Introduction

In the present work we continue to investigate the wear process which has been considered in papers [har02, har04a, har04b, har04d]. In these works there were proposed inverse processes with independent positive increments as models of wear processes. For brevity we will call such a process as I-process. A partial case of such a process is gamma process [har04b], which has some advantages comparatively with other I-processes. Varying a wear diagnostic parameter under law of I-process seams to be completely justified, because the sense of this assumption reduces to the condition for times of wearing of non-overlapping portions of material to be independent random values. Such a model combines two necessary properties of a practical model: "good" features of the process realizations (continuity, monotonicity), and sufficiently simple mathematical apparat. Advantages of the model explicitly appears for non-standard registration of wear data.

In this work two ways for gathering information about I-processes are investigated. The first one is the classical (direct) way, when an observer can measure (random) portions of material which had been worn during determinate time intervals. And the second one is the inverse way, when he finds (random) time intervals having been spent by determinate portions of worn material. Both ways have technical and organizational base and their sphere of application.

In the recent time there arises possibility to record practically the whole continuous wear curve, for example, the acoustical method (see [fad04]). In this case the statistical analysis option depends only on computation possibility of the observer.

For the first variant of data gathering in works [har04b, har04c] there were derived formulae for both one-dimensional, and multi-dimensional distribution densities, which can be used for determination of maximum likelihood estimates and criterions. In the present work we revise and supplement results of these works.

In the second variant we propose for the time increment distributions to be known to within finite number of parameters, and for the increments themselves to be mutually independent. Thus one can use methods of classical mathematical statistics.

Under recording a continuous trajectory of wear it seems to be reasonable to quantize date in the second manner. Under this choice the problem arises how one should split all the interval of wear for obtaining corresponding family of wearing time increments which possesses optimal statistical properties. In the work we discuss the optimal choice of partition fineness with regard to variance of estimate and computation expenditure.

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