# How to estimate full distributions in Cox Proportional Hazard model?

A Cox Proportional Hazard model outputs both the coefficients $$\beta$$ and the baseline hazard $$\lambda_0$$ (eg, Breslow's method). From these two quantities one can estimate the distribution of time to event ($$T$$) for a specific subject with covariate vector $$x_i$$, ie, $$\hat{S}(t|x_i)$$ (or $$\hat{S}_i(t)$$).

However, this estimated distribution is truncated: it is not defined beyond the study period. That is, if a study lasts 10 years, then $$\hat{S}_i(t)$$ is not defined beyond 10 years, even though $$T_i$$ can have nonzero probability mass beyond 10 years (ie, $$\hat{S}_i(10) \ne 0$$).

This is fundamentally due to the nonparametric nature regarding the $$\lambda_0$$ part of the model. In order to estimate the full distribution of $$T_i$$, defined on $$[0, \infty)$$, one has assume a parametric form for $$\lambda_0$$ (amounting to assuming a specific distribution for $$T_i$$).

My questions are: What are the choices of the parametric form for $$\lambda_0$$, and what are their pros and cons? What are the most expressive parametric models? (My dataset is large so can cope with a model with many parameters.)

• If you fixed a model based on the data from 0-3 years old children, my guess that probability of 60 year old man living for other 100 year > 90% if exploration based on fitted model. – user158565 Oct 6 '18 at 5:11