For to obtain maximum likelihood estimates one can search the maximum of this function by something suitable evaluating method. The analytical search of maximum is being reduced to search of roots of a system of two equations arising as a result of partial differentiating of the function with respect to its parameters. For evaluating the partial derivative of gYt(Sx) with respect to parameter S one can use formula (7), which can be rewritten as d dyfy(t\x) = J (fy(t — s\x) — fy(t\x)) v(s,V) ds, taking into account that fy(t\x) =0 as t < 0. For evaluating the partial derivative of gYt (Sx) with respect to parameter y one can use theorem 1, because analytical representation of fx (t) is considerably simpler than that of gt(x).
Essential difficulties arise under dynamical method of registration of wear data. In this case increments of wear values are not independent. Hence the likelihood function represents the whole multi-dimensional joint density (12) for values of the process at the fixed time epochs. Operation time for evaluating this 2n-dimension integral increases exponentially as n increases. Apparently it is impossible to use this formula for obtaining maximum likelihood estimates with reasonable precise. That is why some approximate methods for parameters estimation deserves attention, in partial, constructing an approximate likelihood function. For this aim one can use the property of decreasing of dependency for increments of the process on time intervals separated by sufficiently long time gaps.
It is well-known (see for example [har01]) that trajectories of inverse gamma process consists of intervals of constancy almost wholly. Therefore for any t > 0 (non-random) the trajectory £ is constant on some interval containing t. It implies dependence of increments of the process £. The nearest regenerative point of the process (the process has Markov property with respect to the point) is the right edge of this interval. From ergodic theory it follows there exists the limit distribution of right parts of such intervals. Let Pst be the stationary distribution of an embedded regenerative process, and R+ be length of the right part of the interval covering the point t. Then r e—yu
(see [har01, c.368]). One have to take into account this interval when evaluating stationary distribution of the increment of the process on given time interval. Hence Pst(£(t) — £(0) G dx) = gt(x) dx, where gt(x) = Pst(r)gt—r(x) dr, ■Jo and
If a statistician has observations on N "small" intervals with lengths ti separated by "large" gaps with lengths Ti he can search approximate maximum likelihood estimates as a point of maximum of the product of stationary densities
The obtained estimate the more precise, the more values Ti.
For /-processes the rate of convergence to the stationary distribution can be derived from general ergodic theorems. In our case this rate can be estimated more precisely using special properties of gamma-process. Let
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