Chapter 7: Problem 10
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with
pdf \(f(x ; \theta)=\theta^{2} x e^{-\theta x}, 0
Chapter 7: Problem 10
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with
pdf \(f(x ; \theta)=\theta^{2} x e^{-\theta x}, 0
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Get started for freeLet \(Y_{1}
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample with the common pdf \(f(x)=\) \(\theta^{-1} e^{-x / \theta}\), for \(x>0\), zero elsewhere; that is, \(f(x)\) is a \(\Gamma(1, \theta)\) pdf. (a) Show that the statistic \(\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}\) is a complete and sufficient statistic for \(\theta\). (b) Determine the MVUE of \(\theta\). (c) Determine the mle of \(\theta\). (d) Often, though, this pdf is written as \(f(x)=\tau e^{-\tau x}\), for \(x>0\), zero elsewhere. Thus \(\tau=1 / \theta\). Use Theorem \(6.1 .2\) to determine the mle of \(\tau\). (e) Show that the statistic \(\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}\) is a complete and sufficient statistic for \(\tau\). Show that \((n-1) /(n X)\) is the MVUE of \(\tau=1 / \theta\). Hence, as usual the reciprocal of the mle of \(\theta\) is the mle of \(1 / \theta\), but, in this situation, the reciprocal of the MVUE of \(\theta\) is not the MVUE of \(1 / \theta\). (f) Compute the variances of each of the unbiased estimators in Parts (b) and (e).
Let \(Y_{1}
Let \(f(x, y)=\left(2 / \theta^{2}\right) e^{-(x+y) / \theta}, 0
Let \(X\) be a random variable with pdf of a regular case of the exponential class. Show that \(E[K(X)]=-q^{\prime}(\theta) / p^{\prime}(\theta)\), provided these derivatives exist, by differentiating both members of the equality $$\int_{a}^{b} \exp [p(\theta) K(x)+S(x)+q(\theta)] d x=1$$ with respect to \(\theta\). By a second differentiation, find the variance of \(K(X)\).
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