By G. George Yin, Qing Zhang
This ebook specializes in the speculation and purposes of discrete-time two-time-scale Markov chains. a lot attempt during this booklet is dedicated to designing procedure types coming up from those purposes, reading them through analytic and probabilistic concepts, and constructing possible computational algorithms with a purpose to decrease the inherent complexity. This ebook offers effects together with asymptotic expansions of chance vectors, structural houses of profession measures, exponential bounds, aggregation and decomposition and linked restrict tactics, and interface of discrete-time and continuous-time systems. one of many salient good points is that it encompasses a diversified diversity of functions on filtering, estimation, keep watch over, optimization, and Markov determination procedures, and monetary engineering. This e-book may be a tremendous reference for researchers within the parts of utilized chance, keep watch over conception, operations study, in addition to for practitioners who use optimization techniques. a part of the publication can be utilized in a graduate process utilized likelihood, stochastic techniques, and applications.
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Extra resources for Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications (Stochastic Modelling and Applied Probability)
Sample text
The model given here is close to the Cox model when t is small. It could be useful in practice when the constructed Cox model is not in accordance with the data for small t. We consider the case when the resource is lognormal, G(t) = Φ(ln t), t ≥ 0, where Φ is the distribution function of the standard normal law. If covariates are constant in time then in terms of survival functions the model GM can be written as Φ −1 (Sx (t)) = ln r (x) + Φ −1 (S0 (t)), x ∈ E 1 . It is the famous generalized probit model, see Dabrowska and Doksum (1988).
Cn . The unique compensator of the counting process N(t) with respect to the filtration F is t A(t) = Y (u) dΛ(u), 0 where Λ(t) = process t 0 λ(u) du is the cumulative hazard function of T . , E{M(t)|Fs } = M(s), for s < t. This property of martingale M(t) means that the expected value of M(t), given its history at time s < t, is equal to its value at time s. 12) 0 where Λ(u) is the cumulative hazard function of T . 11) as observation = model + error. 11) holds even when the function Λ(·) is not continuous.
The at-risk process of the ith person is defined as Yi (t) = 1{Xi ≥t} , t ≥ 0. 9) It equals to 1 when the ith person is still under observation at time t−. 3 Censored Data 21 n N(t) = Ni (t), t ≥ 0, i=1 be the total number of failures observed in the interval [0, t] and n Yi (t), t ≥ 0, Y (t) = i=1 be the number of subjects at risk for failure just prior to the moment t. More precisely, for any t the value Y (t) gives the number of patients who are at risk for failure during a small time interval (t − ε, t] for an arbitrarily small positive ε, since any unit that fails exactly at time t must be both in the risk set at the failure time and known to be at risk before the failure occurred.
