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A series for infinite time ruin probabilities
Insurance: Mathematics North-Holland
and Economics
129
4 (1985) 129-134
A series for infinite time ruin probabilities dent of { X, } with
,John A. BEEKMAN Dept. of Mothemo~icol IN 47306. USA
Sciences,
Boll Store Unic~ersiy, Mum-ie,
P{N(t)=n}=e-‘t’*/n!, Let 8>0,
~20,
anddenote
n=O,
1,2,
E(X)
byp,.
... . Let
Received 6 October 1984
This paper presents a series method for calculating the infinite time ruin func?ion. The terms of the series involve convolutions related to the claim size distribution. Approximations to the series are presented. with their error analyses. Three detailed examples are given. two of which involve the inverse Gaussian distribution. A discussionof that distribution is made. including the maximum likelihood estimators of its parameters. The relevance of the Poisson model for numbers of claims stochastic process is considered. Evidence from two very large studies is presented to support that model. at least for some portfolios.
Ktywor& distribution.
Infinite time ruin function. Inverse Poisson model for numbers of claims.
Define a sequence of convo!utions: 1
forxlo.
0
forx
1
fq(xj=
H:(x) = H*(x) = , i /
/
:[I
dy
-P(Y)]
\0
Gaussian
forxr0, forx
lH~_,(x-t)dH:(t) \O
forxz0,
nil.
forx
Then
1. Introduction
\k(u)=l-One of the three principal problems in coilective risk theory is the determination of the infinite time ruin probability. In the lengthy history of risk theory, several methods have been advanced. Sometimes a method which was expressed through one or more theorems has seemed impractical for a long time, until new mathematics was developed, allowing the method to become more practical. The purpose of this paper is to present such a method, a series technique for calculating the infinite time ruin function. Approximations to the series are given, with their error analyses. Three detailed examples are given, two of which involve the inverse Gaussian distribution.
&
,;,
(j+4W.
Remark 1. This theorem is proved in Beekman (19681, where the further assumption was made that P(x) was such that .,Jirnmx[l - P(x)]
= 0.
But from Theorem 3.1 of Bowers et al. (1983) extended to a mixed typed distribution, xjl -P(x)]
=xlPdP(t) .x
Since E(X)
i MJ, then
dP(t).
lim ?dP(t)=O. I ,t I -+ CCJ
2. Principal theorem
and hence lim .Y-.zx[l -0x)1=0.
Theorem 1. Assume Ihar { X, ) are non-negafiue
Remark 2. Again from Theorem 3.1 (lot. cit.) extended to a mixed type distribution,
valued, independetlr random variables, each with disrriburion P(x), wirh E(X) < 00. Further assume that { N(t), t 2 O> is a stochastic process indepen0167-6687/85/S3.30
@ 1985. Else&r
Science Publishers
E(X)
B.V. (North-Holland)
=iWx
d@(x) =ll”[l
- P(x)]
dx.
J.A. Beckman
130
/
Infinite
mne ruin prohahiliries
3. Approximation,
Therefore,
and error analysis
An important question is how does the series perform for more realistic claim distributions? This is answered in the following results.
forx 2 0.
2. Assume the same hypotheses Theorem 1. For any positive integer N,
Theorem Remark 3. The assumption that E{ N(r)) = t, rather than Xt for some X > 0, has not lost generality. We are assuming operational time, in which the unit of time is that amount of calendar time during which one claim is expected.
with
H;(u)/(l
Error<
If we view the random number N of claims for a single period as having a geometric distribution Remark
+B)N+‘.
4.
P[N=n]=&-j
]
II 1
&
,
and the claim distribution then
n=0,1,2,
. ..%
actually being H*(x),
“H;(x).
Proof.
The truncation error is
-!i[&l”H~(u)* i+e ,,=gL For notational purposes, let W,, i = 1, 2, . . . , be independent random variables, each with distribution function H*(x). Now Hz(u)< H;(u) for n=N, N+l, . . ..since H,:(u)=P{W,+
One cannot conclude that e(u) = 1 - F( t(, 1) because the claim distribution for the X, random variables within ‘k(u) is P(x), not the H*(x) distrPution of the & random variables in F( u, 1). Example
1.
Let P(x)=
1 -eeA’,
x 2 0, A > 0.
.*.
+W,+
WN+,
+ *** +?qIu} IP{W,-t
.**
-I-Wx
= H;(u).
Using this result, we obtain
TruncationError5
z 1
Then
H;(u)
E
+
8
,r=N+,
(1
+
e)n
H;(u)
H*(x)=AJle-Ald?,=l-e-~‘=P(x) 0 for all x L 0. In a later section we will return to the relation between P(x) and H*(x) for a more general P(x). The nth convolution is
~16’”“(n_l)! )#-
H”*(x)
I
dw.
= (1 + aIN+’ ’ Remark.
if
l-1+8Ln$O[&]“H;(u)=&. then ‘k(u) -< E, and Truncation
With an interchange operations, \k(u)
as for
=
1 _
of the series and integral
.& -
Error I E.
As a function of N, 1 - Above Sum is decreasing in N. Furthermore, Proof.
lim [ 1 - Above Sum] = !P( u).
N--w
=A
exp(-$). 4. Use of inverse Gaussian distribution
This agrees with the result obtained by inverting a suitable Laplace transform (see pages 44-45 of Beekman (1974)).
We will now illustrate results obtainable if one assumes that H*(x) is an inverse Gaussian distri-
J. A. Beckman
/
Injinite
bution. References for this section are ter Berg (1980), and Seal (1978). The inverse Gaussian distribution B(x) can be expressed as follows:
where x > 0, p > 0, X > 0, and @( .) is the standardized Normal distribution function. The mean and variance for E(x) are I_Land p3/X. As X/p increases,
131
rime ruin prohrrhili~ie.r
Table 1 I,
1 2 3 4 5 6 7 8 9 10
B”*(300)
[1/1.6]“B”‘(300)
1.oooo 0.9998 0.9864 0.9713 0.9429 0.8962 0.8289 0.7357 0.6217 0.5000
0.6250 0.3905 0.2408 0.1482 0.0899 0.0534 0.0309 0.0171 0.0090 0.0045
BJ*(l5O)=@[[;]“‘?[!$!_l]]
h x-/J -I” J- P
+e”(l
approaches a standarized normal random variable. For ourpurposes, what is really important is that the convolutions of B(x) are easy to calculate. To be specific, the k-th convolution
-@[[;)]“Z[~~+l]])
+ 0.8940, B4(150)
+ 0.6736
(with the braced B5*(150) Finally,
term
+ 0), Bh*(150)
= 0.5000, we obtain
YP(150) + 1 - y$ { 1 + i$ (0.9906)
For computation purposes, notice that the quantity within braces is monotonically decreasing in k. We will now calculate several values of ‘k( u j by assuming the claim distribution function P(y) is such that H*(x)=;jn’[l-P(y)] Then
H’*(xj=
d,r=B(x). B’*(x).
Example 2. Let p = 30 units (e.g. $30000.00). Assume X=30, and B=O.6. Let u= 150 (=x). To show the ease of the method, we will begin by showing all of the calculations. The we will shorten what is displayed. 8(150)=@[[#‘2(5-1)]
+ 0.3264.
+ [ A] 2(0.9590)
-I- [ &] ‘(0.8940)
+ [ $Z]4(0.6736)
+ [#(0.5000)
+ [ +$(0.3264)}
+ 0.1070 with Truncation
0.3264 Error 5 I_ = 0.0122. (1.6)’
Next, let u = 300 (=x). summarized in Table 1. It follows that *(300)+ tion Error < 0.0028.
The calculations 0.0215,
with Trunca-
Example 3 (motivated by values for p and h on page 5 1 of Seal (1978)). Choose p= 30, X= 60, and 8=0.6. Let u = 150. By utilizing values for five convolutions, ~P(150) + 0.1063, with Truncation Error < 0.0298. Next, choose u = 225. After computing eight convolutions, one finds that 9(225) + 0.0372, with Truncation Error 2 0.0058.
+e’(l-@[[+]“‘(5+1)]) + 0.9906,
5. Properties of our assumed claim distribution
B2*(15o)=(P[[4]“2(~-1)] What does it require +e4(1 + 0.9590,
are
of P(y)
-@[[$]‘“[$$+I]]) $i’!l 1
- P(y)]
dy = B(x)?
for
J.A. Beekman / Infinite time ruin probabilities
132
If one differentiates one obtains
both sides of that equation,
for x > 0, p > 0, X > 0. This expresses the tail of the claim distribution (divided by p,) as the inverse Gaussian density. A further fact is gained by differentiating both sides again, and simplifying to obtain P(X) -----b(x) PI
I
$+i3;--$ , [
x > 0.
This expresses the claim density in terms of the inverse Gaussian density. A sample of claim values can be used to estimate p,. Therefore, if y and h are estimated, we have an approximation to the claim density p(x). Estimators for those parameters and other results are presented by Tweedie (1957) and Schrodinger (1915). The maximum like lihood estimator of p is the sample mean x,,. The parameter A is an inverse measure of relative dispersion. If a sample produces values x,. x2, . . . , x,,, then the maximum likelihood estimator of l/X is given by
0
H*(x)=$lX[l
-P(y)]
1.0 of b(y)
curves.
1.5
2.0
dy.
1
However, there is a more helpful approach to the estimation problem, based on ideas from risk theory. As explained in Section 12.5, and Problem 12 for that section of Bowers et al. (1983), H*(x) is
c
0.5
Fig. 1. Characteristics
The concavity of the function x-’ guarantees that A-’ will be positive. Although he would not have referred to them as maximum likelihood estimators, the previous estimators r; and 1-l for p and X-’ were presented in the 1915 paper by Schrodinger which concerned the probability distribution of the first passage time in Brownian motion. One could now say his paper derived the probability distribution for the first passage time of a Wiener process with constant drift. The unbiased estimator for X-’ is the above expression with l/n replaced by l/(n - 1). Confidence intervals for X can be obtained, using the fact that l/x = (l/ nX)x*(n - 1 d.f.). Thus for suitable values A and B from a Chi-square distribution with n - 1 degrees of freedom, [X/n)A, (X/n)B] could be a 95% confidence interval for X. A portion of Figure 1 (Tweedie (1957)) reveals the characteristics of b(y) curves. In each case p = 1 which for our purposes corresponds to using the mean claim amount (p,) as the unit of money. The non-negative, skewed characteristics of the densities are valuable in modelling claim densities. A problem still exists as the actuary would be obtaining sample values for P(x), not
2.5
Y
J.A. Beekman /
Infinite rime ruin prohahiliiies
exactly the distribution for the first surplus below the initial level. Moreover,
Table 2 Distribution
E[W]=~mxdH*(x)=~
Number per day
and
65 66 71 76 81 86 91 96 101 106
where pi = l,“x’d P( x). By equating H*(x) to B(x), one obtains kzp
and
133
by number
of deaths
or Iess to 70 to75 to 80 tow to 90 to95 to 100 to 105 or more
of deaths
per day
Number of days Expected number of days, observed in 1964 Poisson distribution 5 19 38 66 78 66 46 32 10 6
6.2 16.5 38.2 63.9 78.2 71.5 49.4 26.4 10.9 4.8
366
366.0
2P, Total
If i)t, fi2, and & represent sample moments of the claim distribution, we obtain the following estimators for p and A-‘:
Another reference to the inverse Gaussian distribution is (Folks and Chhikara (1978)). In addition to a comprehensive review of the history and theory of this distribution, the authors fit such distributions to four sets of data relating to (a) shelflife of a food product, (b) fracture toughness of MIG welds, (c) precipitation at one geographic site, (d) runof’ amounts at Jug Bridge, Maryland. The discussants also provide examples where the inverse Gaussian distribution was fitted to survival times in some cancer studies, and used in the analyses of two other clinical trials.
per day was 84.4 with an actual variance of 89.4. Table 2 shows the actual data and expected values from a Poisson distribution, with mean = 84.4. A chi-square test was made of the fit of the Poisson distribution. The differences were not significant. The Poisson model for deaths was judged to be adequate. A second reference for this subject is (Lewis (1972)). That paper describes an analysis of 8 683 paid death claims for 1965 and the corresponding 6939 deaths for that block of business. The conclusion was that “Under the 5% Chi Square test the Poisson assumption cannot be rejected as a means of approximating the distribution of the number of deaths or number of paid death claims within a single calendar year.”
6. Relevance of the Poisson model
Acknowledgements
for {N(t),
220)
Seal (1983) casts doubt on the relevance of the Poisson model for { N(t), t 2 O}. This author continues to place faith in that model for some insurance portfolios because of the following two significant studies. On October 29.1968 at a Claim Fluctuations Workshop held at the Annual Meeting of the Society of Actuaries in Washington, DC, David G. Halmstad reported on a mammoth study performed at the New York Life Insurance Company. One of the purposes of the study was to determine the adequacy of a Poisson distribution in modelling the number of deaths incurred by a life insurance company. The study involved 30 892 deaths in 1964. The actuai mean number of deaths
The author wishes to express his gratitude to Peter ter Berg for suggesting the use of the inverse Gaussian distribution in this research. I would also like to acknowledge the help of Ball State University through an Academic Year Research Grant. A preliminary version of this paper was presented at an Insurance Research Seminar at the University of Waterloo, Ontario, Canada, and Harry Panjer and Gordon Willmot made several helpful suggestions at that time.
References Beekman,
J.A. (1968). Collective
Society of Actuaries
risk results.
20, 182-199.
Transactions of the
134
J.A. Beekman / Infinite rime ruin probabilities
Beekman, J.A. (1974). Two Stochasric Processes. Almqvist and Wiksell. Stockholm; also Halsted Press (c/o John Wiley and Sons), New York. Berg. P. ter (1980). Two pragmatic approaches to loglinear claim cost analysis. Astin BuNerin 11, 77-90. Bowers, N.L.. Gerber, H.U.. Hickman. J.C.. Jones. D.A., and Nesbitt. C.J. (1983). Arruarial Marhemarics. Soceity of Actuaries. Itasca. IL. Folks, J.L.. and Chhikara, R.S. (1978). The inverse Gaussian distribution and its statistical application - A review (with discussion). Journal OJ the Royal Sratisrical Society Ser. B 40. 263-289. Halmstad, D.G. (1968). Presentation at Claim Fluctuations Workshop of Society of Actuaries in Washington, DC (unpublished).
Lewis, J.L. Jr. (1972). Poisson Deaths Assumption 1000 Companies and Four Seasons Test. Acruarial Research Clearing House, Issue 1972.4, Society of Actuaries, Itasca, IL. Schriidinger, E. (1915). Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung, Physikalische Zeitschrift 16. 289-295. Seal, H.L. (1978). From aggregate claims distribution to probability of ruin. Astin Bulletin 10, 47-53. Seal, H.L. (1983). The Poisson process: Its failure in risk theory. Insurance: Marhematics and Economics 2 (1983) 287-288. Tweedie, M.C.K. (1957). Statistical properties of inverse Gaussian distributions. I and II. Annals of Marhematical Statistics 28. 362-377 and 696-705.
and Economics
129
4 (1985) 129-134
A series for infinite time ruin probabilities dent of { X, } with
,John A. BEEKMAN Dept. of Mothemo~icol IN 47306. USA
Sciences,
Boll Store Unic~ersiy, Mum-ie,
P{N(t)=n}=e-‘t’*/n!, Let 8>0,
~20,
anddenote
n=O,
1,2,
E(X)
byp,.
... . Let
Received 6 October 1984
This paper presents a series method for calculating the infinite time ruin func?ion. The terms of the series involve convolutions related to the claim size distribution. Approximations to the series are presented. with their error analyses. Three detailed examples are given. two of which involve the inverse Gaussian distribution. A discussionof that distribution is made. including the maximum likelihood estimators of its parameters. The relevance of the Poisson model for numbers of claims stochastic process is considered. Evidence from two very large studies is presented to support that model. at least for some portfolios.
Ktywor& distribution.
Infinite time ruin function. Inverse Poisson model for numbers of claims.
Define a sequence of convo!utions: 1
forxlo.
0
forx
1
fq(xj=
H:(x) = H*(x) = , i /
/
:[I
dy
-P(Y)]
\0
Gaussian
forxr0, forx
lH~_,(x-t)dH:(t) \O
forxz0,
nil.
forx
Then
1. Introduction
\k(u)=l-One of the three principal problems in coilective risk theory is the determination of the infinite time ruin probability. In the lengthy history of risk theory, several methods have been advanced. Sometimes a method which was expressed through one or more theorems has seemed impractical for a long time, until new mathematics was developed, allowing the method to become more practical. The purpose of this paper is to present such a method, a series technique for calculating the infinite time ruin function. Approximations to the series are given, with their error analyses. Three detailed examples are given, two of which involve the inverse Gaussian distribution.
&
,;,
(j+4W.
Remark 1. This theorem is proved in Beekman (19681, where the further assumption was made that P(x) was such that .,Jirnmx[l - P(x)]
= 0.
But from Theorem 3.1 of Bowers et al. (1983) extended to a mixed typed distribution, xjl -P(x)]
=xlPdP(t) .x
Since E(X)
i MJ, then
dP(t).
lim ?dP(t)=O. I ,t I -+ CCJ
2. Principal theorem
and hence lim .Y-.zx[l -0x)1=0.
Theorem 1. Assume Ihar { X, ) are non-negafiue
Remark 2. Again from Theorem 3.1 (lot. cit.) extended to a mixed type distribution,
valued, independetlr random variables, each with disrriburion P(x), wirh E(X) < 00. Further assume that { N(t), t 2 O> is a stochastic process indepen0167-6687/85/S3.30
@ 1985. Else&r
Science Publishers
E(X)
B.V. (North-Holland)
=iWx
d@(x) =ll”[l
- P(x)]
dx.
J.A. Beckman
130
/
Infinite
mne ruin prohahiliries
3. Approximation,
Therefore,
and error analysis
An important question is how does the series perform for more realistic claim distributions? This is answered in the following results.
forx 2 0.
2. Assume the same hypotheses Theorem 1. For any positive integer N,
Theorem Remark 3. The assumption that E{ N(r)) = t, rather than Xt for some X > 0, has not lost generality. We are assuming operational time, in which the unit of time is that amount of calendar time during which one claim is expected.
with
H;(u)/(l
Error<
If we view the random number N of claims for a single period as having a geometric distribution Remark
+B)N+‘.
4.
P[N=n]=&-j
]
II 1
&
,
and the claim distribution then
n=0,1,2,
. ..%
actually being H*(x),
“H;(x).
Proof.
The truncation error is
-!i[&l”H~(u)* i+e ,,=gL For notational purposes, let W,, i = 1, 2, . . . , be independent random variables, each with distribution function H*(x). Now Hz(u)< H;(u) for n=N, N+l, . . ..since H,:(u)=P{W,+
One cannot conclude that e(u) = 1 - F( t(, 1) because the claim distribution for the X, random variables within ‘k(u) is P(x), not the H*(x) distrPution of the & random variables in F( u, 1). Example
1.
Let P(x)=
1 -eeA’,
x 2 0, A > 0.
.*.
+W,+
WN+,
+ *** +?qIu} IP{W,-t
.**
-I-Wx
= H;(u).
Using this result, we obtain
TruncationError5
z 1
Then
H;(u)
E
+
8
,r=N+,
(1
+
e)n
H;(u)
H*(x)=AJle-Ald?,=l-e-~‘=P(x) 0 for all x L 0. In a later section we will return to the relation between P(x) and H*(x) for a more general P(x). The nth convolution is
~16’”“(n_l)! )#-
H”*(x)
I
dw.
= (1 + aIN+’ ’ Remark.
if
l-1+8Ln$O[&]“H;(u)=&. then ‘k(u) -< E, and Truncation
With an interchange operations, \k(u)
as for
=
1 _
of the series and integral
.& -
Error I E.
As a function of N, 1 - Above Sum is decreasing in N. Furthermore, Proof.
lim [ 1 - Above Sum] = !P( u).
N--w
=A
exp(-$). 4. Use of inverse Gaussian distribution
This agrees with the result obtained by inverting a suitable Laplace transform (see pages 44-45 of Beekman (1974)).
We will now illustrate results obtainable if one assumes that H*(x) is an inverse Gaussian distri-
J. A. Beckman
/
Injinite
bution. References for this section are ter Berg (1980), and Seal (1978). The inverse Gaussian distribution B(x) can be expressed as follows:
where x > 0, p > 0, X > 0, and @( .) is the standardized Normal distribution function. The mean and variance for E(x) are I_Land p3/X. As X/p increases,
131
rime ruin prohrrhili~ie.r
Table 1 I,
1 2 3 4 5 6 7 8 9 10
B”*(300)
[1/1.6]“B”‘(300)
1.oooo 0.9998 0.9864 0.9713 0.9429 0.8962 0.8289 0.7357 0.6217 0.5000
0.6250 0.3905 0.2408 0.1482 0.0899 0.0534 0.0309 0.0171 0.0090 0.0045
BJ*(l5O)=@[[;]“‘?[!$!_l]]
h x-/J -I” J- P
+e”(l
approaches a standarized normal random variable. For ourpurposes, what is really important is that the convolutions of B(x) are easy to calculate. To be specific, the k-th convolution
-@[[;)]“Z[~~+l]])
+ 0.8940, B4(150)
+ 0.6736
(with the braced B5*(150) Finally,
term
+ 0), Bh*(150)
= 0.5000, we obtain
YP(150) + 1 - y$ { 1 + i$ (0.9906)
For computation purposes, notice that the quantity within braces is monotonically decreasing in k. We will now calculate several values of ‘k( u j by assuming the claim distribution function P(y) is such that H*(x)=;jn’[l-P(y)] Then
H’*(xj=
d,r=B(x). B’*(x).
Example 2. Let p = 30 units (e.g. $30000.00). Assume X=30, and B=O.6. Let u= 150 (=x). To show the ease of the method, we will begin by showing all of the calculations. The we will shorten what is displayed. 8(150)=@[[#‘2(5-1)]
+ 0.3264.
+ [ A] 2(0.9590)
-I- [ &] ‘(0.8940)
+ [ $Z]4(0.6736)
+ [#(0.5000)
+ [ +$(0.3264)}
+ 0.1070 with Truncation
0.3264 Error 5 I_ = 0.0122. (1.6)’
Next, let u = 300 (=x). summarized in Table 1. It follows that *(300)+ tion Error < 0.0028.
The calculations 0.0215,
with Trunca-
Example 3 (motivated by values for p and h on page 5 1 of Seal (1978)). Choose p= 30, X= 60, and 8=0.6. Let u = 150. By utilizing values for five convolutions, ~P(150) + 0.1063, with Truncation Error < 0.0298. Next, choose u = 225. After computing eight convolutions, one finds that 9(225) + 0.0372, with Truncation Error 2 0.0058.
+e’(l-@[[+]“‘(5+1)]) + 0.9906,
5. Properties of our assumed claim distribution
B2*(15o)=(P[[4]“2(~-1)] What does it require +e4(1 + 0.9590,
are
of P(y)
-@[[$]‘“[$$+I]]) $i’!l 1
- P(y)]
dy = B(x)?
for
J.A. Beekman / Infinite time ruin probabilities
132
If one differentiates one obtains
both sides of that equation,
for x > 0, p > 0, X > 0. This expresses the tail of the claim distribution (divided by p,) as the inverse Gaussian density. A further fact is gained by differentiating both sides again, and simplifying to obtain P(X) -----b(x) PI
I
$+i3;--$ , [
x > 0.
This expresses the claim density in terms of the inverse Gaussian density. A sample of claim values can be used to estimate p,. Therefore, if y and h are estimated, we have an approximation to the claim density p(x). Estimators for those parameters and other results are presented by Tweedie (1957) and Schrodinger (1915). The maximum like lihood estimator of p is the sample mean x,,. The parameter A is an inverse measure of relative dispersion. If a sample produces values x,. x2, . . . , x,,, then the maximum likelihood estimator of l/X is given by
0
H*(x)=$lX[l
-P(y)]
1.0 of b(y)
curves.
1.5
2.0
dy.
1
However, there is a more helpful approach to the estimation problem, based on ideas from risk theory. As explained in Section 12.5, and Problem 12 for that section of Bowers et al. (1983), H*(x) is
c
0.5
Fig. 1. Characteristics
The concavity of the function x-’ guarantees that A-’ will be positive. Although he would not have referred to them as maximum likelihood estimators, the previous estimators r; and 1-l for p and X-’ were presented in the 1915 paper by Schrodinger which concerned the probability distribution of the first passage time in Brownian motion. One could now say his paper derived the probability distribution for the first passage time of a Wiener process with constant drift. The unbiased estimator for X-’ is the above expression with l/n replaced by l/(n - 1). Confidence intervals for X can be obtained, using the fact that l/x = (l/ nX)x*(n - 1 d.f.). Thus for suitable values A and B from a Chi-square distribution with n - 1 degrees of freedom, [X/n)A, (X/n)B] could be a 95% confidence interval for X. A portion of Figure 1 (Tweedie (1957)) reveals the characteristics of b(y) curves. In each case p = 1 which for our purposes corresponds to using the mean claim amount (p,) as the unit of money. The non-negative, skewed characteristics of the densities are valuable in modelling claim densities. A problem still exists as the actuary would be obtaining sample values for P(x), not
2.5
Y
J.A. Beekman /
Infinite rime ruin prohahiliiies
exactly the distribution for the first surplus below the initial level. Moreover,
Table 2 Distribution
E[W]=~mxdH*(x)=~
Number per day
and
65 66 71 76 81 86 91 96 101 106
where pi = l,“x’d P( x). By equating H*(x) to B(x), one obtains kzp
and
133
by number
of deaths
or Iess to 70 to75 to 80 tow to 90 to95 to 100 to 105 or more
of deaths
per day
Number of days Expected number of days, observed in 1964 Poisson distribution 5 19 38 66 78 66 46 32 10 6
6.2 16.5 38.2 63.9 78.2 71.5 49.4 26.4 10.9 4.8
366
366.0
2P, Total
If i)t, fi2, and & represent sample moments of the claim distribution, we obtain the following estimators for p and A-‘:
Another reference to the inverse Gaussian distribution is (Folks and Chhikara (1978)). In addition to a comprehensive review of the history and theory of this distribution, the authors fit such distributions to four sets of data relating to (a) shelflife of a food product, (b) fracture toughness of MIG welds, (c) precipitation at one geographic site, (d) runof’ amounts at Jug Bridge, Maryland. The discussants also provide examples where the inverse Gaussian distribution was fitted to survival times in some cancer studies, and used in the analyses of two other clinical trials.
per day was 84.4 with an actual variance of 89.4. Table 2 shows the actual data and expected values from a Poisson distribution, with mean = 84.4. A chi-square test was made of the fit of the Poisson distribution. The differences were not significant. The Poisson model for deaths was judged to be adequate. A second reference for this subject is (Lewis (1972)). That paper describes an analysis of 8 683 paid death claims for 1965 and the corresponding 6939 deaths for that block of business. The conclusion was that “Under the 5% Chi Square test the Poisson assumption cannot be rejected as a means of approximating the distribution of the number of deaths or number of paid death claims within a single calendar year.”
6. Relevance of the Poisson model
Acknowledgements
for {N(t),
220)
Seal (1983) casts doubt on the relevance of the Poisson model for { N(t), t 2 O}. This author continues to place faith in that model for some insurance portfolios because of the following two significant studies. On October 29.1968 at a Claim Fluctuations Workshop held at the Annual Meeting of the Society of Actuaries in Washington, DC, David G. Halmstad reported on a mammoth study performed at the New York Life Insurance Company. One of the purposes of the study was to determine the adequacy of a Poisson distribution in modelling the number of deaths incurred by a life insurance company. The study involved 30 892 deaths in 1964. The actuai mean number of deaths
The author wishes to express his gratitude to Peter ter Berg for suggesting the use of the inverse Gaussian distribution in this research. I would also like to acknowledge the help of Ball State University through an Academic Year Research Grant. A preliminary version of this paper was presented at an Insurance Research Seminar at the University of Waterloo, Ontario, Canada, and Harry Panjer and Gordon Willmot made several helpful suggestions at that time.
References Beekman,
J.A. (1968). Collective
Society of Actuaries
risk results.
20, 182-199.
Transactions of the
134
J.A. Beekman / Infinite rime ruin probabilities
Beekman, J.A. (1974). Two Stochasric Processes. Almqvist and Wiksell. Stockholm; also Halsted Press (c/o John Wiley and Sons), New York. Berg. P. ter (1980). Two pragmatic approaches to loglinear claim cost analysis. Astin BuNerin 11, 77-90. Bowers, N.L.. Gerber, H.U.. Hickman. J.C.. Jones. D.A., and Nesbitt. C.J. (1983). Arruarial Marhemarics. Soceity of Actuaries. Itasca. IL. Folks, J.L.. and Chhikara, R.S. (1978). The inverse Gaussian distribution and its statistical application - A review (with discussion). Journal OJ the Royal Sratisrical Society Ser. B 40. 263-289. Halmstad, D.G. (1968). Presentation at Claim Fluctuations Workshop of Society of Actuaries in Washington, DC (unpublished).
Lewis, J.L. Jr. (1972). Poisson Deaths Assumption 1000 Companies and Four Seasons Test. Acruarial Research Clearing House, Issue 1972.4, Society of Actuaries, Itasca, IL. Schriidinger, E. (1915). Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung, Physikalische Zeitschrift 16. 289-295. Seal, H.L. (1978). From aggregate claims distribution to probability of ruin. Astin Bulletin 10, 47-53. Seal, H.L. (1983). The Poisson process: Its failure in risk theory. Insurance: Marhematics and Economics 2 (1983) 287-288. Tweedie, M.C.K. (1957). Statistical properties of inverse Gaussian distributions. I and II. Annals of Marhematical Statistics 28. 362-377 and 696-705.