Generalization of NBUAFR properties of distribution functions (Q1842401)

We shall attempt to find a general mechanism that applies to all so-called aging objects. We shall examine an element in good working order with the reliability function \(\overline F(x)\), \(\overline F(0) = 1\), a finite mean \(\mu_F\), and a failure rate at zero (initial value) \(\lambda_F = - \lim_{x \to + 0} (1/x) \ln \overline F(x)\). We shall assume that the element operates for a fairly long time as follows: in the case of failure, its operating ability is instantaneously restored to its original level and its functioning continues. It is known that at an ``infinitely distant'' time the failure rate of this element is \((1/ \mu_F)\). It is natural to assume that if the failure rate at zero is smaller than (exceeds) the stationary failure rate, the element is mainly aging (negative aging). In fact, it has been found that most aging principles adhere to this mechanism. The most complete class in the left branch is class \({\mathfrak L}\). We shall show that each distribution \(F\) of \({\mathfrak L}\) satisfies the general principle of aging, i.e., (1) \(\lambda_F \mu_F \leq 1\). In fact, let \(F \in {\mathfrak L}\). This means, by definition, that \(\widehat F(s) \leq 1/(1 + s \mu_F)\), where \(\widehat F\) is the Laplace-Stieltjes transform of the distribution function \(F (DF)\), or \(s \cdot \widehat F(s) \leq s/(1 + s \mu_F) \to 1/ \mu_F\) for \(s \to \infty\); the left side of the inequality (according to the Tauberian theorem) converges to \(\lambda_F : s \widehat F(s) \to \lambda_F\), \(\sigma \to \infty\). Thus we obtain inequality (1). It is also easy to establish property (1) for the DF of the most general class on the right side of the branch, NBUAFR. However, while class \({\mathfrak L}\) can be considered an operator extension of class HNBUA, there is no such extension of class NBUAFR. It can be expected of this class that: property (1) is realized in it, it is related in a certain way to class \({\mathfrak L}\), it is broader than class NBUAFR, and it makes it possible to obtain estimates that, along with estimates in class \({\mathfrak L}\), form the confidence interval of the reliability of shock models of random action.