Existence of the limit lim E(t) = lim X/ ln(1 + X) = 1.
follows from Tauberian theorem (see [fel67, c.513]). In work [har04b] it has been shown that convergency rate of E(t) to its limit determines the value of mistake when we substitute the product of one-dimensional stationary densities instead of the proper likelihood function.
In this case we propose there exists a table of fixed wear levels and corresponding increments of hitting times for these levels. In frames of the theory of inverse gamma processes for to find reasonable estimates of two its parameters it is sufficient to know two first distribution moments. Consistent estimates can be obtain by the method of moments. More precise estimates can be obtain with the help of maximum likelihood method. Let, for example, levels of wear be (xi,x2,...,xn) fixed and their corresponding hitting times be measured. Taking into account independence along the time axis we construct likelihood function, i.e. the joint density with unknown parameters:
L(ti, ...,tn; y,S) = J] fyi (s^, Y, S) = 7 n JJ-wf1)-'
where Sk = tk — tk-i, yk = xk — xk-i (to = 0, xo = 0). The maximum likelihood estimates one can obtain either by standard analytical method, or with the help of computer.
Inverse way of data gathering when dealing with a continuous wear curve
In accordance with non-contact (non-stop) methods of wear registration, which recently have increasing expansion (see [fad04]), the problem arises how to estimate parameters when dealing with continuous trajectory of the process. Reducing the continuous record £(t) (0 < t < T) to a finite sample of n independent and identically distributed random values can be obtain by splitting the realized wear interval (0, x) on n equal parts by points (x/n, 2x/n, .. ., x(n — 1)/n) and determining n the first hitting times (ti,t2,... ,tn-i,tn) of the boundaries of these intervals, where tk = TXk/n(£). The first problem is to choose the number n in a reasonable way. In the case of inverse gamma process the random value tk — tk-i has distribution with the density fx(t; y,S). Thus EoTx = xS/y, Do(tx) = Eo(tx — EoTx)2 = xS/y2. It serves base for application of the method of moments for estimating both the ratio S/y, and S/y2, and corresponding parameters of the process with accordance to formulae
As follows from these formulae, the estimate of the ratio S/S does not depend on n. For reasonable choice of n we have to find variance of the estimate s/y2:
By the way of not-difficult but awkward transformations we obtain the formula
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