# Mohammad, Khandoker Akib

## Efficient Estimation For The Cox Proportional Hazard (PH) Cure Model

#### School of Mathematics and Statistics, Victoria University of Wellington

While analysing time-to-event data, it is usually assumed that a certain fraction of subjects will never experience the event of interest and they are said to be cured. When this important feature of survival models is taken into account then the models are commonly referred to as cure models. In the presence of covariates, the conditional survival function of the population can be modelled by using cure model which depends on the probability of being uncured (incidence) and the conditional survival function of the uncured subjects (latency). Usually in the Cox PH cure model, a combination of logistic regression and Cox PH regression is used to model the incidence probability and latency respectively. Here profile likelihood approach has been used to estimate the cumulative hazard and the regression parameters. However, the estimator of the baseline hazard is an implicit function that is why it is very difficult to find an estimate of variance of profile likelihood estimator in the Cox PH cure model. We can solve of the problem of implicit function by considering the ‘statistical generalized derivative’ which is used to calculate the score function and do not require differentiability of the score function to show the asymptotic normality of the profile likelihood estimator. We applied the theory of ‘statistical generalized derivative’ in the Cox PH cure model to calculate the efficient score function and establish the asymptotic normality of the estimator without assuming the derivative of the score function in the model.

This presentation is eligible for the NZSA Student Prize.