Chapter 7

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 \(f_{Y}(y)=\theta e^{-\theta y}\), for \(y \geq 0\), zero elsewhere. 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.

Show that \(Y=|X|\) is a complete sufficient statistic for \(\theta>0\), where \(X\) has the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta

Let \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pmf \(p(x ; \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots\), 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 \(\theta\).

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution with pdf \(f(x ; \theta)=(1 / 2) \theta^{3} x^{2} e^{-\theta x}, 0

Let \(X_{1}, \ldots, X_{n}\) be a random sample from a distribution of the continuous type with cdf \(F(x)\). Let \(\theta=P\left(X_{1} \leq a\right)=F(a)\), where \(a\) is known. Show that the proportion \(n^{-1} \\#\left\\{X_{i} \leq a\right\\}\) is the MVUE of \(\theta\).

The pdf depicted in Figure \(7.9 .1\) is given by $$ f_{m_{2}}(x)=e^{-x}\left(1+m_{2}^{-1} e^{-x}\right)^{-\left(m_{2}+1\right)}, \quad-\infty0\) (the pdf graphed is for \(m_{2}=0.1\) ). This is a member of a large family of pdfs, \(\log F\) -family, which are useful in survival (lifetime) analysis; see Chapter 3 of Hettmansperger and McKean (2011). (a) Let \(W\) be a random variable with pdf \((7.9 .2)\). Show that \(W=\log Y\), where \(Y\) has an \(F\) -distribution with 2 and \(2 m_{2}\) degrees of freedom. (b) Show that the pdf becomes the logistic \((6.1 .8)\) if \(m_{2}=1\). (c) Consider the location model where $$ X_{i}=\theta+W_{i} \quad i=1, \ldots, n $$ where \(W_{1}, \ldots, W_{n}\) are iid with pdf (7.9.2). Similar to the logistic location model, the order statistics are minimal sufficient for this model. Show, similar to Example \(6.1 .2\), that the mle of \(\theta\) exists.