Vilijandas Bagdonavicius, Mikhail Nikulin's Accelerated Life Models: Modeling and Statistical Analysis PDF

By Vilijandas Bagdonavicius, Mikhail Nikulin

ISBN-10: 1420035878

ISBN-13: 9781420035872

ISBN-10: 1584881860

ISBN-13: 9781584881865

The authors of this monograph have constructed a wide and demanding classification of survival research versions that generalize lots of the present types. In a unified, systematic presentation, this monograph totally info these types and explores components of speeded up existence checking out frequently simply touched upon within the literature.Accelerated existence types: Modeling and Statistical research offers types, equipment of information assortment, and statistical research for failure-time regression facts in sped up lifestyles checking out and for degradation info with explanatory variables. as well as the classical effects, the authors dedicate massive cognizance to types with time-varying explanatory variables and to tools of semiparametric estimation. in addition they learn the simultaneous research of decay and failure-time information while the intensities of failure in several modes depend upon the extent of deterioration and the values of explanatory variables.The authors stay away from technical info by way of explaining the information and relating assets the place thorough research are available. even if used for educating, learn or basic reference, sped up existence versions: Modeling and Statistical research offers new and identified versions and glossy tools of sped up existence facts research.

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The PH and AFT models are equivalent on E1 if and only if the failure-time distribution is Weibull for all x ∈ E1 Proof. Sufficiency. If the failure-time distribution is Weibull and the PH model holds on E1 then for all x ∈ E1 t Sx (t) = e−( θ(x) ) α(x) = S0 (t)r(x) , Taking twice the logarithm of both sides, we obtain that for all t > 0 α(x)(ln t − ln θ(x)) = ln r(x) + ln(− ln S0 (t)). e. the AFT model holds on E1 . If the failure-time distribution is Weibull and the AFT model holds then t α Sx (t) = e−( θ(x) ) , Ax (t) = ( t α ) , θ(x) αx (t) = r(x) α0 (t), −α and α0 (t) = tα .

37) where S0 (t) = exp − t 0 t α0 (u)du , A0 (t) = 0 α0 (u)du = − ln S0 (t). 38) where ρ(x0 , x) = r(x)/r(x0 ). 2 Definition of the model for time-varying stresses In the statistical literature the following formal generalization of the PH model to the case of time-varying stresses is used. 3. 40) This definition implies that t Ax(·) (t) = 0 r{x(u)}dA0 (u). 41) In terms of survival functions the PH model is written : Sx(·) (t) = exp − t 0 r{x(u)}dA0 (u) . 40) is not natural when units are aging under constant stresses.

40) This definition implies that t Ax(·) (t) = 0 r{x(u)}dA0 (u). 41) In terms of survival functions the PH model is written : Sx(·) (t) = exp − t 0 r{x(u)}dA0 (u) . 40) is not natural when units are aging under constant stresses. Indeed, denote by xt constant in time stress equal to the value of time-varying stress x(·) at the moment t. Then αxt (t) = r{x(t)} α0 (t), which implies αx(·) (t) = αxt (t). 43) For any t the hazard rate under the time-varying stress x(·) at the moment t does not depend on the values of the stress x(·) before the moment t but only on the value of it at this moment.

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Accelerated Life Models: Modeling and Statistical Analysis by Vilijandas Bagdonavicius, Mikhail Nikulin

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