  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.
