ABSTRACT
The time transformation introduced in Chapter 5 defines a “metric”, whose properties are now studied in this chapter. Specifically, the metric is defined by the derivative https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003250685/2cc273d1-d7ad-4f7b-9a9d-732ca34bd245/content/EQN_C006_EQ_0047.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> This derivative is expressed in terms of the mean residual life (MRL), V(t), which allows to determine V(t) as the solution of a linear differential equation when g(t) is known; or conversely, to determine g(t) as the solution of a linear differential equation when V(t) is known. Then the study of the second derivative of g(t) shows that g(t) is either a concave function (in the case of the Pareto distribution for instance), or an S-shaped curve (such as in the cases of Weibull and Gamma distributions). When the g(.) transformation corresponds to an S-shaped curve, an upper bound for the absolute value of the time derivative of the MRL(“the average loss rate”) is given by k, for time values greater than the time corresponding to the inflection point of g(.).
