Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 2, 174-188

PARAMETRIC ESTIMATION OF HAZARD FUNCTIONS WITH STOCHASTIC COVARIATE PROCESSES

By

SIMEON M. BERMAN, * Courant Institute of Mathematical Sciences, New York*

and

HALINA FRYDMAN,
* Stern School of Business, New York*

SUMMARY. Let $X(t),\; t\ge 0$, be a real or vector valued stochastic process and $T$ a random killing-time of the process which generally depends on the sample function. In the context of survival analysis, $T$ represents the time to a prescribed event (e.g.~system failure, time of disease symptom, etc.) and $X(t)$ is a stochastic covariate process, observed up to time $T$. The conditional distribution of $T$, given $X(t),\;t\ge 0$, is assumed to be of a known functional form with an unknown vector parameter $\theta$; however, the distributions of $X(\cdot)$ are not specified. For an arbitrary fixed $\alpha >0$ the observable data from a single realization of $T$ and $X(\cdot)$ is $\min (T,\,\alpha)$, $X(t),\;0\le t\le \min(T,\,\alpha)$. For $n\ge 1$ the maximum likelihood estimator of $\theta$ is based on $n$ independent copies of the observable data. It is shown that solutions of the likelihood equation are consistent and asymptotically normal and efficient under specified regularity conditions on the hazard function associated with the conditional distribution of $T$. The Fisher information matrix is represented in terms of the hazard function. The form of the hazard function is very general, and is not restricted to the commonly considered cases where it depends on $X(\cdot)$ only through the present point $X(t)$. Furthermore, the process $X(\cdot)$ is a general, not necessarily Markovian process.

*AMS (1991) subject classification.* 62F12, 62M09, 62P10.

*Key words and phrases. *Asymptotic distribution of estimators,
hazard function, killing-time, likelihood function, stochastic marker
process.