Chapter 7

Show that the product of the sample observations is a sufficient statistic for \(\theta>0\) if the random sample is taken from a gamma distribution with parameters \(\alpha=\theta\) and \(\beta=6\).

Let \(Y_{1}

Let \(X\) have the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta0\) (a) Is the statistic \(Y=|X|\) a sufficient statistic for \(\theta ?\) Why? (b) Let \(f_{Y}(y ; \theta)\) be the pdf of \(Y\). Is the family \(\left\\{f_{Y}(y ; \theta): \theta>0\right\\}\) complete? Why?

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(N(\mu, \theta), 0<\theta<\infty\), where \(\mu\) is unknown. Let \(Y=\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} / n\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2}\). If we consider decision functions of the form \(\delta(y)=b y\), where \(b\) does not depend upon \(y\), show that \(R(\theta, \delta)=\left(\theta^{2} / n^{2}\right)\left[\left(n^{2}-1\right) b^{2}-2 n(n-1) b+n^{2}\right]\). Show that \(b=n /(n+1)\) yields a minimum risk decision function of this form. Note that \(n Y /(n+1)\) is not an unbiased estimator of \(\theta\). With \(\delta(y)=n y /(n+1)\) and \(0<\theta<\infty\), determine \(\max _{\theta} R(\theta, \delta)\) if it exists.

Let \(Y_{1}

Let \(X\) have the \(\operatorname{pmf} p(x ; \theta)=\frac{1}{2}\left(\begin{array}{c}n \\ |x|\end{array}\right) \theta^{|x|}(1-\theta)^{n-|x|}\), for \(x=\pm 1, \pm 2, \ldots, \pm n\), \(p(0, \theta)=(1-\theta)^{n}\), and zero elsewhere, where \(0<\theta<1\) (a) Show that this family \(\\{p(x ; \theta): 0<\theta<1\\}\) is not complete. (b) Let \(Y=|X| .\) Show that \(Y\) is a complete and sufficient statistic for \(\theta\).

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of \(Y_{1}=\sum_{1}^{n} K\left(X_{i}\right)\) is of the form \(f_{Y_{1}}\left(y_{1} ; \theta\right)=R\left(y_{1}\right) \exp \left[p(\theta) y_{1}+n q(\theta)\right]\). Hint: Let \(Y_{2}=X_{2}, \ldots, Y_{n}=X_{n}\) be \(n-1\) auxiliary random variables. Find the joint pdf of \(Y_{1}, Y_{2}, \ldots, Y_{n}\) and then the marginal pdf of \(Y_{1}\).

What is the sufficient statistic for \(\theta\) if the sample arises from a beta distribution in which \(\alpha=\beta=\theta>0 ?\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(b(1, \theta), 0 \leq \theta \leq 1 .\) Let \(Y=\sum_{1}^{n} X_{i}\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2} .\) Consider decision functions of the form \(\delta(y)=b y\), where \(b\) does not depend upon \(y .\) Prove that \(R(\theta, \delta)=b^{2} n \theta(1-\theta)+(b n-1)^{2} \theta^{2} .\) Show that $$ \max _{\theta} R(\theta, \delta)=\frac{b^{4} n^{2}}{4\left[b^{2} n-(b n-1)^{2}\right]}, $$ provided that the value \(b\) is such that \(b^{2} n>(b n-1)^{2}\). Prove that \(b=1 / n\) does not \(\operatorname{minimize} \max _{\theta} R(\theta, \delta)\)

Let \(Y\) denote the median and let \(\bar{X}\) denote the mean of a random sample of size \(n=2 k+1\) from a distribution that is \(N\left(\mu, \sigma^{2}\right)\). Compute \(E(Y \mid \bar{X}=\bar{x})\). Hint: See Exercise \(7.5 .4 .\)