The multivariate De Pril transform
q
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 the definition of the De Pril transform to a class of multivariate functions and discuss various properties of this multivariate De Pril transform. In particular, we show that, like in the univariate case, it is additive for convolutions, and discuss De Pril transforms of compound functions and higher order functions. Finally, we introduce a multivariate Dhaene–De Pril transform, which we compare with the De Pril transform. © 2000 Elsevier Science B.V. All rights reserved.
MSC: M10; M11
Keywords: Multivariate functions; Recursions; De Pril transforms; Convolutions; Compound distributions
1. Introduction
1A. During the last two decades an extensive literature on recursive methods for exact and approximate evaluation of aggregate claims distributions has grown up, starting with Panjer (1980, 1981). Most of it has been confined to univariate distributions. However, multivariate aggregate claims distributions are also of interest in insurance, e.g. when considering the joint distribution of aggregate claims of bodily injury and material damage in motor insurance. Further applications are discussed in Sundt (1999a). Multivariate extensions of Panjer’s (1981) recursion for compound distributions have been presented in two directions, for references and discussion, see Sundt (2000a).
1B. Inspired by in particular De Pril (1989) and Dhaene and De Pril (1994), Sundt (1995) defined and discussed the De Pril transform of a univariate distribution as a tool for recursive evaluation of aggregate claims distributions. In Dhaene and Sundt (1998) the definition was extended to more general univariate functions in connection with approximate evaluation of distributions. Properties of the De Pril transform are also discussed in Sundt et al. (1998), Sundt (1998) and Sundt and Ekuma (1999).
1C. In the present paper we extend the definition of the De Pril transform to multivariate functions. In Section 2 we recapitulate some definitions and results from the univariate case in a form that will simplify the multivariate extension performed in Section 3. As a starting point for the multivariate extension we apply the multivariate
q
The present paper is dedicated to the memory of Nelson De Pril on whose research the paper heavily leans.
∗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.sundt@vital.no (B. Sundt).
extension of Sundt (1999a) to the recursion of Sundt (1992). We show that, like in the univariate case, the multivariate De Pril transform is additive for convolutions, and discuss De Pril transforms of compound functions, higher order functions, and infinitely divisible distributions. We also express the De Pril transform of the distribution of a vector of independent sub-vectors of random variables by the De Pril transforms of the marginal distributions of these sub-vectors. Finally, in Section 4 we introduce the Dhaene–De Pril transform as a multivariate version of a function discussed by Dhaene and De Pril (1994) in the univariate case, and discuss its advantages and disadvantages compared to the De Pril transform.
Sundt (1999b) discusses some results related to the present paper.
1D. In the present paper we shall work on distributions in the representation of their probability functions. Therefore, we shall for simplicity refer to probability functions as distributions.
2. The univariate De Pril transform
2A. LetN1denote the set of non-negative integers andP10the class of distributions onN1with a positive mass at zero. For a positive integerkand functionsaandbwe defineRk[a, b] to be the distributionp ∈P10 given by the recursion
p(n)=
k
X
i=1
a(i)+b(i)
n
p(n−i) (n=1,2, . . . ), (2.1)
withp(n)=0 for alln <0. Following Sundt (1992), we letRk denote the class of all such distributions for fixed
k; we allowk= ∞. We have thatR∞=P10. For simplicity, we leta(i)=b(i)=0 fori=k+1, k+2, . . .
2B. LetN1+denote the set of positive integers andP1+ the class of distributions onN1+. Forp ∈ P10 and
h∈P1+, the compound distributionp∨his given by
p∨h=
∞
X
n=0
p(n)hn∗. (2.2)
As
hn∗(x)=0 (x∈N1; n=x+1, x+2, . . . ), (2.3)
we obtain
(p∨h)(x)=
x
X
n=0
p(n)hn∗(x) (x∈N1),
where the infinite summation in (2.2) has been reduced to a finite summation. In particular we have that(p∨h)(0)=
p(0) >0. Thus,p∨h∈P10.
Sundt (1992) showed that ifpisRk[a, b], then
(p∨h)(x)=
x
X
y=1
(p∨h)(x−y)
k
X
i=1
a(i)+b(i)
i y x
hi∗(y) (x∈N1+), (2.4)
i.e.p∨hisRk[c, d] with
c(y)=
k
X
i=1
a(i)hi∗(y), d(y)=y
k
X
i=1
b(i)
i h
i∗(y) (y ∈N
2C. From (2.1) we see that the distributionR∞[0, b] satisfies the recursion
p(n)= 1
n
n
X
i=1
b(i)p(n−i) (n∈N1+),
and by solving forb(n)we obtain
b(n)= 1
p(0) np(n)−
n−1 X
i=1
b(i)p(n−i) !
(n∈N1+).
Thus, for anyp∈R∞there exists a unique functionbsuch thatpcan be represented asR∞[0, b]. We call thisb
the De Pril transform ofpand denote it byϕp. For convenience we extend the definition of the De Pril transform
withϕp(0)=0. We now have
p(n)= 1
n
n
X
i=1
ϕp(i)p(n−i) (n∈N1+) (2.6)
ϕp(n)=
1
p(0) np(n)−
n−1 X
i=1
ϕp(i)p(n−i)
!
(n∈N1), (2.7)
making the convention thatPs
i=r =0 whens < r.
From (2.5) we see that ifp∈P10andh∈P1+, then
ϕp∨h(y)=y x
X
i=1
ϕp(i)
i h
i∗(y) (y∈N
1+), (2.8)
which was given by Sundt (1995). He also showed that forf1, f2, . . . , fr ∈P10
ϕ∗r j=1fj =
r
X
j=1
ϕfj. (2.9)
LetpbeRk[a, b]. Sundt (1995) showed that then
ϕp(n)=na(n)+b(n)+ k
X
i=1
a(i)ϕp(n−i) (n∈N1+), (2.10)
witha(n)=b(n)=0 for alln > kandϕp(n)=0 for all negativen. From this and (2.5) we obtain that
ϕp∨h(x)=x k
X
y=1
a(y)+b(y)
y
hy∗(x)+
x−1 X
y=1
ϕp∨h(x−y) k
X
z=1
a(z)hz∗(y) (x∈N1). (2.11)
This recursion is discussed by Sundt and Ekuma (1999).
Sundt (1995) showed that a distribution inP10has a non-negative De Pril transform if and only if it is infinitely divisible, or, equivalently, that it can be represented as a compound Poisson distribution.
2D. Formulae (2.7), (2.9) and (2.6) can be applied for numerical evaluation of convolutions of distributions in
Unfortunately, for largex, numerical evaluation of (2.8) can be rather time-consuming. Therefore, methods have been developed under which the De Pril transform of the counting distribution is replaced with a function that is equal to zero for all values of the argument larger than some integerr. Such approximations were first discussed within the terminology of De Pril transforms by Dhaene and Sundt (1998). Of earlier references, we mention De Pril (1989) and Dhaene and De Pril (1994). With such approximations to De Pril transforms, the resulting approximations to distributions are not necessarily distributions themselves. Dhaene and Sundt (1998) therefore extended the definition (2.7) of the De Pril transform to functions inF10, the class of functions onN1with a positive mass at zero, and they discussed properties of the De Pril transform within this framework. In particular they showed that the additivity property (2.9) still holds for functions inF10, and that (2.8) holds whenp∈F10andh∈F1+, being the class of functions onN1+.
From (2.6) we see that the De Pril transform of a function inF10determines the function only up to a multiplicative constant. However, a distribution inP10is uniquely determined by its De Pril transform as it should sum to one.
As the recursion (2.1) determines a functionp ∈ F10 only up to a multiplicative constant, Dhaene and Sundt (1998) definedpto be in the formRk[a, b] if it satisfies that recursion, and showed that (2.10) still holds for such
functions. Extension of (2.11) to the case whenp∈F10is in the formRk[a, b] andh∈F1+, is trivial.
2E. Recursions like (2.4) were originally developed for probability functions. Dhaene et al. (1999) discuss how one can deduce, from recursions for probability functions, recursions for cumulations like cumulative distribution functions and functions of even higher order. In this connection they defined for functionsf ∈F10the cumulation operatorŴby
Ŵf (x)=
x
X
y=0
f (y) (x ∈N1).
We see that we haveŴf =f ∗u, whereu∈F10is defined by
u(x)=1 (x∈N1). (2.12)
From (2.7) we obtain that
ϕu(x)=1 (x∈N1+), (2.13)
and application of (2.9) gives
ϕŴtf(x)=ϕf(x)+t (x∈N1+; t∈N1), (2.14)
which was proved by Dhaene et al. (1999).
3. Multivariate De Pril transform
3A. In this section we shall extend the definition of the De Pril transform to multivariate functions. As indicated in Section 2, the univariate De Pril transform could be developed from the framework of distributions in theRk
classes. We shall apply this framework as the basis also for the development of the multivariate De Pril transform. We shall start within the framework of multivariate distributions. However, as also in the multivariate case it seems desirable to be able to work with approximations to distributions where the approximations are not necessarily distributions themselves, we shall extend the framework to more general functions. Such approximations are further discussed in Sundt (2000b).
where all elements are non-negative integers, and introduceNm+ =Nm ∼ {0}with 0 denoting them×1 vector with all elements equal to zero. For x,y ∈ Nm, by y ≤ x we shall mean that x−y ∈ Nmand by y < x that x−y∈Nm+. Forj =1, . . . , mwe define ej as them×1 vector whosejth element is 1 and all the other elements
are 0. Strictly speaking, we should have indicated the dimensionmin the notation ej and 0, but as the dimension
will normally be clear, we shall drop that. When indicating the range for a vector, we tacitly assume that all its elements are non-negative integers.
We letPmandFmdenote the classes of distributions and functions, respectively, onNm,Pm0andFm0the classes of distributions and functions, respectively, onNm with a positive mass at 0, andPm+ andFm+ the classes of distributions and functions, respectively, onNm+.
The convolutionh1∗h2of two functionsh1, h2∈Fmis defined by with (2.2) we pointed out that, although that formula involved an infinite summation for the functionp∨h, this summation became finite for values of the function. This is also the case in the multivariate situation. Corresponding to (2.3) we have
hn∗(x)=0 (x∈Nm;n=x·+1, x·+2, . . . ), (3.1)
and insertion in (2.2) gives
(p∨h)(x)=
Both these recursions reduce to (2.4) whenm=1.
To develop a recursion forp∨hbased on the De Pril transform ofp, we can letk= ∞, a=0, andb=ϕp.
Because of (3.1), lettingk = ∞does not create any problem as we still have a finite number of non-zero terms in the inner summations. We obtain that for allp∈P10andh∈Pm+
We see that both (3.4) and (3.5) are very similar to (2.6). In particular, if we let
in (3.5), then we obtain
f (x)= 1
x·
X
0 0 0<y≤x
ϕf(y)f (x−y) (x∈Nm+), (3.7)
which is a multivariate extension of (2.6). As anyf ∈Pm0can be represented asp∨hwithp∈P10andh∈Pm+
given by
p(1)=1−p(0)=π =1−f (0), (3.8)
h(x)= f (x)
π (x∈Nm+), (3.9)
we have now defined the De Pril transformϕf for allf ∈Pm0. By lettingϕf(0)=0 and solving (3.7) forϕf(x)
we obtain
ϕf(x)=
1
f (0)
x·f (x)− X
0 0 0<y<x
ϕf(y)f (x−y)
(x∈Nm), (3.10)
which is a multivariate extension of (2.7). Like in the univariate case we see thatϕf is uniquely determined by (3.7).
Analogously we could have developed a multivariate De Pril transform from (3.4). However, then both the multivariate De Pril transform and the recursions would have depended onj. In particular, this would have implied that iff (x)is symmetric in the elements of x, that would not have been the case withϕf(x), and that does not seem
logical. This will be further discussed in Section 4.
Let us more generally consider (3.10) as definition of the De Pril transform of a functionf ∈Fm0. Then (3.7) is satisfied. Like in the univariate case we see that the De Pril transform determinesf up to a multiplicative constant, and iff is a distribution, then it is uniquely determined by its De Pril transform.
When considering whether (3.10) makes sense as a definition of the multivariate De Pril transform, we have to check to what extent the properties of the De Pril transform in the univariate case extend to the multivariate case. In particular, we want the additivity property (2.9) to hold. Furthermore, we wonder whether (3.6) still holds in the more general case whenp∈F10andh∈Fm+. In the following two subsections we shall prove that the answer to each of these questions is affirmative.
3E. To prove the additivity property we extend the deductions in subsection 5A of Dhaene and Sundt (1998). For a functionh∈Fmwe introduce the transformh∈Fmgiven by
h(x)=x·h(x) (x∈Nm).
Forf ∈Fm0we can now reformulate (3.7) asf =ϕf ∗f. This relation determinesϕf uniquely. Lemma 5.1 in
Dhaene and Sundt (1998) says that
h1∗h2=h1∗h2+h1∗h2 (3.11)
for allh1, h2∈F1; the proof is easily extended toh1, h2∈Fm. The additivity property (2.9) for functions inFm0 is easily proved in the same way as in the proof of Theorem 5.1 in Dhaene and Sundt (1998) under application of (3.11).
3F. From Lemma 6.1 in Dhaene and Sundt (1998) we obtain that ifh∈F1, then
h(n+i)∗=n+i
i h
i∗∗hn∗ (n, i∈N
1+) (3.12)
The proof of the following theorem is a modification of the proof of Theorem 3.1 in Sundt et al. (1998) for the
Insertion of (3.12) gives
f (x)=
from which we obtain (3.3).
The proof of (3.2) goes analogously. However, there we will also have to prove (3.11) and (3.12) withx·h(x)
replaced withxjh(x)in the definition ofh. These modifications are trivial.
This completes the proof of Theorem 1.
From Theorem 1 we immediately see that (3.4) and (3.5) hold forf =p∨hwithp∈F10andh∈Fm+, and thusϕf is given by (3.6).
The following theorem gives a multivariate version of (2.11).
Theorem 2. Iff =p∨hwithh∈Fm+andp∈F10in the formRk[a, b], then
from which we obtain
3G. In this subsection we shall extend the characterisation of distributions with non-negative De Pril transforms in terms of compound Poisson distributions from the univariate case to the multivariate case. Our deduction is parallel to the deduction of Sundt (1995) for the univariate case.
Theorem 3. A distribution inPm0has a non-negative De Pril transform if and only if it can be represented as a compound distributionp∨h, whereh∈Pm+and p is a Poisson distribution.
Proof. We first assume thatf =p∨hwithh∈Pm+andpis the Poisson distribution given by
i.e.ϕf is non-negative.
We now assume thatϕf is non-negative and introduce
λ= X
x∈Nm+ ϕf(x)
x·
.
Application of (3.10) gives
As (3.15) holds and a distribution in Pm0 is uniquely determined by its De Pril transform, we must have that
f =p∨h, i.e. every distribution inPm0can be represented in the formf =p∨h, whereh∈ Pm+andpis a
Poisson distribution.
This completes the proof of Theorem 3.
Sundt (2000c) utilises recursions for multivariate distributions to show that a distribution inPm0is infinitely divisible if and only if it can be represented in the formp∨h, whereh ∈ Pm+andpis a Poisson distribution. This result also follows from Theorem 2.1 in Horn and Steutel (1978). Together with Theorem 3 it implies that a distribution inPm0is infinitely divisible if and only if its De Pril transform is non-negative. This result is closely related to Corollary 2.2 in Horn and Steutel (1978).
3H. The main purpose of this subsection is to express the De Pril transform of the distribution of a vector of independent sub-vectors of random variables by the De Pril transforms of the marginal distributions of these sub-vectors. However, on the way we shall also show some other results.
We express x∈ Nmas(x(1)′, . . . ,x(k)′)with x(i)∈Nm
i fori=1, . . . , k with Pk
i=1mi =m; analogous for y.
We also introduce
Ci = {x∈Nm: x(j )=0;j 6=i}, Ci+=Ci ∼ {0},
and for functionsf ∈Fm0, we definefˆi ∈Fmi0by
ˆ
fi(x(i))=f (x) (x∈Ci).
For the following it will be convenient to rewrite (3.10) as
ϕf(x)=
1
f (0)
x·f (x)− X
0 0 0<y<x
ϕf(x−y)f (y)
(x∈Nm). (3.16)
Lemma 1. Forf ∈Fm0we have
ϕf(x)=ϕfˆi(x
(i)) (x∈C
i+; i=1, . . . , k). (3.17)
Proof. From (3.16) we obtain that for x∈Ci+,
ϕf(x)=
1 ˆ
fi(0)
x·(i)fˆi(x(i))−
X
{yyy∈Ci+:000<y(i)<x(i)}
ϕf(x−y)fˆi(y(i))
.
This is the same recursion as the recursion (3.16) forϕfˆ
i, and (3.17) follows.
Lemma 2. Assume thatf ∈Fm0satisfies
f (x)=0 (x∈Nm∼Ci) (3.18)
for somei∈ {1, . . . , k}. Then
ϕf(x)=
ϕfˆ
i(x
(i)) (x∈C i+)
0 (x∈Nm∼Ci+).
(3.19)
(3.16) and (3.18) give
ϕf(x)= −
1
f (0) X
{yyy∈Ci:000<y<x}
ϕf(x−y)f (y). (3.20)
When x ∈ Nm ∼ Ci and y∈ Ci, we cannot have x−y ∈ Ci, and, thus, (3.20) gives a recursion forϕf(x)for
x∈Nm∼Ci, from which we see thatϕf(x)=0 for all x∈Nm∼Ci.
This completes the proof of Lemma 2.
Theorem 4. Iff ∈Fm0can be written in the form
f (x)=
k
Y
i=1
fi(x(i)) (x∈Nm) (3.21)
withfi ∈Fmi,0fori=1, . . . , k, then ϕf(x)=
ϕfi(x
(i)) (x∈C
i+; i=1, . . . , k)
0 (x∈Nm∼ ∪k
i=1Ci+).
(3.22)
Proof. In the present case we havef = ∗k
i=1f˙i with ˙
fi(x)=
fi(x(i)) (x∈Ci)
0 (x∈Nm∼Ci),
fori=1, . . . , m. Lemma 2 gives that fori=1, . . . , m,
ϕf˙i(x)=
ϕfi(x
(i)) (x∈C i+)
0 (x∈Nm∼Ci+),
and (3.22) follows by insertion in (2.9).
Corollary 1. A functionf ∈Fm0can be expressed in the form (3.21) if and only if
ϕf(x)=0 (x∈Nm∼ ∪ki=1Ci+). (3.23)
Proof. From Theorem 4 it follows that iff can be expressed in the form (3.21), then (3.23) holds. Let us now assume thatfsatisfies (3.23). We define the functionfi ∈Fmi0byfi =f (0)
(1/ k)−1fˆ
ifori=1, . . . , k
andg∈Fm0by
g(x)=
k
Y
i=1
fi(x(i)) (x∈Nm).
Theng(x)=f (x)when x∈ ∪k
i=1Ci, and from Lemma 1, we obtain thatϕg(x)=ϕf(x)when x∈ ∪ki=1Ci+. From
Theorem 4 it follows thatϕg(x)=0 =ϕf(x)when x∈Nm ∼ ∪ki=1Ci+. Hence,gandf have the same De Pril
transform, and asg(0)=f (0), we must haveg=f, i.e.f can be expressed in the form (3.21).
This completes the proof of Corollary 1.
The following corollary is a reformulation of Theorem 4 in the special case whenk=mandm1=m2= · · · =
mm=1; the other results in this subsection can be reformulated analogously.
Corollary 2. Iff ∈Fm0can be written in the form
f (x)=
m
Y
j=1
withfj ∈F10forj =1, . . . , m, then
ϕf(x)=
ϕfj(xj) (x=xjej; xj ∈N1+;j =1, . . . , m)
0 (all other x∈Nm).
Now letf ∈ Pm0be the distribution of a random vector(X(1)′, . . . ,X(k)′)′, where X(i)is anmi ×1 vector of
non-negative integer-valued random variables (i=1, . . . , k). The random vectors X(1), . . . ,X(k)are independent if and only iff can be expressed in the form (3.21) withfi being the marginal distribution of X(i), and therefore
Theorem 4 expresses the De Pril transform of the distribution of a random vector of independent sub-vectors by the De Pril transforms of the marginal distributions of these sub-vectors. Corollary 1 gives a necessary and sufficient condition in terms of the De Pril transform of the distribution of the total vector for the sub-vectors to be independent.
3I. In this subsection we shall apply Corollary 2 to generalise (2.14) to the multivariate case. Forf ∈ Fm0we define the cumulation operatorŴby
Ŵf (x)= X 0 00≤y≤x
f (y) (x∈Nm).
ThenŴf =f ∗vwithv∈Fm0defined by
v(x)=
k
Y
j=1
u(xj)=1 (x∈Nm)
withu∈F10given by (2.12), and consequentlyŴtf =f ∗vt∗for allt ∈N1. From (2.9) we obtain that
ϕŴtf =ϕf +t ϕv. (3.24)
It remains to find an expression forϕv. Application of (2.13) and Corollary 2 gives
ϕv(x)=
1 (x=xjej; xj ∈N1+;j =1,2, . . . , m)
0 (all other x∈Nm+), (3.25)
and by insertion in (3.24) we obtain
ϕŴtf(x)=
ϕf(x)+t (x=xjej; xj ∈N1+;j =1,2, . . . , m)
ϕf(x) (all other x∈Nm).
3J. At the end of subsection 3H we considered the De Pril transform of the distribution of a random vector with independent elements. Let us now go in the opposite direction and consider the De Pril transform of the distribution of a random vector with strongly dependent elements.
Letp∈F10and c∈Nm+. We definef ∈Fm0by
f (x)=
p(n) (x=nc; n∈N1)
0 (all other x∈Nm). (3.26)
Ifpis the distribution of a random variableN, thenf is the distribution of the random vectorNc.
We can expressf as the compound functionp∨h, whereh ∈ Pm+is the distribution concentrated in c. For
n∈N1we have
hn∗(x)=
1 (x=nc)
and insertion in (3.6) gives
Pril transforms characterises the class of functions that can be expressed in the form (3.26).
Example. LetZ, Y1, Y2, . . . , Ym be independent random variables with distributions inP10, and letg and fj
denote the distributions ofZandYj (j =1, . . . , m), respectively. We introduce the randomm×1 vector X with
Xj = Z+Yj (j = 1, . . . , m) and denote its distribution byf. By application of (3.27), Corollary 2, and the
additivity property (2.9), we obtain
ϕf(x)=
Insertion in (3.7) gives
f (x)= 1
the latter formula is a slight modification of a recursion introduced by Teicher (1954).
4. Multivariate Dhaene–De Pril transform
4A. For a functionf ∈F10, Dhaene and De Pril (1994) considered instead of the De Pril transform the function
ψf given by
Let us callψf the Dhaene–De Pril transform off.
the most important tools when applying the transform for numerical calculations, and by using the De Pril transform instead of the Dhaene–De Pril transform one avoids some multiplications. Furthermore, the De Pril transform seems to be more in line with the representation (2.1) ofRk[a, b] as introduced in Sundt (1992).
On the other hand, the Dhaene–De Pril transform also has advantages. In particular the value ofψf(0)ensures
that the transform determines functions inF10uniquely. Also some formulae and deductions look more tidy with the Dhaene–De Pril transform. In the next subsection we shall extend the definition of the Dhaene–De Pril transform to the multivariate case, and we shall see that in that case it has an additional advantage that does not appear in the univariate case.
4B. Any functionf ∈ Fm0withf (0) <1 can be represented asp∨hwithp ∈ P10 andh∈ Fm+given by (3.8) and (3.9). Application of (4.1) in (3.4) and (3.5) gives
f (x)= 1
Insertion in (4.2) and (4.3) gives
f (x)= 1
By rescalingf we obtain that (4.4)–(4.7) also hold whenf (0) ≥ 1. Thus, we see that in the multivariate case the Dhaene–De Pril transform gives more flexibility than the De Pril transform with regard to how to recursively evaluate the function from the transform and the transform from the function.
Acknowledgements
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