Some Nonparametric Techniques for Estimating the Intensity Function of a Cancer Related Nonstationary Poisson Process

A penalty method for nonparametric estimation of the intensity function of a counting process

“…A discussion of the existence of the MPL estimator of the intensity function of a nonhomogeneous Poisson process is given in Section 4 of Bartoszynski et al (1981), drawing on material from Tapia and Thompson (1978). Since, in that work, the penalty functional is a norm (and not a seminorm), the question of existence of our estimator is a little more delicate.…”

“…Estimation procedures for the intensity function of point processes have been analyzed in a number of papers, for instance, those written by Ogata (1978), Kutoyants (1979), Lin'kov (1981), Sagalovsky (1983) and Konecny (1984) in the parametric case, Aalen (1975Aalen ( , 1978, Bartoszynski et al (1981), Leadbetter and Wold (1983), Ramlau-Hansen (1983) and Karr (1987) in the nonparametric case, to cite only few. For the multiplicative intensity model, Aalen (1978) provided an estimator for the cumulative hazard function *This work was conducted while the author was visiting the Department of Mathematics, University of California at lrvine.…”

“…Our interest in the multiplicative intensity counting process model and the questions addressed here was motivated by the work of Bartoszynski et al (1981). The basic problem they treat is estimation (from i.i.d, realizations) of the intensity function of a nonhomogeneous Poisson process on ~+.…”

Another Look at Nonparametric Estimation for Trend Renewal Processes

“…Boswell [10] proposes a nonparametric maximum likelihood estimator for NHPPs whose mean value functions are unknown. Latter, Bartozynski et al [7] derive three nonparametric maximum likelihood estimation algorithms (constrained maximum likelihood estimation, penalized maximum likelihood estimation, and an application to a Cox process [16]) and apply them to the melanoma metastatic data. Barlow and Davis [6] and Kvaloy and Lindqvist [26] consider another nonparametric estimator based on the total time on test concept with multiple lifetime data sets of a repairable system.…”

“…Heggland and Lindqvist [21] assume a Weibull failure rate function and propose a constrained nonparametric maximum likelihood estimator (CNPMLE) when the time transformation function called the trend function, which is similar to the intensity function of an NHPP, is monotone but unknown. The basic idea is due to the CNPMLE for an NHPP by Boswell [10] and Bartozynski et al [7]. In this paper we develop another CNPMLE for TRPs, under the assumption that the failure rate function of the underlying RP is unknown but the form of trend function is known.…”