The paper considers linear degradation and failure time models with multiple failure modes. Dependence of traumatic failure intensities on the degradation level are included into the models. Estimators of traumatic event cumulative intensities, and of various reliability characteristics are proposed. Prediction of residual reliability characteristics given a degradation value at a given moment is discussed. Non-parametric, semiparametric and parametric estimation methods are given. Theorems on simultaneous asymptotic distribution of random functions characterising degradation and intensities of traumatic events are proposed. Asymptotic properties of unconditional and residual reliability characteristics estimators are given. Real tire wear and failure time data are analysed.
We continue investigation of the ARCH(∞) model begun in Giraitis, Kokoszka, and Leipus
(2000, Econometric Theory 16, 3–22). Nonrestrictive
conditions for the existence of a strictly stationary solution
are established. The paper generalizes the results of Nelson
(1990, Econometric Theory 6, 318–334) and Bougerol
and Picard (1992, Journal of Econometrics 52,
115–127) to the ARCH(∞) model.
A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.
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