Problem 1
In the linear model (5.3), suppose that \(n=2 r\) is an even integer and define \(W_{j}=Y_{n-j+1}-\) \(Y_{j}\) for \(j=1, \ldots, r\). Find the joint distribution of the \(W_{j}\) and hence show that $$ \tilde{\gamma}_{1}=\frac{\sum_{j=1}^{r}\left(x_{n-j+1}-x_{j}\right) W_{j}}{\sum_{j=1}^{r}\left(x_{n-j+1}-x_{j}\right)^{2}} $$ satisfies \(\mathrm{E}\left(\tilde{\gamma}_{1}\right)=\gamma_{1} .\) Show that $$ \operatorname{var}\left(\tilde{\gamma}_{1}\right)=\sigma^{2}\left\\{\sum_{j=1}^{n}\left(x_{j}-\bar{x}\right)^{2}-\frac{1}{2} \sum_{j=1}^{r}\left(x_{n-j+1}+x_{j}-2 \bar{x}\right)^{2}\right\\}^{-1} $$ Deduce that \(\operatorname{var}\left(\tilde{\gamma}_{1}\right) \geq \operatorname{var}\left(\widehat{\gamma_{1}}\right)\) with equality if and only if \(x_{n-j+1}+x_{j}=c\) for some \(c\) and all \(j=1 \ldots, r\)
Problem 2
What natural exponential families are generated by (a) \(f_{0}(y)=e^{-y}, y>0\),
and (b) \(f_{0}(y)=\) \(\frac{1}{2} e^{-|y|},-\infty
Problem 2
Suppose \(Y=\tau \varepsilon\), where \(\tau \in \mathbb{R}_{+}\)and \(\varepsilon\) is a random variable with known density \(f\). Show that this scale model is a group transformation model with free action \(g_{\tau}(y)=\tau y\). Show that \(s_{1}(Y)=\bar{Y}\) and \(s_{2}(Y)=\left(\sum Y_{j}^{2}\right)^{1 / 2}\) are equivariant and find the corresponding maximal invariants. Sketch the orbits when \(n=2\).
Problem 3
Show that the geometric density $$ f(y ; \pi)=\pi(1-\pi)^{y}, \quad y=0,1, \ldots, 0<\pi<1 $$ is an exponential family, and give its cumulant-generating function. Show that \(S=Y_{1}+\cdots+Y_{n}\) has negative binomial density $$ \left(\begin{array}{c} n+s-1 \\ n-1 \end{array}\right) \pi^{n}(1-\pi)^{s}, \quad s=0,1, \ldots $$ and that this is also an exponential family.
Problem 3
Suppose that \(\varepsilon\) has known density \(f\) with support on the unit circle in the complex plane, and that \(Y=e^{i \theta} \varepsilon\) for \(\theta \in \mathbb{R}\). Show that this is a group transformation model. Is it transitive? Is the action free?
Problem 3
Consider data from the straight-line regression model with \(n\) observations and $$ x_{j}= \begin{cases}0, & j=1, \ldots, m \\ 1, & \text { otherwise }\end{cases} $$ where \(m \leq n .\) Give a careful interpretation of the parameters \(\beta_{0}\) and \(\beta_{1}\), and find their least squares estimates. For what value(s) of \(m\) is \(\operatorname{var}\left(\widehat{\beta}_{1}\right)\) minimized, and for which maximized? Do your results make qualitative sense?
Problem 3
Use the relation \(\mathcal{F}(y)=\exp \left\\{-\int_{0}^{y} h(u) d u\right\\}\) between the survivor and hazard functions to find the survivor functions corresponding to the following hazards: (a) \(h(y)=\lambda\); (b) \(h(y)=\lambda y^{\alpha}\); (c) \(h(y)=\alpha y^{\kappa-1} /\left(\beta+y^{k}\right) .\) In each case state what the distribution is. Show that \(\mathrm{E}\\{1 / h(Y)\\}=\mathrm{E}(Y)\) and hence find the means in (a), (b), and (c).
Problem 4
(a) Suppose that \(Y_{1}\) and \(Y_{2}\) have gamma densities (2.7) with
parameters \(\lambda, \kappa_{1}\) and \(\lambda, \kappa_{2}\). Show that the
conditional density of \(Y_{1}\) given \(Y_{1}+Y_{2}=s\) is
$$
\frac{\Gamma\left(\kappa_{1}+\kappa_{2}\right)}{s^{\kappa_{1}+\kappa_{2}-1}
\Gamma\left(\kappa_{1}\right) \Gamma\left(\kappa_{2}\right)}
u^{\kappa_{1}-1}(s-u)^{\kappa_{2}-1}, \quad 0
Problem 4
The mean excess life function is defined as \(e(y)=\mathrm{E}(Y-y \mid Y>y)\). Show that $$ e(y)=\mathcal{F}(y)^{-1} \int_{y}^{\infty} \mathcal{F}(u) d u $$ and deduce that \(e(y)\) satisfies the equation \(e(y) Q^{\prime}(y)+Q(y)=0\) for a suitable \(Q(y)\). Hence show that provided the underlying density is continuous, $$ \mathcal{F}(y)=\frac{e(0)}{e(y)} \exp \left\\{-\int_{0}^{y} \frac{1}{e(u)} d u\right\\} $$ As a check on this, find \(e(y)\) and hence \(\mathcal{F}(y)\) for the exponential density. One approach to modelling survival is in terms of \(e(y)\). For human lifetime data, let \(e(y)=\gamma(1-y / \theta)^{\beta}\), where \(\theta\) is an upper endpoint and \(\beta, \gamma>0\). Find the corresponding survivor and hazard functions, and comment.
Problem 5
Show that the inverse Gaussian density $$ f(y ; \mu, \lambda)=\left(\frac{\lambda}{2 \pi y^{3}}\right)^{1 / 2} \exp \left\\{-\lambda(y-\mu)^{2} /\left(2 \mu^{2} y\right)\right\\}, \quad y>0, \lambda, \mu>0 $$ is an exponential family of order \(2 .\) Give a general form for its cumulants.
