The standard cure model with a linear hazard

In this paper we introduce a mixture cure model with a linear hazard rate regression model for the event times. Cure models are statistical models for event times that take into account that a fraction of the population might never experience the event of interest, this fraction is said to be {`}cured{'}. The population survival function in a mixture cure model takes the form $S(t) = 1 - \pi + \pi\exp(-\int_0^t\alpha(s)\,d s)$, where $\pi$ is the probability of being susceptible to the event under study, and $\alpha(s)$ is the hazard rate of the susceptible fraction. We let both $\pi$ and $\alpha(s)$ depend on possibly different covariate vectors $X$ and $Z$. The probability $\pi$ is taken to be the logistic function $\pi(X^{\prime}\gamma) = 1/\{1+\exp(-X^{\prime}\gamma)\}$, while we model $\alpha(s)$ by Aalen's linear hazard rate regression model. This model postulates that a susceptible individual has hazard rate function $\alpha(t;Z) = \beta_0(t) + \beta_1(t)Z_1 + \cdots + Z_{q-1}\beta_{q-1}(t)$ in terms of her covariate values $Z_1,\ldots,Z_{q-1}$. The large-sample properties of our estimators are studied by way of parametric models that tend to a semiparametric model as a parameter $K \to \infty$. For each model in the sequence of parametric models, we assume that the data generating mechanism is parametric, thus simplifying the derivation of the estimators, as well as the proofs of consistency and limiting normality. Finally, we use contiguity techniques to switch back to assuming that the data stem from the semiparametric model. This technique for deriving and studying estimators in non- and semiparametric settings has previously been studied and employed in the high-frequency data literature, but seems to be novel in survival analysis.