Non-standard asymptotics in an inhomogeneous gamma process |
| |
Authors: | Nibedita Bandyopadhyay Ananda Sen |
| |
Institution: | (1) Department of Mathematics and Statistics, Oakland University, 48309 Rochester, MI, USA;(2) Glaxo Smith Kline, One Franklin Plaza, FP 1375, 19101 Philadelphia, PA, USA;(3) Center for Statistical Consultation and Research, University of Michigan, 48109 Ann Arbor, MI, USA |
| |
Abstract: | Nonhomogeneous Poisson process (NHPP) is a commonly used stochastic model that is utilized to describe the pattern of repeated
occurrence of certain events or conditions. Aninhomogeneous gamma process evolves as a generalization to NHPP, where the observed failure epochs correspond to every successive κ-th event of the underlying
Poisson process, κ being an unknown parameter to be estimated from the data. This article focuses on a special class of inhomogeneous
gamma process, calledmodulated power law process (MPLP) that assumes the Weibull form of the intensity function. The traditional power law process is a popular stochastic
formulation of certain empirical relationships between the time to failure and the cumulative number of failures, often observed
in industrial experiments. The MPLP retains this underlying physical basis and provides a more flexible modeling environment
potentially leading to a better fit to the failure data at hand. In this paper, we investigate inference issues related to
MPLP. The maximum likelihood estimators (MLE’s) of the model parameters are not in closed form and enjoy the curious property
that they are asymptotically normal with a singular variance-covariance matrix. Consequently, the derivation of the large-sample
results requires non-standard modifications of the usual arguments. We also propose a set of simple closed-form estimators
that are asymptotically equivalent to the MLE’s. Extensive simulation results are carried out to supplement the theoretical
findings. Finally, we implement our inference results to a failure dataset arising from a repairable system. |
| |
Keywords: | and phrases" target="_blank"> and phrases Asymptotics maximum likelihood estimation modulated power law process power law process recurrent event |
本文献已被 SpringerLink 等数据库收录! |
|