Greatest Lower Bounds of System Failure Probability on a Time Interval Under Incomplete Information About the Distributi
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GREATEST LOWER BOUNDS OF SYSTEM FAILURE PROBABILITY ON A TIME INTERVAL UNDER INCOMPLETE INFORMATION ABOUT THE DISTRIBUTION FUNCTION OF TIME TO FAILURE L. S. Stoikova1 and S. N. Krasnikov2
UDC 519.2
Abstract. The authors solve problems of finding the greatest lower bounds for the probability F ( u ) - F ( u), 0 < u < u < ¥, in the set of distribution functions F ( x ) of nonnegative random variables with unimodal density with mode m, u < m < u, and fixed two first moments. Keywords: extremum of a linear functional, unimodal distribution function with mode m and fixed two first moments, partition of the domain of parameters. INTRODUCTION The present paper continues the studies [1–3]. In [1], the least upper bounds of a functional are found and related bibliography is collected; auxiliary transformations reduce the problem to a simpler one. In [2], greatest lower bounds of functional are found under the constraint m < u . The concept of a boundary distribution function is introduced, which facilitates finding the partitions of the domain of parameters. It was for the first time that the distribution function G 5 was considered as an extremum one and its existence conditions were analysed. In [3], the same problem is solved for the case where u < m, which has revealed for the first time the possibility of passage from one subdomain of parameters into more than one adjacent subdomains of parameters. (Finding “adjacent” distribution functions and “adjacent” domains of parameters is outlined in [4].) In [3], it is also shown that to find the infimum (supremum) of one and the same functional, we can obtain several partitions of the entire given domain of parameters. This principal new result needs further investigations, in particular, the proof that all possible numerical values of parameters are enveloped by different partitions. We can explain the appearance of several partitions in the domain of parameters as follows. As shown in [4], the distribution function (d.f.) G 3 can transfer into the following adjacent d.f.: G 2 (if the sign of expression L ( B ( u ), u - 0) < 0 is changed to the opposite: L ( B ( u ), u - 0) > 0 ); G 4 (if the sign varies in the inequality M (0, B ( u ), u ) > 0 : M (0, B ( u ), u ) < 0 ); G 6 (if the sign of the inequality M ( x 36 , B ( u ), u ) > 0 changes to the opposite). Therefore, depending on the numerical value of the vector of parameters and on which of the functions, L ( B ( u ), u - 0), M (0, B ( u ), u ) or M ( x 36 , B ( u ), u ) , changes its sign first when parameter u varies, passage to the function G 2 , G 4 or G 6 , respectively, is made. In the present paper (the case u < m < u), we will obtain the results for two variants of mutual arrangement of five parameters of problem (2), (3). Since the problem is complicated by various possible partitions, which depend on the mutual arrangement of parameters, its purely analytical solution would be rather difficult since it is necessary to trace many variants. However, the numerical calculations by the co-author S. Krasnikov have
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