Insurance: Mathematics and Economics 53 (2013) 230–236
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Insurance: Mathematics and Economics
journal homepage: www.elsevier.com/locate/ime
A note on the family of extremality stochastic orders
María Concepción López-Díaz
a
, Miguel López-Díaz
b,∗
a
Departamento de Matemáticas, Universidad de Oviedo, C/Calvo Sotelo s/n. E-33007 Oviedo, Spain
b
Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, C/Calvo Sotelo s/n. E-33007 Oviedo, Spain
highlights
• Each extremality stochastic order is generated by a partial order and its upper quadrant sets.
• The extremality stochastic orders are order-isomorphic, in particular order-isomorphic to the upper orthant order.
• The maximal generator of each extremality order is obtained.
• A new family of stochastic orders is defined by means of the partial orders which determine the extremality orders.
article info
Article history:
Received September 2012
Received in revised form
April 2013
Accepted 24 April 2013
JEL classification:
C02
MSC:
60E15
Keywords:
Extremality order
Maximal generator
Order-isomorphism
Order-preserving mapping
Partial order
abstract
The family of extremality stochastic orders was introduced in Laniado et al. (2012) (Portfolio selection
through an extremality stochastic order. Insurance: Mathematics and Economics 51, 1–9), as an extension
of the upper and lower orthant orders, having important applications in the research of optimal allocations
of wealth among risks in single period portfolio problems. In this paper we analyze some properties of
such a family of stochastic orders, namely we prove that any extremality stochastic order is generated by
a partial order on the Euclidean space and the class of upper quadrant sets of the partial order, showing
that all the extremality orders are order-isomorphic. The above analysis will lead to the determination
of the maximal generator of each extremality order by means of the maximal generator of the upper
orthant order. Moreover we introduce a new family of stochastic orders which arises from the previous
construction.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
A method to extend a pre-order on a set to a stochastic order
on the class of probabilities associated with such a set, is proposed
in Giovagnoli et al. (2008). The method is based on the so-called
upper quadrant sets of the pre-order. If ≼ stands for a pre-order on
a set X and x ∈ X, the upper quadrant set of x, denoted by Q
≼
x
, is
the set Q
≼
x
={y ∈ X | x ≼ y}.
The class of mappings
{ I
Q
≼
x
| x ∈ X },
where I
A
stands for the indicator function of the set A ⊂ X,
generates a stochastic order on the class of probabilities on X, we
∗
Corresponding author. Tel.: +34 985103362; fax: +34 985103354.
E-mail addresses: cld@uniovi.es (M.C. López-Díaz), mld@uniovi.es
(M. López-Díaz).
will denote it by
≼, defined by: given P
1
and P
2
probabilities,
P
1
≼P
2
when
X
f dP
1
≤
X
f dP
2
for any mapping f = I
Q
≼
x
with x ∈ X, that is, when
P
1
(Q
≼
x
) ≤ P
2
(Q
≼
x
)
for any x ∈ X. Roughly speaking this means that P
2
concentrates
more on probability than P
1
in the ‘‘higher’’ parts of the pre-order.
The above procedure allows the generation of stochastic orders
by means of ‘‘non-stochastic’’ orders. It is worth noting that
in Giovagnoli et al. (2008) the authors introduce a inter-rater
agreement stochastic order by means of this technique.
The family of extremality stochastic orders has been recently
published (see Laniado et al., 2012) as a generalization of the
upper and lower orthant orders, which allows comparisons of
random vectors in different directions determined by a unit vector.
0167-6687/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.insmatheco.2013.04.009