c Allerton Press, Inc., 2008.
ISSN 1066-5307, Mathematical Methods of Statistics, 2008, Vol. 17, No. 1, pp. 35–43. ⃝
Nonparametric Estimation of Ruin Probabilities
Given a Random Sample of Claims
R. Mnatsakanov1* , L. L. Ruymgaart, and F. H. Ruymgaart2**
1
2
Dept. of Statist., West Virginia Univ., USA
Dept. of Math. and Statist., Texas Tech Univ., USA
Received September 26, 2007; in final form, January 30, 2008
Abstract—In this paper the well-known insurance ruin problem is reconsidered. The ruin probability is estimated in the case of an unknown claims density, assuming a sample of claims is given.
An important step in the construction of the estimator is the application of a regularized version of
the inverse of the Laplace transform. A rate of convergence in probability for the integrated squared
error (ISE) is derived and a simulation study is included.
Key words: nonparametric estimation, ruin probabilities, regularized inverse Laplace transform.
2000 Mathematics Subject Classification: primary 62G05; secondary 91B30.
DOI: 10.3103/S1066530708010031
1. INTRODUCTION AND PRELIMINARIES
The insurance ruin problem will be considered here in the nonparametric situation, where the claims
density is assumed to be entirely unknown, but a random sample from the density is given. Based on this
sample the ruin probability function will be estimated. A rate of convergence in probability of the ISE
over an interval of arbitrary finite length will be obtained for this estimator. The method of the Laplace
transform will be employed, where this transform is considered as a mapping of L2 (0, ∞) into itself. On
the one hand the Laplace transform of the claims density is not known in the present case, and can be
only estimated from the data. On the other hand the inverse Laplace transform, which has to be applied
to recover the ruin probabilities, is unbounded. It is therefore of crucial importance that a regularized
version of this inverse be used. Here the spectral-cut-off regularization will be applied (Kress [1]).
To describe the problem more precisely let us make the usual assumptions that claims come in
according to a Poisson process N (t), t ≥ 0, with rate λ > 0, and that the claim amounts X1 , X2 , . . .
are independent and identically distributed with density f with respect to Lebesgue measure on [0, ∞),
having finite mean µ > 0 and variance σ 2 > 0. The process N is supposed to be independent of the
sequence X1 , X2 , . . . If the company receives income from the policies at a constant rate c > 0 per unit
time, the probability we are interested in is
N (t)
!
#
"
G(u) = P u + ct −
Xk ≥ 0 ∀ t ≥ 0 ,
k=1
u ≥ 0.
(1.1)
Two simple assumptions will be made without further reference: in the first place
λµ
< 1,
c
and secondly G has a measurable, bounded derivative
0<ρ=
0 ≤ G′ (u) = g(u),
*
**
g(u) ≤ M < ∞
E-mail: [email protected]
E-mail: [email protected]
35
(1.2)
∀ u ≥ 0.
(1.3)
36
MNATSAKANOV et al.
Note that G′ (u) ≥ 0 if it exists, since G is nondecreasing, and that (1.2) entails positivity of the
expectation of the stochastic process in the curly brackets in (1.1).
It is well known that G satisfies the integro-differential equation
$u
λ
λ
′
G (u) = G(u) −
G(u − x)f (x) dx,
u ≥ 0,
(1.4)
c
c
0
with
G(0) = 1 − ρ ≤ G(u) ≤ 1,
u ≥ 0,
(1.5)
and G nondecreasing on [0, ∞). For an elementary introduction to the ruin problem see, for instance,
Ross [2]; see also Beekman [3], Booth et al. [4], and Jarman [5]. For any suitable h : [0, ∞) → R let us
define the Laplace transform L as usual by
$∞
s ≥ 0,
(1.6)
(Lh)(s) = hL (s) = e−su h(u) du,
0
and observe that application of L to (1.4) yields the equivalent relation
GL (s) =
1−ρ
,
D(s)
D(s) = s −
λ
{1 − fL (s)},
c
s ≥ 0.
(1.7)
One usually assumes the claims density f to be given and tries to obtain G explicitly from either (1.4)
or (1.7) for special choices of f (see, for instance, Jarman [5]). In such instances the exact inverse of the
Laplace transform can be employed. During the last five decades many methods have been developed for
calculating G(u). For example, Thorin and Wikstad [6] used Piessens [7] inversion method of the Laplace
transform. Other results using the numerical inversions of the Laplace and Fourier transforms can be
found, for example, in Seal [8, 9], Lima et al. [10], and Embrechts et al. [11]. Let us mention also the
articles of Embrechts and Veraverbeke [12], and Usabel [13], where the approximation of G(u) for large
values of the claims sizes are obtained. In Asmussen and Binswanger [14] three algorithms of simulation
of the ruin probabilities in the classical risk model with initial large reserve u and subexponential
distribution of claims have been proposed.
In this paper we will construct an estimator, respectively an approximation of G, if either f is
unknown, but a sample
X1 , . . . , Xn
i.i.d. (f )
(1.8)
of size n from f is given, or f is still known, but G is hard to explicitly solve from (1.4) or (1.7). Both
cases will be dealt with in essentially the same manner. Our main purpose is, however, to consider the
random case when the sample in (1.8) is given; it should once more be noted that here regularization of
the inverse Laplace is a very important issue.
If f is not known, fL and ρ can be estimated by their empirical counterparts
n
1 " −sXi
fˆL (s) =
e
,
n
i=1
and thus an estimator
ĜL (s) =
1 − ρ̂
D̂(s)
,
D̂(s) = s −
ρ̂ =
λ
X,
c
λ
{1 − fˆL (s)},
c
(1.9)
s ≥ 0,
(1.10)
of GL in (1.7) is obtained. Although ĜL is close to GL , there is no guarantee that L−1 ĜL will be close
to G, because typically L−1 is not continuous. More specifically, L as an operator mapping L2 (0, ∞)
into itself is known to have an unbounded inverse. Hence recovering a function from a noisy image is illposed and requires application of a regularized inverse. The functions GL and ĜL will be slightly modified
so as to ensure they are in L2 (0, ∞), and then G will be recovered by first applying a regularized inverse
to this modified ĜL ; see Section 2 for details.
MATHEMATICAL METHODS OF STATISTICS
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NONPARAMETRIC ESTIMATION
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A tractable regularized inverse of the Laplace transform L has been constructed in Chauveau et
al. [15] by relating L to convolution on the space of functions on (0, ∞) that are square integrable with
respect to the Haar measure on (0, ∞) as a group under multiplication. See also Gilliam et al. [16]. This
construction is briefly reviewed in Section 2.
Even in the case, where f is known and hence GL is exactly known, so that the recovery procedure
is not ill-posed, application of a regularized inverse might be a useful alternative to employing the exact
inversion formula, which is often awkward to deal with. In Section 3 we also briefly comment on the
approximation of G for known f by using a regularized inverse.
In Section 4 we derive a speed of convergence in probability of the ISE for the estimator of G when
f is unknown and briefly consider the performance of the approximation of G for known f . Finally, in
Section 5 some simulations are performed that turn out to be quite satisfactory.
2. A REGULARIZED INVERSE OF THE LAPLACE TRANSFORM
Let us consider (0, ∞) as a locally compact Abelian group under multiplication, written additively:
a · b = a ⊕ b for a, b ∈ (0, ∞). The Haar measure H on this group has density% (dH/dλ)(x) = x−1 ,
∞
x > 0, with respect to Lebesgue measure. Let ψ : (0, ∞) → R be measurable with 0 |ψ| dH < ∞, and
%
∞
L2H (0, ∞) = {h : (0, ∞) → R measurable, 0 h2 dH < ∞}. Then a convolution operator
Kψ : L2H (0, ∞) → L2H (0, ∞)
can be defined by
$∞ & '
x
1
(Kψ h)(x) = (ψ ! h)(x) = ψ
h(y) dy,
y
y
0
x ∈ (0, ∞), h ∈ L2H (0, ∞).
(2.1)
The operator Kψ can be reduced to a multiplication operator by applying a suitable unitary Fourier–
Plancherel transform. The operator Kψ is then essentially multiplication by the Fourier transform ψ̃ of
the kernel ψ. If ψ̃ ̸= 0 a.e., the inverse of Kψ exists and reduces to multiplication by the reciprocal 1/ψ̃.
Now let us consider the Laplace transform L as a mapping of L2 (0, ∞) into itself. A change of
variables yields the identity (cf. (1.6))
'&
& ''
$∞ &(
1
s −s/u
1
1
1
√ h
(Lh)(s) = √
e
du,
s ≥ 0,
(2.2)
s
u
u
u
u
0
L2 (0, ∞),
h∈
which shows that apart from some isometries the Laplace transform is a convolution in
L2H (0, ∞) with the kernel
√
ψ(x) = x e−x ,
x > 0.
(2.3)
Exploiting this relation it is possible to construct a family of operators
2
2
L−1
α : L (0, ∞) → L (0, ∞),
α > 0,
(2.4)
with the property that each L−1
α is bounded with
∥L−1
α ∥
√
1
π
≤ ∥L∥ =
,
α
α
α > 0,
(2.5)
and such that
∥L−1
α Lh − h∥ → 0
as α ↓ 0 ∀ h ∈ L2 (0, ∞).
(2.6)
Each L−1
α is an integral operator explicitly given by
(L−1
α h)(u)
1
= 2
π
$∞ $∞
0
1
Rα (s) √ e−uts h(t) dt ds,
s
0
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u ∈ (0, ∞),
(2.7)
38
MNATSAKANOV et al.
h ∈ L2 (0, ∞), where
Rα (s) =
1
2
$
(cosh πx) cos(x log s) dx,
s ∈ (0, ∞),
(2.8)
{ρ(x)≥α}
with
π
,
x ∈ R.
cosh πx
For details and further information see Gilliam et al. [16] and Chauveau et al. [15].
ρ(x) =
(2.9)
3. RECOVERING THE PROBABILITY FUNCTION
The Laplace transform fL in (1.6) satisfies
$∞
dk
k
fL (s) = (−1)
e−sx xk f (x) dx, k = 1, 2,
fL′ (0) = −µ,
fL′′ (0) = σ 2 + µ2 ,
dsk
0
and, since f ≥ 0,
By Taylor expansion we obtain
σ 2 + µ2 ≥ fL′′ (s) ≥ 0 ∀ s ≥ 0.
1
1 λ 2 ′′
D(s) = D(0) + s · D′ (0) + s2 D′′ (s̃) = (1 − ρ)s +
s fL (s̃),
2
2c
which entails (cf. (1.2))
D(s) ≥ (1 − ρ)s
and, because 0 ≤ fL (s) ≤ 1 for all s ≥ 0,
D(s) ≤ s
(3.1)
0 ≤ s̃ ≤ s,
∀ s ≥ 0,
(3.2)
∀ s ≥ 0.
(3.3)
It is obvious from (1.5) that G ∈
/ L2 (0, ∞) and it follows from (1.7) and (3.3) that GL ∈
/ L2 (0, ∞) due
to the behavior of D near 0. Similarly we have (1 − ρ̂)s ≤ D̂(s) ≤ s, for all s ≥ 0, and we may draw the
same conclusion for ĜL as for GL . Hence the L2 -inversion method of Section 2 does not apply at once.
For arbitrary fixed θ > 0 let us define
Gθ (u) = e−θu G(u),
It is clear that Gθ ∈ L2 (0, ∞) and that
u ≥ 0.
(3.4)
Gθ,L (s) = GL (s + θ),
s ≥ 0,
(3.5)
Ĝθ,L (s) = ĜL (s + θ),
s ≥ 0,
(3.6)
so that now also Gθ,L ∈ L2 (0, ∞). As an estimator of Gθ,L we introduce (cf. (1.10))
with Ĝθ,L ∈
Writing
L2 (0, ∞)
a.s., for n sufficiently large.
Ĝθ,α = L−1
α Ĝθ,L
for suitable α > 0,
and L−1
α determined by (2.7), we finally arrive at
Ĝα (u) = eθu Ĝθ,α (u),
(3.7)
u ≥ 0,
(3.8)
u ≥ 0,
(3.9)
as an estimator of G.
In the case of known f the situation is somewhat simpler and an approximation of G would be given
by
G̃α (u) = eθu Gθ,α (u),
where
Gθ,α = L−1
α Gθ,L
for suitable α > 0.
MATHEMATICAL METHODS OF STATISTICS
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NONPARAMETRIC ESTIMATION
39
4. THE INTEGRATED SQUARED ERROR
Again first the case of unknown f will be discussed. Since G ∈
/ L2 (0, ∞) we cannot compare G and
2
Ĝα in the L -norm over the entire positive half-line. Therefore an arbitrary fixed 0 < B < ∞ will be
chosen and the comparison will be carried out in L2 (0, B) by restricting the functions to [0, B]. Let us
write ∥ · ∥ and ∥ · ∥B for the norm in L2 (0, ∞) and L2 (0, B), respectively.
It should also be noted that E((1 − ρ̂)−2 ) = ∞, unless a very restrictive condition is imposed on the
claims density f . It will become apparent from the expressions below that for this reason it will be hard to
deal with the mean integrated squared error (MISE). Therefore we will satisfy ourselves with considering
the ISE.
Theorem 4.1. Under assumptions (1.2), (1.3), and (1.7) we have, for unknown f ,
&
'
1
2
∥Ĝα − G∥B = Op
as n → ∞,
log n
)
provided that α = α(n) = (log n)/n as n → ∞.
(4.1)
Proof. Let us first observe that (see (3.10))
*
+
2
2
2
∥Ĝα − G∥2B ≤ e2θB ∥Ĝθ,α − Gθ ∥2 ≤ 2e2θB ∥L−1
α ∥ ∥Ĝθ,L − Gθ,L ∥ + ∥Gθ,α − Gθ ∥ .
(4.2)
In order to deal with the bias part, i.e., the second term on the right in (4.2), let us write G′θ = gθ and
note that
$∞
$∞
$∞
*
+
√
1
′ 2 1
2
u · ( u Gθ (u))
du ≤
Gθ (u) du + 2 u2 gθ2 (u) du
u
2
0
0
≤
1
2
$∞
0
e−2θu du + 2
0
1
≤
+4
4θ
≤
$∞
0
$∞
*
+2
u2 g(u)e−θu − θG(u)e−θu du
−2θu
u g (u)e
2 2
du + 4θ
0
2
$∞
u2 G2 (u)e−2θu du
0
1
+ 4(M 2 + θ 2 )
4θ
$∞
0
u2 e−2θu du < ∞.
(4.3)
Therefore, condition (3.16) for application of (3.21) in Chauveau et al. [15] is satisfied and we may
conclude that
,
1
as α ↓ 0.
(4.4)
∥Gθ,α − Gθ ∥2 = O
log(1/α)
Next, let us consider the first term on the right in (4.2), the variance part, and recall that (cf. (2.5))
2
2
∥L−1
α ∥ ≤ π/α . It is immediate from (1.7), (3.5) and (1.9), (3.6) that
∥Ĝθ,L − Gθ,L ∥ =
2
$∞ .
(1 − ρ){D̂(θ + s) − D(θ + s)} + (ρ̂ − ρ)D(θ + s)
D̂(θ + s)D(θ + s)
0
/2
ds.
(4.5)
Exploiting (3.2), its empirical counterpart, and the fact that P{ρ̂ = 1} = 0, it follows after some algebra
that, a.s., the right-hand side in (4.5) is bounded by
2(ρ̂ − ρ)2
2
I
+
I2 ,
1
(1 − ρ̂)2
(1 − ρ̂)2 (1 − ρ)2
MATHEMATICAL METHODS OF STATISTICS
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(4.6)
40
MNATSAKANOV et al.
where
I1 =
$∞
{D̂(θ + s) − D(θ + s)}2
ds,
(θ + s)4
I2 =
0
$∞
D2 (θ + s)
ds.
(θ + s)4
(4.7)
0
By the law of large numbers, assumption (1.2), and the central limit theorem it follows that
,12
2(ρ̂ − ρ)2
=
O
(1),
=
O
as n → ∞.
p
p
(1 − ρ̂)2
(1 − ρ̂)2 (1 − ρ)2
n
(4.8)
$∞
(4.9)
Application of (3.3) entails at once that
I2 ≤
0
1
ds < ∞.
(θ + s)2
Regarding I1 let us denote the cdf of the claims distribution by F and the empirical cdf of the sample
in (1.8) by F̂n . Also, let us write
Ψs+θ (x) = 1 − e−(s+θ)x ,
x ≥ 0,
(4.10)
for the cdf of the exponential distribution with mean 1/(s + θ). We have
& '2
+2 *
+2
c *
D̂(θ + s) − D(θ + s) = fˆL (θ + s) − fL (θ + s)
λ
. $∞
/2 . $∞
/2
−(s+θ)x
=
e
d{F̂n (x) − F (x)} =
{F̂n (x) − F (x)}dΨs+θ (x)
0
0
$∞
$∞
≤ {F̂n (x) − F (x)}2 dΨs+θ (x) = (s + θ) {F̂n (x) − F (x)}2 e−(s+θ)x dx
0
0
≤ (s + θ) sup{F̂n (x) − F (x)}2
x≥0
$∞
e−θx dx = (s + θ)Op
0
It follows that
I1 = Op
,1-
n
Combining (4.6), (4.8), (4.9), and (4.12) yields
∥Ĝθ,L − Gθ,L ∥2 = Op
,1n
as n → ∞.
as n → ∞.
,1n
as
n → ∞.
(4.11)
(4.12)
(4.13)
Combining (4.12) with (4.4), (4.2), and (4.5) yields
, 1 ,
1
∥Ĝα − G∥2B = Op
+
O
.
(4.14)
nα2
log(1/α)
)
Choosing α = α(n) ≍ (log n)/n balances the two terms on the right in (4.14) and yields the rate
claimed in the theorem.
Remark A. Although the theoretical rate is low due to the severe ill-posedness of the inverse of the
Laplace, the simulations in Section 5 show rather good results, even for relatively small sample sizes.
Remark B. It is possible to let θ depend on n in such a way that θ = θ(n) → 0 as n → ∞. Because then
Gθ(n) (u) → G(u) as n → ∞, for each u ≥ 0, we may consider Ĝθ(n),α as an estimator of G rather than
Ĝα in (3.8). It would have been easy to specify the dependence on θ(n) in (4.14) and determine an overall
rate for these estimators.
MATHEMATICAL METHODS OF STATISTICS
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NONPARAMETRIC ESTIMATION
41
Remark C. As usual only a rate of convergence is given for the smoothing parameter α = α(n) in
Theorem 1, and this does not provide any information about how to choose α for a given sample size n
in a specific situation. The simulations in Section 5 will give some insight into this matter.
Theorem 4.2. Under assumptions (1.2) and (1.3) we have, for known f ,
,
1
2
∥G̃α − G∥B = O
as α ↓ 0.
log(1/α)
(4.15)
Proof. In this case we simply have
∥G̃α − G∥2B ≤ 2e2θB ∥Gθ,α − Gθ ∥2 ,
(4.16)
and the result follows at once from (4.4).
Remark D. When f is known, there is no sample and the sample size n does not play a role as the
most important parameter. The only parameter present is the smoothing parameter α. Even though f
is known, there will in general be inaccuracies in the numerical evaluation of GL and Gθ,L that might
warrant the application of a regularized inverse L−1
α for some α > 0 and provide a useful alternative
numerical procedure.
5. A SIMULATION STUDY
It is known (see, for instance, Ross [2]) that for the claims density
f (x) =
1 −x/µ
e
1[0,∞) (x),
µ
x ∈ R,
(5.1)
u ≥ 0.
(5.2)
the solution G that satisfies (1.4) or (1.7) is
G(u) = 1 − µe−u(1−µ)/µ ,
Choosing µ = 1/2 and λ/c = 1 we arrive at a situation where conditions (1.2) and (1.3) are met.
First we will follow the procedure (3.8) for unknown f . For this purpose samples of size n = 50,
n = 100, and n = 300 are simulated from the exponential density f in (5.1) for µ = 1/2. Since the value
of α > 0 in (2.8) is not specified, we might as well integrate over an interval [−A, A] for some A > 0 and
write RA rahter than Rα . The integration in (2.8) can be actually carried out, which yields
*
+
1
RA (s) = 2
π(sinh πA) cos(A log s) + (log s)(cosh πA) sin(A log s) ,
s > 0, (5.3)
2
π + (log s)
see Chauveau et al. [15]. The corresponding estimator will be written ĜA , and is computed for
(i) A = 2.5, θ = 0.095, when n = 50,
(ii) A = 2.2, θ = 0.075, when n = 100,
(iii) A = 2.1, θ = 0.075, when n = 300.
The comparisons of the true function G in (5.2) with µ = 1/2 (the solid line) and ĜA (the dashed line)
are summarized in Figs. 1 (a)–(c), respectively.
Next procedure (3.9) for known f will be considered. Again we change from α to A, compute G̃A (the
dotted-dashed line) for
(i′ ) A = 2.5, θ = 0.095,
(ii′ ) A = 2.3, θ = 0.09,
(iii′ ) A = 2.1, θ = 0.085,
and plot the corresponding graphs in Figs. 2 (a)–(c), respectively. The true function G (the solid line) is
plotted as well.
MATHEMATICAL METHODS OF STATISTICS
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42
MNATSAKANOV et al.
Fig. 1. (a) n = 50, A = 2.5, θ = 0.095; (b) n = 100, A = 2.2, θ = 0.075; (c) n = 300, A = 2.1, θ = 0.075.
Fig. 2. (a) A = 2.5, θ = 0.095; (b) A = 2.3, θ = 0.09; (c) A = 2.1, θ = 0.085.
ACKNOWLEDGMENTS
The last author is grateful to Ben Duran for introducing him to the insurance ruin problem. The
authors are also thankful to Mathew McMahon for his help in conducting the simulations.
Research partly supported by NSF grant DMS-0203942.
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MATHEMATICAL METHODS OF STATISTICS
Vol. 17
No. 1
2008