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# Parametric and non-parametric inference for Geometric Process

Abstract (Summary)
(Uncorrected OCR) Abstract of the thesis entitled PARAMETRIC AND NON-PARAMETRIC INFERENCE FOR GEOMETRIC PROCESS submitted by Ho Pak Kei for the degree of Master of Philosophy at The University of Hong Kong in July 2005 Because of accumulated wears, the successive operating times of a repairable system after repairs decrease stochastically. One approach of modeling trend data is to use the non-homogeneous Poisson process (NHPP) with monotone hazard rates. Lam (1988) first proposed modeling directly the monotone trend by a monotone process called Geometric Process (GP). By definition, a stochastic process {Xi,i = 1,2,...} is a GP if there exists a positive real number a = 0 such that {Yi = ai-lXi} generates a renewal process (RP) with a mean ? The real number a is called the ratio of the GP and it measures the direction and strength of the trend. Traditionally, GP models have been applied extensively to model the inter-arrival times in the reliability and maintenance problems. In this thesis, we aim to extend the GP models to the generalized GP (GGP) models by incorporating covariates in the mean function of ?and to the multiple GP (MGP) models by adopting multiple GPs for multiple trends data when data are measured on a continuous scale. The resulting process {Yi = ai-1Xi} in the GGP models is no longer identically distributed but it is a stochastic process (SP) in general. Moreover we also generalized the GP models to binary GP (BGP) models for binary data by assuming an underlying GP defined in terms of the binary observations Wi = I(Xi > 1) = I(Yi > ai-1) where I is a indicator function. The statistical inference for these extended models was studied by using non-parametric, maximum likelihood and Bayesian methods and the modeling was demonstrated through simulations and real data analyses. In non-parametric inference, parameters were estimated by minimizing the least-squared error on Xi or ln Xi. However, in maximum likelihood and Bayesian inferences, the corresponding SP {Yi} in the GP models were modeled within the exponential family of sampling distributions in the Generalized Linear Mixed Models (GLMM) framework by a lifetime distribution, say exponential, gamma, Weibull and lognormal distributions. In particular, the means of these distribution was linked to some linear function of covariates by a log function for the GGP models. These models extensions will definitely make significant and pioneering contribution to modeling methodologies for longitudinal data with monotone trends. The performances of the GGP and BGP models using different methodologies were evaluated through simulation studies. Real life applications in epidemiology, business and system maintenance were studied thoroughly. Some discussions and suggestions were made on each type of the extended models.
Bibliographical Information:

School:The University of Hong Kong

School Location:China - Hong Kong SAR

Source Type:Master's Thesis

Keywords:nonparametric statistics poisson processes distribution probability theory

ISBN:

Date of Publication:01/01/2005