta)], a
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These are the key concepts you need to understand to accurately answer the question.
Chapter 7: Problem 6
Given that \(f(x ; \theta)=\exp [\theta K(x)+S(x)+q(\theta)], a
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeLet \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N\left(\theta_{1}, \theta_{2}\right)\) distribution. (a) Show that \(E\left[\left(X_{1}-\theta_{1}\right)^{4}\right]=3 \theta_{2}^{2}\). (b) Find the MVUE of \(3 \theta_{2}^{2}\).
Let \(Y_{1}
Prove that the sum of the observations of a random sample of size \(n\) from a Poisson distribution having parameter \(\theta, 0<\theta<\infty\), is a sufficient statistic for \(\theta\).
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots\), zero elsewhere, where \(0 \leq \theta \leq 1\) (a) Find the mle, \(\hat{\theta}\), of \(\theta\). (b) Show that \(\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta\). (c) Determine the MVUE of \(\theta\).
Consider the situation of the last exercise, but suppose we have the following two independent random samples: (1). \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample with the common pdf \(f_{X}(x)=\theta^{-1} e^{-x / \theta}\), for \(x>0\), zero elsewhere, and (2). \(Y_{1}, Y_{2}, \ldots, Y_{n}\) is a random sample with common pdf \(f_{Y}(y)=\tau e^{-\tau y}\), for \(y>0\), zero elsewhere. Assume that \(\tau=1 / \theta\) The last exercise suggests that, for some constant \(c, Z=c \bar{X} / \bar{Y}\) might be an unbiased estimator of \(\theta^{2}\). Find this constant \(c\) and the variance of \(Z\). Hint: Show that \(\bar{X} /\left(\theta^{2} \bar{Y}\right)\) has an \(F\) -distribution.
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