Question

\(\mathrm{T}\) is a failure time following a Weibull distribution. Consider \(\mathrm{Y}=\log \mathrm{T}\) where \(\mathrm{Y}\) has an extreme value distribution with survival function

$$ S_{Y}(y)=e^{-\epsilon^{\frac{n \mu}{\sigma}}} $$

where \(-\infty<\mu<\infty\) is the location parameter and \(\sigma>0\) is the scale parameter. Expressing with parameters \(\mu\) and \(\varphi=\log \sigma\). Assume that failure times of subjects under study arise from Weibull distribution. Let \(x_{1}, \ldots, x_{n}\) be the observed failure or right censoring times for n subjects. Each subject i (i = \(1, \ldots, \mathrm{n}\) ) has a censoring indicator \(\delta_{\text {, }}\) taking values 1 if \(\mathrm{x}_{\mathrm{i}}\) is a failure time and 0 if \(\mathrm{x}_{\mathrm{i}}\) is a right censoring time. Consider the log time scale \(\mathrm{y}_{\mathrm{i}}=\log \mathrm{x}_{\mathrm{i}}\). The observed data consist of \(\mathrm{n}\) pairs \(\left(\mathrm{y}_{1}, \delta_{1}\right), \ldots,\left(\mathrm{y}_{\mathrm{n}}, \delta_{\mathrm{n}}\right)\)

(a) With parameters \(\mu\) and \(\varphi\), write out the likelihood function \(\mathrm{L}(\mu, \varphi)\) for the observed data.

b) Derive the score functions for \(\mu\) and \(\varphi\).