Random Point Processes and Models of Cumulated Damage

The contribution deals with a stochastic process which cumulates random increments at random moments, the cumulative process. A multiplicative model of rate of cumulation, with regression on covariate processes, is studied. We show the consistency of estimators and then the asymptotic normality of the process of residuals. An application dealing with the process of growing damage of a technical device is considered. We also discuss the problem of prediction of the cumulative process behaviour. 1 The model of cumulative process In the present paper, we consider random process called the compound or also cumulative process: C(t) = Z t 0 Y (s) dN(s); (C(0) = 0); (1) where N(t) is a counting process and Y (t) is a set of random variables-increments. Its model is suitable for the description of many real-world technological, environmental, biological and also nancial processes (Rolski et al, 1999, Volf, 2000). In all elds, one of main problems is the problem of process prediction, which is equivalent to the problem of assessing the probability of crossing a certain level, the failure or ruin probability problem. Actually, satisfactory solutions are (up to now) derived just for the simplest forms of process considering Poisson process of times and i.i.d. increments. We propose a model allowing for the dependence of distribution of Y (t) on S(t ?), where S(t) is a corresponding ltration, so that its left-continuous version S(t ?) is a 'history'. Intensity of N(t) is (t) = h(t; Z(t))J(t), cumulative intensity L(t) = R t 0 (s)ds, h(t; z) is a hazard function, J(t) and Z(t) are S(t ?) measurable predictable processes, e.g. an indicator of observability of C(t) and a covariate process. As regards the distribution of random variables Y (t), we assume that the conditional distribution of Y (t), given S(t ?), can be described via a density function f(y; t; Z(t)), and it possesses the rst and second conditional moments E(Y (t)jS(t ?)) = (t; Z(t)), var(Y (t)jS(t ?)) = 2 (t; Z(t)). These deenitions imply that the processes N(t); C(t) depend on S(t ?) through Z(t) and J(t). The description given above is rather general, that is why we shall consider a more speciied multi-plicative form of the model (actually the model of Cox's type), even for the rate of growth of C(t). We shall deal with the following problems: 1. Compensator { martingale decomposition of C(t). 2. Estimation of the rate of …