= Solution
Fit a <Poisson regression> to artificial responses $Z_i=1$ with means
$$
\mu_i=\Lambda(Y_i)e^{\beta^Tx_i}.
$$
With the logarithmic link, this is the <generalized linear model>
$$
\log\mu_i=\beta^Tx_i+\log\Lambda(Y_i),
$$
where $\log\Lambda(Y_i)$ is a known offset. Its log likelihood, apart from constants, is
$$
\ell_P(\beta)
=\sum_i[\log\Lambda(Y_i)+\beta^Tx_i-\Lambda(Y_i)e^{\beta^Tx_i}].
$$
The difference
$$
\ell(\beta)-\ell_P(\beta)
=\sum_i[\log\lambda(Y_i)-\log\Lambda(Y_i)]
$$
does not depend on $\beta$. \b[The Poisson surrogate therefore has exactly the same maximum-likelihood estimate of $\beta$.] This is the <Poisson surrogate for an uncensored proportional-hazards likelihood>.
Back to article page
