Consider the semimartingale regression model $X(t)$ = $X(0)$ + $\int\sbsp{0}{t}$ $Y(s)\alpha(s,Z(s))$ $ds + M(t)$, where $Y, Z$ are observable covariate processes, $\alpha$ is a (deterministic) function of both time and the covariate process $Z$, and $M$ is a square integrable martingale. Under the assumption that i.i.d. copies of $X, Y, Z$ are observed continuously over a finite time interval, inference for the function $\alpha(t,z)$ is investigated. Applications of this model include hazard function estimation for survival analysis and inference for the drift function of a diffusion process. / An estimator $\ A$ for the time integrated $\alpha(t,z)$ and a kernel estimator of $\alpha(t,z)$ itself are introduced. For $X$ a counting process, $\ A$ reduces to the Nelson-Aalen estimator when $Z$ is not present in the model. Various forms of consistency are proved, rates of convergence and asymptotic distributions of the estimators are derived. Asymptotic confidence bands for the time integrated $\alpha(t,z)$ and a Kolmogorov-Smirnov-type test of equality of $\alpha$ at different levels of the covariate are given. / For the case $Y$ $\equiv$ 1 we introduce an estimator $\{\cal A}$ of the time and space integrated $\alpha(t,z)$. The asymptotic distribution of the estimator $\{\cal A}$ is derived under the assumption that the covariate process $Z$ is $\cal F\sb0$-adapted, where ($\cal F\sb{t}$) is the filtration with respect to which $M$ is a martingale. In the counting process case this amounts to assuming that $X$ is a doubly stochastic Poisson process. Weak convergence of the appropriately normalized time and state indexed process $\{\cal A}$ to a Gaussian random field is shown. As an application of this result, confidence bands for the covariate state integrated hazard function of a doubly stochastic Poisson process whose intensity does not explicitly depend on time are derived. / Source: Dissertation Abstracts International, Volume: 49-03, Section: B, page: 0816. / Major Professor: Ian W. McKeague. / Thesis (Ph.D.)--The Florida State University, 1987.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_76268 |
| Contributors | Utikal, Klaus Johannes., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 58 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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