M2(x) = a2(x) = E0(tx - A1(x))2 = u2n(du x [0,x)), (2)
M4(x) = E0(tx - A1(x))4 = u4n(du x [0,x)) + 3M%(x). (4)
It is not difficult to evaluate other moments. Inverse gamma process
In the works [har04a, har04d] and others we gave arguments justifying I-process to be used in reliability problems. We showed examples, where I-process analytical properties are useful in optimization problems of prophylaxis and reservation. For practical aim one should consider more narrow class of the processes with a finite number of parameters. In [har04b, har04c] such processes were shown. Now we consider such a process, namely gamma process, in more details.
Inverse (homogeneous) gamma process is an I-process where the function tx is distributed according to gamma distribution: Px(Ty G dt) = fy(t| x) dt (x < y), where fy(t| x) = fy-x(tl 0) = fy-x(t) and fx(t) = Tk)(Yt)xS-1e-Yt (x> 0), r(■) is the gamma function (r(x) = tx-1e-t dt) , S > 0 is a form parameter, and y > 0 is a scale parameter. Evidently, fx(t) = fx(t; Y,S) = Yfsx(lt; 1,1) = YfL^t), where f0(t) = r(x)tx-1e-t.
The Levy exponent of the gamma process is of the form xS ln(Y + ty/j. The view of its Levy measure follows from the formula
(see [har01]). Thus the density of Levy measure has representation:
In applications they consider more general class of inverse gamma processes when parameters y and S depend on position x. This class of I-processes seems to be naturally called as class of generalized inverse gamma processes in spite of the distribution of the value tx in this case is not gamma distribution. The generating function of the first exit time of such a more general process has the form
E0(e-XTx) = exp (-JX 5(y) ln X d^j , which implies formulae for moments:
The main advantage of the inverse gamma process comparatively with other I-processes are both its simplicity, and flexibility due to its two parameters. For a fixed wear level the gamma distribution is used for description of a random failure time in work [gk66]. In work [bn01] the direct (proper) gamma process had been considered as a model of wear.
In work [har04b] we show how a continuous strictly increasing function can be approximated by I-processes (partially, gamma processes). This property of I-processes permits to deny the determinate component in Levy representation in any case when such a component does not have explicit interpretation. In turn the denial of the determinate component makes more simple using of absolute continuous property for I-processes (see [sko64]) and analysis of its finite-dimensional densities.
Let us consider I-process without determinate component. For such a process r y r m b(\,x,y)= / (1 - e-Xu) v(u,s) duds. (6)
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