138 B. Sundt Insurance: Mathematics and Economics 27 2000 137–144
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 =
x
1
, . . . , x
m ′
and x
·
= P
m j =
1
x
j
. For each positive integer m we denote by N
m
the set of all m × 1 vectors where all elements are non-negative integers, and introduce N
m+
= N
m
∼ {0} with 0 denoting the m × 1 vector with
all elements equal to zero. For x, y ∈ N
m
, by y ≤ x we shall mean that x − y ∈ N
m
and by y x that x − y ∈ N
m+
. When indicating the range for a vector, we tacitly assume that all its elements are non-negative integers.
Let P
m
and F
m
denote the classes of distributions and functions, respectively, on N
m
; P
m
and F
m
the classes of distributions and functions, respectively, on N
m
with a positive mass at 0, and P
m+
and F
m+
the classes of distributions and functions, respectively, on N
m+
. 2B. Sundt 2000 defined the De Pril transform ϕ
f
of a function f ∈ F
m
by the recursion ϕ
f
x =
1
f 0
x
·
f x −
X
0yx
ϕ
f
yf x − y
x ∈ N
m
. When ϕ
f
is given, we can evaluate f recursively by
f x =
1 x
·
X
0y≤x
ϕ
f
yf x − y x ∈ N
m
. 2.1
2C. The convolution f ∗ g of two functions f, g ∈ F
m
is defined by
f ∗ gx =
X
0≤y≤x
f ygx − y x ∈ N
m
. Sundt 2000 showed that if f, g ∈ F
m
, then ϕ
f ∗g
= ϕ
f
+ ϕ
g
. 2.2
2D. For p ∈ F
10
and h ∈ F
m+
we define the compound function p ∨ h ∈ F
m
by
p ∨ hx =
x
·
X
n=
pnh
n∗
x x ∈ N
m
. Sundt 2000 showed that
ϕ
p∨h
x = x
· x
·
X
y= 1
ϕ
p
y y
h
y∗
x x ∈ N
m+
. 2.3
3. Approximations
3A. For j = 1, . . . , t, let f
j
= p
j
∨ h
j
with p
j
∈ P
10
and h
j
∈ P
m+
. We want to evaluate f = ∗
t j =
1
f
j
. From 2.2 and 2.3, we obtain
ϕ
f
x =
t
X
j = 1
ϕ
f
j
x = x
· t
X
j = 1
x
·
X
y= 1
ϕ
p
j
y y
h
y∗ j
x x ∈ N
m+
. 3.1
Thus, we can first find ϕ
f
by 3.1 and then evaluate f recursively by 2.1. However, evaluation of ϕ
f
by 3.1 can be rather time-consuming; when x
·
is large, we need to perform long summations involving high order convolutions
B. Sundt Insurance: Mathematics and Economics 27 2000 137–144 139
of the h
j
’s. Therefore it is tempting to, for each j , approximate p
j
by some function q
j
∈ F
10
such that ϕ
q
j
y = for all y greater than some positive integer r. Let g
j
= q
j
∨ h
j
and g = ∗
t j =
1
g
j
. Then ϕ
g
x =
t
X
j = 1
ϕ
g
j
x = x
· t
X
j = 1
r
X
y= 1
ϕ
q
j
y y
h
y∗ j
x x ∈ N
m+
. 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 the p
i
’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 each j, ϕ
p
j
y with zero for all y greater than r and keep p
j
unchanged. 2. The Kornya approximation is a rescaling of the De Pril approximation such that for each j, q
j
sums to one like a probability distribution.
3. In the Hipp approximation we determine q
j
such that its moments up to order r match the corresponding moments of p
j
assuming that these moments exist and are finite, that is
∞
X
x=
x
i
q
j
x =
∞
X
x=
x
i
p
j
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 function f ∈ F
m
by another function g ∈ F
m
, we want some idea of the accuracy of the approximation. As a measure of accuracy we introduce ǫf, g =
P
x∈N
m
| f x − gx|
. 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 measure ǫ into 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 for ǫf, 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
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