Statistical Analysis of Stochastic Processes in Time

Abstract

This book was first published in 2004. Many observed phenomena, from the changing health of a patient to values on the stock market, are characterised by quantities that vary over time: stochastic processes are designed to study them. This book introduces practical methods of applying stochastic processes to an audience knowledgeable only in basic statistics. It covers almost all aspects of the subject and presents the theory in an easily accessible form that is highlighted by application to many examples. These examples arise from dozens of areas, from sociology through medicine to engineering. Complementing these are exercise sets making the book suited for introductory courses in stochastic processes. Software (available from www.cambridge.org) is provided for the freely available R system for the reader to apply to all the models presented.

Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint

“…However, in posterior sampling of η, an M-H sampling algorithm may be used when the conditional posterior density Equation (18) (µ, µ 0 , τ, η), implementation of the Gibbs (or Metropolis-within-Gibbs) sampling algorithm consists of drawing repeatedly from distributions Equation (15) through Equation (18). The R package tmvtnorm and the R package mvtnorm can be used to sample from the conditionals and to calculate δ for a given α from Equation (11).…”

“…When the LNFT model is assumed, Bayesian estimates of reliability measures can be simply obtained by setting κ(η (k) ) = 1 to Equation (18). That is, the reliability at time y iŝ…”

“…In cases where η ∼ Gamma(ν/2, ν/2) with E[η] = 1 and κ(η) = 1/η are chosen, the SMLNFT model changes to log-t ν failure time (Lt ν FT) or log-Cauchy failure time (LCFT ≡ Lt 1 FT) models, allowing for the regulation of model tail distribution by means of the degrees of freedom. The LCFT model has been particularly used for certain survival processes where significant outliers or extreme results may occur (see, e.g., [18]). We also see that the SMLNFT model approximately reduces to log-logistic failure time (LLFT) model (see [19]), provided that the choices are κ(η) = 4η 2 and η 2 ∼ IG(2.5, 1.233), where IG(α, β) is an inverse gamma distribution with a probability function f (x) = β α x −(α+1) e β/x /Γ(α).…”

“…Otherwise, we can define a Metropolis-within-Gibbs sampler that uses an M-H (Metropolis-Hastings) step to sample from the conditional in Equation (18). See [26], for the Metropolis-within-Gibbs sampler.…”

Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint

“…In cases where η ∼ Gamma(ν/2, ν/2) with E[η] = 1 and κ(η) = 1/η are chosen, the SMLNFT model changes to log-t ν failure time (Lt ν FT) or log-Cauchy failure time (LCFT≡ Lt 1 FT) models, allowing for the regulation of model tail distribution by means of the degrees of freedom. The LCFT model has been particulary used for certain survival processes where significant outliers or extreme results may occur (see, e.g., [17]). We also see that the SMLNFT model approximately reduces to log-logistic failure time (LLFT) model (see, [18]), provided that the choices are κ(η) = 4η 2 and η 2 ∼ IG(2.5, 1.233), where IG(α, β) is an inverse gamma distribution with a probability function f (x) = β α x −(α+1) e β/x /Γ(α).…”