Nonparametric Estimation of Time Trend for Repairable Systems Data

Abstract

The trend-renewal-process (TRP) is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a nonhomogeneous Poisson process (NHPP). A nonparametric maximum likelihood estimator of the trend function of a TRP can be obtained much in the same manner as for the NHPP using kernel smoothing. But for a TRP one must consider the simultaneous estimation of the renewal distribution, which is here assumed to belong to a parametric class such as the Weibull-distribution. A weighted kernel estimator for λ(·) is derived using a general approach for kernel smoothing in counting processes.

Testing for Monotone Trend in Recurrent Event Processes

“…A limitation of these tests is the assumption that the data arise from HPPs in the absence of trend. They are sensitive to departures from this assumption, and one can falsely conclude there is a trend when the processes are RPs but not HPPs (e.g., Lindqvist, Kjønstad, and Meland 1994;Lawless and Thiagarajah 1996). The same criticism applies to tests based on total time on test (TTT) statistics (e.g., Kvaløy and Lindqvist 1998;Kvist et al 2008).…”

Nonparametric estimation in trend-renewal processes

“…Later, Lindqvist [23] considered the case when λ(·) is a general nonnegative function, thus extending earlier work on nonparametric estimation in NHPPs, e.g., Bartozinski et al [4]. A review of these studies can be found in the monograph by Gamiz, Kulasekera, Limnios and Lindqvist [13].…”