有相依关系的成败型元件串联系统可靠性经典精确最优置信下限

THE CLASSICAL EXACT OPTIMAL CONFIDENCE LIMITS OF RELIABILITY FOR THE SERIES SYSTEM OF THE DEPENDENT PASS-FAIL COMPONENTS

设有由Ｋ个成败型元件Ａ１，Ａ２，…，Ａｋ组成的串联系统，设Ａ１，Ａ２，…，Ａｋ之间有相依关系，要Ａ１成功的条件下Ａ２才可以试验，在Ａ１，Ａ２，都成功的条件下Ａ３才可以试验，…在Ａ１，Ａ２…，Ａｋ－１都成功的条件下Ａｋ才可以试验，设条件概率Ｐ（Ａｉ｜Ａ１Ａ２…Ａｉ－１）＝ｐｉ，ｐｉ未知，ｉ＝１，２…，Ｋ。设有样本（Ｘ１，Ｘ２，…。Ｘｋ）本文用样本点排序法求此串联系统可靠性之经典精确置信下限，并证明了此置信限之最优性。

Let {A1, A2, … ,Ak} be a series system consisting of pass-fail component A1 , A2, … , AK. Suppose that one can test Ai if and only if tests A1,A2, … , Ai-1 success.Set P(A1) = p1, P(A2| A1) = p2, … , P(Ai| A1A2 … Ai-1) = pi, … , P(Ak| A1A2 … Ak-1)= pk and the p1, p2, …,pK are unknown. Suppose that there is the sample (X1, X2, … ,Xk) if one makes n tests.In this paper, we discuss the classical exact optimal confidence limits of reliability R = p1p2 … pk, and prove that this confidence limits is the best.