Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 13J / c / Solution

Fit a Poisson regression to artificial responses $Z_i=1$ with meansWith the logarithmic link, this is the generalized linear modelwhere $\log \Lambda (Y_i)$ is a known offset. Its log likelihood, apart from constants, isThe differencedoes not depend on $\beta$. The Poisson surrogate therefore has exactly the same maximum-likelihood estimate of $\beta$. This is the Poisson surrogate for an uncensored proportional-hazards likelihood.