Stat Papers (2017) 58:947–949
Belzunce, F., Martínez-Riquelme, C. and J. Mulero:
An Introduction to Stochastic Orders. Academic Press,
New York, 2016, 174 pp., EUR 53.95 (print),
Received: 24 April 2017 / Revised: 28 April 2017 / Published online: 10 May 2017
© Springer-Verlag Berlin Heidelberg 2017
The literature on stochastic orders has been growing rapidly over the last 20 years.
With dozens (if not hundreds) of ordering concepts, with often subtle differences,
it is becoming increasingly difﬁcult to ﬁnd a suitable point of entry for newcomers
to the ﬁeld. Of course, there is the standard reference by Shaked and Shanthikumar
(2007), but this is an encyclopaedic treatment providing comprehensive coverage of
the literature up to about 10 years ago. The book under review attempts to provide
an introductory account in just 150 pages. The presentation is accessible for readers
with a fairly modest background in applied probability and statistics. This implies that
results are usually not given under weakest conditions; an advantage is that it was
possible to include a large amount of interesting and useful material.
There are three chapters: the ﬁrst (25pp) mainly presents background material, the
second (90pp) studies univariate and the third (35pp) multivariate stochastic orders.
More speciﬁcally, the ﬁrst chapter collects some standard characteristics of univariate
and multivariate distributions along with concepts such as stop-loss transforms, mean
residual life, total time on test, risk measures based on quantiles, concentration and
deprivation, and also selected notions of dependence. In addition, there are a few key
results on total positivity, needed later for some proofs, as well as various families
of statistical distributions. The latter are used for illustrations, on which more below.
The second and main chapter presents a number of popular stochastic orders based
on comparisons of distribution functions, hazard rates, mean residual lives, integrated
tails, total times on test, as well as differences and ratios thereof. For each order, it
gives the main characterizations, sufﬁcient conditions and preservation theorems (e.g.,
with respect to mixing, transformations, or convolution) as well as a limited number of
Universität Basel, Basel, Switzerland