INTEGRATION OF THE NORMAL POWER
APPROXIMATION
GOTTFRIED BERGER
Stamford, U.S.A.
1. Consider the set of functions
M*) = }(t-xydF(t),j
= 0,1, ...
(i)
Obviously, •K\[X) represents the net premium of the excess cover
over the priority x, and o2(x) = 7T2(#) — TZ\ (x) the variance thereof.
If a distribution function F(x) = 1 — no{x) is given, the set (1)
can be generated by means of the recursion formulae
n}{x) =—j
TCy,! (x), j =1,2,
...
(2)
2. Let us study the special class of d.fs. F(x) which satisfy
F(x) = ®(y) ^ - ^ = J e-1^ dt,
(3a)
* = A(y) = ^0 + ^ 1 ^ + . . . + hyk-
(3b)
where
If these conditions are met, the integrals (1) have the solution:
n,(x) = Ai(y) • (1 - Q(y)) + B,(y) • W(y).
(4)
A)(y) and B][y), respectively, are polynomials of rank jh and
jk — 1. Their coefficients are determined by the equations:
A'j(y)=-jti(y)-Aj_1(y),
B'}{y) = - ; A'(y) • B^^y) + A}{y) + yB}(y),
My) = 1,
(
B0(y) = 0.
)
^5j
The system (5) is obtained by differentiation of (4) with respect
to y, and observing (2).
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NORMAL POWER APPROXIMATION
O.I
3. The idea behind the normal power expansion is to apply (3 a)
as approximation, subject to a transformation x = &(y). Preferably
the parameters of A(y) should not depend on the particular choice
of y or x, but only on general characteristics of the d.f. F(x), such as
E = TTI(O), G = CT(O), yi = skewness and y2 = excess.
Kauppi and Ojantakanen [1] have tackled the problem to define
functions x = A(y), which make (3a) a reasonable approximation.
They found three suitable expressions A(y), one of them—credited
to Loimaranta—has the form (3b) and this one became known as
the normal power expansion. Under this method (see BeardPentikaeinen-Pesonen [2]) the coefficients (3* of (3b) are determined
by reversion of the Edgeworth expansion as follows:
x—E
yi
+ • ••
(6)
We may denote by NPk the normal power approximation, which
uses the first k terms of (6). Then, NPi corresponds to the well
known normal approximation.
NP2 uses the first line of (6) only, and NP3 everything which is
written out. Thus, NP3 requires the solution of a cubic equation.
4. The methods NP2 and NP3 were programmed in APL. This
required about 20 lines, including the subprograms to solve (5), the
quadratic or cubic equation (6), and to determine O(y) and &'{y)The cubic equation for NP3 has in some relevant cases 3 real
roots. It is necessary therefore to program rules to select the meaningful of several real roots y.
On an IBM 370, the CPU time needed to calculate no(x), ni(x)
and a(x) for a set of 6 values x was 1 second for NP2, and 2.4 seconds
for NP3.
5. The NP approximations were applied first to a life insurance
distribution similar to the one used by Ammeter [3]. The result is
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92
NORMAL POWER APPROXIMATION
shown in Table i. The exact values were obtained by another APL
program, the CPU time needed was:
28 seconds for t = 100
3.2 seconds for t = 10
1 seconds for t =
1
Thus, the approximation technique makes economical sense only,
if the number t of expected claims is at least 10 or more.
As another example, the non-industrial fire distribution from the
work of Bohman-Escher [4] was chosen. Table 2 shows a comparison
with correct values from [4], Table 3 some additional comparisons
with numerical results from Seal [5].
6. The comparisons contained in the Tables 1 to 3 point out the
following suggestions:
a) The integration does not seem to enlarge the error margin.
Thus, the NP technique can be applied to estimate stop loss gross
premiums.
b) NP2 yields quite reasonable results, if yi < 2. This corresponds
with previous experience.
c) NP3 does not generally produce better results than NP2. It
appears that NP3 is preferable only for lower values of x (say
x <E + 2a).
d) NP3 yields reasonable results even in the Life case with
yi = 4.3, but not in the Fire cases with yi = 3.5 and 3.8 (not even
in the vicinity of x = E). It may be that not only yi, but also the
relation Eyi/a is a criterion of goodness of fit.
REFERENCES
[1] KAUPPI, OJANTAKANEN (1969): "Approximations of the generalized
Poisson function"; Astin Bulletin.
[2] BEARD. PENTIKAEINEN, PESONEN (1969): "Risk Theory"; Methuen,
London.
[3] Ammeter (1955). "The calculation of premium rates for excess of loss and
stop loss reassurance treaties"; Arithbel, Brussels.
[4] BOHMAN, ESCHER (1964): "Studies in Risk Theory..."; Skand. Aktu.
Tidskr.
[5] SEAL (1971): "Numerical calculation of the Bohman-Escher family convolution-mixed negative binomial distribution functions"; MVSM.
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94
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NORMAL POWER APPROXIMATION
95
Table 3
Non-industrial fire distribution
x = E + \ "
/
/JO
IOOO
i
CT
7to(#) =
?
#/£
.5
1.0
0
I
3
5
100
I
.5
0
1.0
1
3
5
i—F(*)
NP3
Exact
(Seal)
NP2
0.59778
36710
13531
1839
0.5645
3805
1587
0.5825
3593
1347
229
194
104
117
124
250
28
29
113
0.4905
3540
1587
0.4846
297
51
238
90
103
129
150
120
in
122
0.5470
3448
1226
198
46
NP3
0/
/o
NP2
/o
3040
1189
56
94
97
98
100
106
117
89
88
97
The total claim distributions being tested have these statistical
measures:
t
ho
c/E
Yi
Y2
•175
•554
i-75i
.427
I-35I
4.271
.246
2-459
24.590
.218
.312
1.024
1.214
.811
2.010
2.624
1153
6.045
Life:
100
10
i
00
00
00
Non-industrial Fire:
1000
00
20
1
100
00
20
1
.690
•725
1.215
3-839
3-5O5
2.614
26.234
22.577
10.729
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