On the matter of evaluation of the variation coefficient of the time to failure based on low-level quantiles

In the context of various tasks related to dependability estimation of systems by probabilistic physical methods the most important a priori information that ensures effective solutions is the information on the variation coefficient of the time to failure. Given the low failure statistics, the estimation of the variation coefficient of the time to failure is complicated due to significant sample censoring. In these cases, methods of variation coefficient evaluation with additional a priori information and the method of quantiles are used. The solution of a number of dependability-related tasks that require taking into consideration various failure distributions is significantly simplified if the functions of such distributions are tabulated in the relative operation time and variation coefficient parameters. An effective solution of dependability-related tasks with the use of tables of DN distribution function was first proposed for the parametrization of distribution in parameters x and v, where x is the scale parameter, relative operation time x = at; v is the shape parameter, variation coefficient v = V; a is the average degradation rate. That allowed performing tabulation out of real time, simplifying function tabulation and its use in a number of dependability-related tasks by method of quantiles. The paper analyzed the effectiveness of the method of quantiles in the estimation of the variation coefficient of the time to failure, that is at the same time the shape parameter of the DN distribution, under scarce failure statistics and based on it proposes a new, more effective, method. The method of estimation of the variation coefficient using low and ultralow-level quantiles is based on the behaviour analysis of function ai = f(t) obtained using the method of quantiles. It is considered that the best choice of the a priori value of v is a choice under which the dependence graph ai = f(t) is most accurately described by a straight horizontal line, which is in complete compliance with the hypothesis of constant degradation rate accepted in the context of DN distribution formalization. In cases when the dependence graph ai = f(t) does not easily allow concluding on the best choice of the a priori value v (it is especially difficult to make a choice based on the statistics of first failures), the following formal criterion can be used: the most acceptable a priori value of the shape parameter v lies within the range of values, where the sign of the trend of the average degradation rate (h) in graph ai = f(t) changes. Studies have established that the most significant errors in the estimation of the variation coefficient are associated with first failures. When processing the results of dependability tests it is assumed the first failures in a sample have the lowest information weight, as their occurrence is due to serious defects not detected by final quality inspection of products. The first failures normally “fall out” of the overall statistical pattern, and it is recommended to omit them from further analysis. The proposed method of estimation of the variation coefficient of the time to failure based on ultralow-level quantiles enables – in the context of limited failure statistics, when other methods are inefficient – for sufficiently accurate identification of not only the variation coefficient of the time to failure and DN distribution parameters, but also make conclusions regarding the feasibility and legitimacy of equalization (description) of the considered sample using this diffusion distribution, i.e. it can be used as a kind of criterion of compliance of the empirical failure distribution under consideration with the chosen theoretical dependability model. The described process of finding the truest values of the variation coefficient of the time to failure using the formal criterion can be computerized.