On the stability of a characterization by the lack of memory property

Abstract

An useful and interesting characterization of the Weibull distribution is its lack of memory (of order α) property, i.e., P(X  ≥ (x^α + y^α)^1/α|X ≥ y) = P(X ≥ x) for all x, y ≥ 0. The characterization holds even in the case when it is required to fulfil this relation not on the entire semi-axis {y|y ≥ 0}, but only at two incommensurable points y₁ and y₂. The stability estimation in this characterization is analyzed.