Insurance: Mathematics and Economics 27 2000 137–144
On error bounds for approximations to multivariate distributions
Bjørn Sundt
a,b,∗
a
University of Bergen, Bergen, Norway
b
University of Melbourne, Melbourne, Australia Received 17 September 1999; received in revised form 5 January 2000; accepted 20 January 2000
Abstract
In the present paper we extend some error bounds developed for approximations to univariate distributions to a multivariate framework. © 2000 Elsevier Science B.V. All rights reserved.
MSC: M10; M11
Keywords: Multivariate distributions; Approximations; Error bounds
1. Introduction
1A. Dhaene and De Pril 1994 presented a framework for approximations to univariate aggregate claims distri- butions, incorporating the approximations of Kornya, De Pril, and Hipp, and within this framework they developed
general results for error bounds. Some of the results of Dhaene and De Pril 1994 were reformulated within the framework of De Pril transforms and further discussed by Dhaene and Sundt 1998. Dhaene and Sundt 1997
discussed some error bounds without introducing the De Pril transform.
Sundt 2000 extended the definition of the De Pril transform to multivariate functions and discussed its properties within that framework.
In the present paper we shall extend some of the approximations and error bounds of Dhaene and De Pril 1994 and Dhaene and Sundt 1997, 1998 to the multivariate case, utilising the multivariate De Pril transform wherever
appropriate. In Section 2, we recapitulate some notation, definitions, and results from Sundt 2000 in connection with
the multivariate De Pril transform. Section 3 gives a general discussion of approximations to aggregate claims distributions. Sections 4 and 5 are devoted to multivariate extensions of error bounds; in Section 4, we consider the
bounds of Dhaene and Sundt 1997, whereas the topic of Section 5 is the bounds of Dhaene and De Pril 1994 and Dhaene and Sundt 1998.
Sundt 1999b discusses some results related to the present paper. 1B. In this paper we shall represent probability distributions by their probability functions. Therefore, for conve-
nience, we shall refer to a probability function as a distribution. We make the convention that
P
i∈S
= 0 and
Q
i∈S
= 1 when the set S is empty.
∗
Present address: Vital Forsikring ASA, PO Box 250, N-1326 Lysaker, Norway. Tel.: +47-67-83-44-71; fax: +47-67-83-45-01. E-mail address:
bjoern.sundtvital.no B. Sundt. 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 4 2 - 1
138 B. Sundt Insurance: Mathematics and Economics 27 2000 137–144
2. The multivariate De Pril transform
