91Ó°ÊÓ

Data is generated from the exponential distribution with density \(f(y)=\) \(\lambda \exp (-\lambda y)\) where \(\lambda, y>0\) (a) Identify the specific form of \(\theta, \phi, a(), b()\) and \(c()\) for the exponential distribution. (b) What is the canonical link and variance function for a GLM with a response following the exponential distribution? (c) Identify a practical difficulty that may arise when using the canonical link in this instance. (d) When comparing nested models in this case, should an \(F\) or \(\chi^{2}\) test be used? Explain. (e) Express the deviance in this case in terms of the responses \(y_{i}\) and the fitted values \(\hat{\mu}_{i}\).

The Conway-Maxwell-Poisson distribution has probability function: \\[ P(Y=y)=\frac{\lambda^{y}}{(y !)^{\mathrm{v}}} \frac{1}{Z(\lambda, v)} \quad y=0,1,2, \dots \\] where \\[ Z(\lambda, v)=\sum_{i=0}^{\infty} \frac{\lambda^{i}}{(i !)^{v}} \\] Place this in exponential family form, identifying all the relevant components necessary for use in a GLM.

Consider the following distributions and determine whether they can be put in exponential family form. If it is not possible, consider whether there is a special case which follows the form. \(\bullet\) The Uniform distribution: \\[ f(y | \alpha, \beta)=\frac{1}{\beta-\alpha} I(y \in[\alpha, \beta]) \\] \(\bullet\) The Weibull distribution: \((y>0)\) \\[ f(y | \alpha, \beta)=\alpha \beta^{-\alpha} y^{\alpha-1} e^{(y / \beta)^{\alpha}} \\] \(\bullet\) The Gumbel distribution: \((y>0)\) \\[ f(y | \alpha, \beta)=\frac{1}{\beta} \exp ((y-\alpha) / \beta-\exp ((y-\alpha) / \beta)) \\]