On error bounds for approximations to multivariate distributions
Bjørn Sundt
a,b,∗ aUniversity of Bergen, Bergen, Norway bUniversity of Melbourne, Melbourne, AustraliaReceived 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 thatP
i∈S=0 and
Q
i∈S =1 when the setSis 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:[email protected] (B. Sundt).
2. The multivariate De Pril transform
2A. In this paper a column vector will be denoted by a bold-face letter and its elements by the corresponding italic with the number of the element indicated by a subscript; subscript ‘·’ denotes the sum of the elements, e.g.
x =(x1, . . . , xm)′andx· =Pmj=1xj. For each positive integermwe denote byNmthe set of allm×1 vectors where all elements are non-negative integers, and introduceNm+=Nm∼ {0}with0denoting them×1 vector with all elements equal to zero. Forx,y∈Nm, byy≤xwe shall mean thatx−y∈Nmand byy<xthatx−y∈Nm+. When indicating the range for a vector, we tacitly assume that all its elements are non-negative integers.
LetPmandFmdenote the classes of distributions and functions, respectively, onNm;Pm0andFm0the classes of distributions and functions, respectively, onNm with a positive mass at0, andPm+ andFm+ the classes of distributions and functions, respectively, onNm+.
2B. Sundt (2000) defined theDe Pril transformϕf of a functionf ∈Fm0by the recursion
ϕf(x)= 1
f (0)
x·f (x)− X
0 0 0<y<x
ϕf(y)f (x−y)
(x∈Nm).
Whenϕf is given, we can evaluatef recursively by
f (x)= 1
x·
X
0 0 0<y≤x
ϕf(y)f (x−y) (x∈Nm). (2.1)
2C. The convolutionf ∗gof two functionsf, g∈Fmis defined by
(f∗g)(x)= X
0 0 0≤y≤x
f (y)g(x−y) (x∈Nm).
Sundt (2000) showed that iff, g∈Fm0, then
ϕf∗g=ϕf +ϕg. (2.2)
2D. Forp∈F10andh∈Fm+we define the compound functionp∨h∈Fm0by
(p∨h)(x)= x·
X
n=0
p(n)hn∗(x) (x∈Nm).
Sundt (2000) showed that
ϕp∨h(x)=x·
x·
X
y=1 ϕp(y)
y h
y∗(x) (x∈N
m+). (2.3)
3. Approximations
3A. Forj =1, . . . , t, letfj =pj∨hjwithpj ∈P10 andhj ∈Pm+. We want to evaluatef = ∗tj=1fj. From
(2.2) and (2.3), we obtain
ϕf(x)= t
X
j=1
ϕfj(x)=x· t
X
j=1
x·
X
y=1 ϕpj(y)
y h
y∗
j (x) (x∈Nm+). (3.1)
of thehj’s. Therefore it is tempting to, for eachj, approximatepjby some functionqj ∈F10such thatϕqj(y)=0 for allygreater than some positive integerr. Letgj =qj∨hj andg= ∗tj=1gj. Then
ϕg(x)= t
X
j=1
ϕgj(x)=x· t
X
j=1
r
X
y=1 ϕqj(y)
y h
y∗
j (x) (x∈Nm+).
3B. In the univariate case (m=1) such approximations have been studied by Dhaene and De Pril (1994), Dhaene and Sundt (1998) and Sundt et al. (1998). For the special case where all thepi’s are Bernoulli distributions, there are several earlier papers; we mention in particular De Pril (1989).
For further discussion on the following three classes of approximations we refer to De Pril (1989) and Dhaene and De Pril (1994):
1. In the De Pril approximation we replace for eachj, ϕpj(y)with zero for ally greater thanr and keeppj(0) unchanged.
2. The Kornya approximation is a rescaling of the De Pril approximation such that for eachj, qjsums to one like a probability distribution.
3. In the Hipp approximation we determineqjsuch that its moments up to orderrmatch the corresponding moments ofpj (assuming that these moments exist and are finite), that is
∞
X
x=0
xiqj(x)=
∞
X
x=0
xipj(x) (i=0,1, . . . , r).
This matching of moments is discussed by Dhaene et al. (1996) and Sundt et al. (1998).
3C. When approximating a functionf ∈Fmby another functiong∈Fm, we want some idea of the accuracy of the approximation. As a measure of accuracy we introduceǫ(f, g)=P
x∈Nm|f (x)−g(x)|. In the univariate case
this measure has been discussed by i.a. De Pril (1989), Dhaene and De Pril (1994), and Dhaene and Sundt (1997, 1998). We shall divide our discussion of the error measureǫinto two sections:
1. In Section 4, we consider bounds in which De Pril transforms do not appear. Here we generalise error bounds discussed by Dhaene and Sundt (1997) in the univariate case.
2. In Section 5, we consider bounds based on the De Pril transform. We first generalise a general bound forǫ(f, g)
deduced by Dhaene and De Pril (1994) in the univariate case. Then we extend some results presented by Dhaene and Sundt (1998) in the univariate case.
4. General error bounds
Forf ∈Fmwe introduceν(f )=P
x∈Nmf (x). When such quantities appear in our formulae, it will always be
tacitly assumed that they exist and are finite. Iff ∈Pm, thenν(f )=1.
Lemma 4.1. Iff, g∈Fmsuch thatν(|f|) <∞andν(|g|) <∞,then
ν(f∗g)=ν(f )ν(g).
Proof. We have
ν(f∗g)=
∞
X
x∈Nm
(f∗g)(x)= X
x∈Nm
X
0≤yyy≤xxx
f (y)g(x−y)= X
y∈Nm
f (y)X
x≥yyy
g(x−y)=ν(f )ν(g).
results that we will present in the following are analogous extensions of univariate results, and if the proofs are equally trivial extensions of the univariate case, then we shall leave the proofs to the readers. This goes in particular for the following lemma, which extends Lemma 4.2 in Dhaene and Sundt (1997).
Lemma 4.2. Forf, g, h∈Fm,we have
ǫ(f ∗h, g∗h)≤ǫ(f, g)ν(|h|).
The following theorem is proved in the univariate case as Theorem 4.1 in Dhaene and Sundt (1997). By application of Lemmas 4.1 and 4.2, the proof trivially extends to the multivariate case.
Theorem 4.1. Forfj, gj ∈Fmforj =1, . . . , t,we have
The following two theorems are trivial multivariate extensions of the univariate Theorems 5.1 and 5.2 in Dhaene and Sundt (1997).
Theorem 4.2. Forp, q∈F1andh∈Fm+withν(|h|)≤1,we have
ǫ(p∨h, q∨h)≤ǫ(p, q). (4.3)
Theorem 4.3. Forp∈F1andh, k∈Fm+withν(|h|)≤1andν(|k|)≤1,we have
ǫ(p∨h, p∨k)≤ν(|p|)ǫ(h, k).
5. Error bounds based on the De Pril transform
5A. Forf, g∈Fm+, let
In the univariate case, these quantities were introduced in Dhaene and Sundt (1998)
Theorem 5.1. Letf ∈Pm0andg∈Fm0.Then
ǫ(f, g)≤eδ(f,g)−1. (5.1)
Proof. Whenδ(f, g)= ∞, the theorem trivially holds. Let us therefore turn to the caseδ(f, g) <∞. We definea ∈Fm0by
from which we obtain
|a(x)| ≤ 1
andpis the Poisson distribution with parameterδ0(f, g), that is,
p(n)= (δ0(f, g)) n
n! e
−δ0(f,g) (n∈N 1).
Then, by formula (3.8) in Sundt (1999a),bsatisfies the recursion
We shall prove by induction that
|a(x)| ≤b(x)eδ(f,g) (x∈Nm). (5.4)
As
|a(0)| =exp
lng(0)
f (0)
≤exp
lng(0)
f (0)
=exp[δ(f, g)−δ0(f, g)]=b(0)eδ(f,g),
the hypothesis holds forx=0. Now let us assume that it holds for allxsuch that0≤x<v. Then
|a(v)| ≤ 1
v·
X
000<y≤v
|ϕa(y)| |a(v−y)| ≤ 1
v·
X
0 00<y≤v
|ϕa(y)|b(v−y)eδ(f,g)=b(v)eδ(f,g),
that is, the hypothesis also holds forx=v, and hence it holds for allx∈Nm. Application of (5.4) gives
X
x∈Nm+
|a(x)| ≤ X
x∈Nm+
b(x)eδ(f,g)=(1−b(0))eδ(f,g) =eδ(f,g)−exp[δ(f, g)−δ0(f, g)],
and by insertion in (5.3), we obtain (5.1).
This completes the proof of Theorem 5.1.
The criticism raised at the end of Section 3 in Dhaene and Sundt (1998) against the univariate case of the error bound (5.1) is also valid in the multivariate case.
5B. Forf ∈Fmwe define the cumulation operatorŴby
Ŵf (x)= X
0 00≤y≤x
f (y) (x∈Nm).
In the univariate case the following theorem was proved as Corollary 1 in Dhaene and De Pril (1994). The proof easily extends to the multivariate case.
Theorem 5.2. Letf ∈Pm0andg∈Fm0.Then
|Ŵf (x)−Ŵg(x)| ≤(eδ(f,g)−1)Ŵf (x) (x∈Nm). (5.5)
AsŴf (x)≤1, (5.5) immediately gives
|Ŵf (x)−Ŵg(x)| ≤eδ(f,g)−1.
Unfortunately the bound in (5.5) depends onŴf (x), that is, the quantity we want to approximate. Therefore Dhaene and De Pril (1994) proved the following corollary in the univariate case as their Corollary 2. The multivariate extension of their proof is trivial.
Corollary 5.1. Letf ∈Pm0andg∈Fm0.Ifδ(f, g) < ln 2,then
|Ŵf (x)−Ŵg(x)| ≤ e
δ(f,g)−1
2−eδ(f,g)Ŵg(x) (x∈Nm). (5.6)
From (5.6) we see thatŴg(x)≥0 for allx∈Nmifδ(f, g) < ln 2.
In the univariate case, the following theorem was contained in Theorem 5.2 in Dhaene and Sundt (1998). The proof easily extends to the multivariate case.
Theorem 5.3. Withfj ∈Pm0andgj ∈Fm0forj =1, . . . , t,we have
On the other hand, insertion of (5.1) in (4.2) gives
ǫ(f, g)≤
The comparison of these two bounds forǫ(f, g)in subsection 5B in Dhaene and Sundt (1998) immediately carries over to the multivariate case.
The proof of the following theorem is a multivariate extension of the corresponding results in the univariate case in Theorem 6.2 in Dhaene and Sundt (1998).
Theorem 5.4. Letp∈P10,q∈F10,andh∈Pm+.Then
δ0(p∨h, q∨h)≤δ0(p, q), (5.8)
δ(p∨h, q∨h)≤δ(p, q). (5.9)
Proof. By using (2.3), we obtain
δ0(p∨h, q∨h)=
that is, (5.8) holds. Furthermore,
We see that both by applying (5.9) in (5.1) and by applying (5.1) in (4.3), we obtain
ǫ(p∨h, q∨h)≤eδ(p,q)−1.
In their Section 7, Dhaene and Sundt (1998) discuss improvements of the bounds in the case when approximating an infinitely divisible distributionf ∈ P10 with a functiong ∈ F10. Extension to the multivariate situation is straightforward and left to the readers.
5D. Let us now return to the situation of subsection 3A. By applying Theorems 5.3 and 5.4 successively we obtain
δ(f, g)≤ t
X
j=1
δ(fj, gj)≤ t
X
j=1
δ(pj, qj).
As we have now got a bound that depends on only the counting distributions and their approximations, the discussions on the univariate case in Section 5 of Dhaene and De Pril (1994) and Section 8 of Dhaene and Sundt (1998) are still valid in the multivariate case. In particular, for the multivariate extension of the approximations of De Pril, Kornya, and Hipp, the error bounds given in Dhaene and De Pril (1994) are still valid.
Acknowledgements
The present research was carried out while the author stayed as GIO Visiting Professor at the Centre for Actuarial Studies, University of Melbourne.
References
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