## generating from a failure rate function [X’ed]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on July 4, 2015 by xi'an

While I now try to abstain from participating to the Cross Validated forum, as it proves too much of a time-consuming activity with little added value (in the sense that answers are much too often treated as disposable napkins by users who cannot be bothered to open a textbook and who usually do not exhibit any long-term impact of the provided answer, while clogging the forum with so many questions that the individual entries seem to get so little traffic, when compared say with the stackoverflow forum, to the point of making the analogy with disposable wipes more appropriate!), I came across a truly interesting question the other night. Truly interesting for me in that I had never considered the issue before.

The question is essentially wondering at how to simulate from a distribution defined by its failure rate function, which is connected with the density f of the distribution by

$\eta(t)=\frac{f(t)}{\int_t^\infty f(x)\,\text{d}x}=-\frac{\text{d}}{\text{d}t}\,\log \int_t^\infty f(x)\,\text{d}x$

From a purely probabilistic perspective, defining the distribution through f or through η is equivalent, as shown by the relation

$F(t)=1-\exp\left\{-\int_0^t\eta(x)\,\text{d}x\right\}$

but, from a simulation point of view, it may provide a different entry. Indeed, all that is needed is the ability to solve (in X) the equation

$\int_0^X\eta(x)\,\text{d}x=-\log(U)$

when U is a Uniform (0,1) variable. Which may help in that it does not require a derivation of f. Obviously, this also begs the question as to why would a distribution be defined by its failure rate function.