Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 46 / 3 / b / i / Solution

Put $d={\sum }_i{v}_i$. With common Weibull distribution parameters and the hazard from part (a), substitution into the general log-likelihood givesThis is up to censoring-mechanism terms independent of $\lambda ,p$. For $d>0$ and fixed $p$, its maximizing rate is $(d/{\sum }_i{x}_i^p{)}^{1/p}$; the shape still requires separate estimation if it is unknown.