Open Journal of Statistics, 2012, 2, 177-183
http://dx.doi.org/10.4236/ojs.2012.22020 Published Online April 2012 (http://www.SciRP.org/journal/ojs)
177
A Bayesian Inference of Non-Life Insurance Based on
Claim Counting Process with Periodic Claim Intensity
Uraiwan Jaroengeratikun, Winai Bodhisuwan, Ampai Thongteeraparp
Department of Statistics, Kasetsart University, Bangkok, Thailand
Received January 22, 2012; revised February 25, 2012; accepted March 10, 2012
ABSTRACT
The aim of this study is to propose an estimation approach to non-life insurance claim counts related to the insurance
claim counting process, including the non-homogeneous Poisson process (NHPP) with a bell-shaped intensity and a
beta-shaped intensity. The estimating function, such as the zero mean martingale (ZMM), is used as a procedure for
parameter estimation of the insurance claim counting process, and the parameters of model claim intensity are estimated
by the Bayesian method. Then, , the compensator of

t


N
t is proposed for the number of claims in a time in-
terval




0,t. Given the process over the time interval 0,t, the situations are presented through a simulation study and
some examples of these situations are also depicted by a sample path relating


N
t

t

to its compensator .
Keywords: Estimating Function; Zero Mean Martingale; Non-Life Insurance Claim Counting Process;
Non-Homogeneous Poisson Process; Bell-Shaped Intensity; Beta-Shaped Intensity
1. Introduction
In the field of non-life insurance, the modeling of claim
counts is a very important component in a risk model
with regard to loss reserving, pricing, underwriting, etc.
The precision of a claim count estimation is the key to
running an insurance business successfully. Jewell [1]
presented the Bayesian credibility model, called the exact
credibility model, for claim counts involving Bayesian
analysis with a natural conjugate prior distribution. The
exact credibility model for claim counts has been used in
the non-life insurance industry as part of the process of
estimating and predicting the expected claim counts in
upcoming periods, using past experience of claims of a
risk class or related risk classes, i.e., Jewell estimated


1
j
j

jt
EN 

1jt
N

1t
  , where is a random
variable representing the claim counts of the jth insurance
contract during the th period with the parameter
j
. The
j
N is independent and Poisson distributed as
an exponential family with mean

j
sj
EN , and vari-
ance


Var
j
sj
N. The prior distribution of
j

has a
Gamma distribution with shape parameter

and scale
parameter

. Also, the risk exposure or the number of
policies of jth contract in each period s is 1, where
and
1,1,2, 3,2,3,,jk,
s
t

. The


 11
,,
1
ˆˆ
,,
j
jjt jt
ENNEN 


jjt
NN

is the es-





1
1
ˆˆ
,, 1
timator, i.e.
j
jt jj
jt
EN NNzzN 
 

where
j
E




 , t zt

 is the credibility factor, and

j
N is the sample mean of
j
N

. Jewell’s Bayesian
credibility model was extended to others, such as the
exact credibility model of Kaas, Dannenburg and Goova-
erts (see [2]) and Ohlsson and Johansson (see [3]). Cal-
culating the expected claim counts with the credibility
approach only depends on the information from past ex-
perience of claim counts, and does not consider the oc-
currence behavior of claim counts over time. Some au-
thors have considered the claim counts relating to a
specified time or their behavior over time. Mikosch [4]
viewed the claim counting process as a homogeneous
Poisson process (HPP) in the Cramér-Lundberg model,
one of the most popular and useful risk models in non-
life insurance. In non-life insurance portfolios, the claim
counts during a time period are caused by periodic phe-
nomena or seasonality. These claim counts are modeled
in terms of a non-Homogeneous Poisson process (NHPP)
with a period time-dependent intensity rate. Morales [5]
studied the periodic risk model consisting of the claim
counting process with a bell-shaped intensity function
(called the Gaussian intensity) of the form
 
2
*
2
11
exp 2
2
11
2π
22
t
t


















, (1)
C
opyright © 2012 SciRes. OJS

U. JAROENGERATIKUN ET AL.
178
for

s
tt  
0
, , ,
and
 
0, 1t0,1, 2,s*0  

, where *

,

and s are the parameters of
model periodic claim intensity and is the standard
normal distribution function. The unknown parameters of
model intensity were estimated by the maximum likeli-
hood estimation (MLE), and the ruin probability model
was evaluated through a simulation study. Furthermore,
Lu and Garrido [6] and Garrido and Lu [7] explored the
periodic NHPP model with a beta-shaped intensity func-
tion,
Φ

 
11
11
pq
ttm
DD
*
*
1
ttm
t






 


, (2)
for ,
21
Dm m
11
**
11
*1
p
q
tm
DD

 


 
 

tm




 is
the scale factor, while * tm
*
1
1
2
Dp
pq



001mm
is the mode
of function, , and 12,
where
 
*
, 1pq

, and are the parameters, p q



is the
greatest integer function, 1 and 2 represent the start-
ing and ending point of the occurrence interval, respec-
tively.
m m
In this study, we present an estimation approach to
non-life insurance claim counts in the claim counting
processes using an estimating function, the zero mean
martingale (ZMM). This approach provides a parameter
estimator, , of process, including the MLE for the
parameter estimation of model claim intensity proposed
by Jaroengeratikun et al. [8]. The can be inter-
preted as the insurance claim counts, , during the
time interval

ˆt

ˆt

Nt


0,t

ˆt

:t0t
. The estimate is also useful
for predicting time of claim occurrences or the claim
counts in the next periods. In this paper, we extend their
approach to estimate clam counts in an NHPP with peri-
odic bell-shaped and beta-shaped intensities. Then, the
Bayesian analysis is used to estimate the parameters of
model claim intensity.
2. Non-Life Insurance Claim Counting
Process
We define ; the cumula-
tive number of insurance claims that have occurred dur-
ing time

Nt


#1
i
i T

0,t, where 1n TW
n
W

 1n; is a
claim arrival time and i is an identically independent
distributed (iid) with Exponential whose parameter is
, called the claim intensity rate.
W

i
w 


;0NNtt

is a claim counting process,


Nt

0
d
t
 can be written as where
 
NuNt


Nt
N
*
is an
increment of in a small fraction period. In this study,
the insurance claim counts are NHPP with periodic claim
intensity rate, i.e. bell-shaped intensity function and beta-
shaped intensity function. The bell-shaped intensity as an
initial season, s = 0, is given in Equation (1) [5], where

is an average number of claims over a period and

is a variability of season. The mean value function of


Nt



has the following representation
 
*
*
1
1
2
2
11
22
tt
tt






















 


 

*
(3)
The beta-shaped intensity function is given in Equa-
tion (2) [6,7], where

is the peak level for claim in-
tensity, Its integrated intensity is


11
11 *
*
1
=
pq
p
ttm ttm
DB DD
t



  

 
 




, (4)

where 1 ,,;
p
ttm
BtBpqBpq D




 
11
1
0
,1d
q
p
Bpqvvv




 
1
1
0
,;1 d
tq
p
Bpqtvv v




,
and
.
Both the bell-shaped and the beta-shaped intensity are
depicted in Figure 1. In Figure 1(a) the claim of occur-
rence in the tail of the period, i.e. left and right tail of
period, changes slowly. While the beta-shaped intensity
is shown in Figure 1(b), the claim of occurrence in the
left and right tail of the period changes quickly.


,
F
P On a probability space



Nt
  
0
d
t
tuuENt   

, is modeled
by NHPP with a mean value function or parameter
. As






 
;0ttktt 

; is called the multiplicative
intensity, where




t  and kt
N
   
 
are defined as the claim
intensity rate and the exposure risk, respectively. We
consider as a non-decreasing right continuous step
function 0 at time t = 0 and jumps of size 1, and


exp
;Pr !
n
tt
PpNt Ntnn



0,1,2,n  and for





 

Pr d1ddNtt tE Nt   

U. JAROENGERATIKUN ET AL.
2012 SciR OJS
179
(a) (b)
Figure 1. Non-life insurance claim intensity function, (a) Bell-shaped intensity


 





t
2
1
82 t
42 exp
2π
 and (b) Beta-
shaped intensity
    
tt10 2  ttt
14 14
1

,
while p = q.


3. Parameter Estimation of the Non-Life
Insurance Claim Counting Process
Copyright ©es.
In this section, we introduce the methods which are use-
ful for parameter estimation of the non-life insurance
claim counting process, including the estimating function,
the martingale method, and the Bayesian estimation ap-
proach (BE).
3.1. Estimating Function
On a probability space
F
P 
, where ,

is
an open interval on the real line, . Sup-
pose that the observation , the estimating func-
tions, , are functions of and the pa-
rameter . By solving
 
;Nt

Nt
Pp


Nt n
 
;gNt 



 
;gNt

0
 
;t
N


 , a so-called es-
timating equation, an estimate of is obtained. Then
is an unbiased estimating function if


 
;t
gN
Eg 0


for all

 (see [9]).
In this study, the estimating function for parameter es-
timation of the insurance claim counting process is pro-
vided by the martingale method.
3.2. The Martingale Method
The martingales are random processes relating to time.
On a probability space

,
F
P

;0t
t

0
, we suppose the in-
creasing family t  , a filtration or history
t, which is the available data at the time . The proc-
ess is a martingale with respect to


;MMtt

if exist, and
 
EMt 


s


tMt0s
 
EMt for all . As a result of
the properties of the martingale,




0EM
EM t

for all , then 0t


00EM

for a zero mean mar-
tingale [10,11].
This study of the martingale method is useful for con-
structing an estimating function for a parameter estima-
tion of the insurance claim counting process. The process
takes place over a small time interval,dtt t

,



ddENt tt  
t and as a result of the meaning
of martingale property, the martingale can be written as




dddNtt tMt  
 


00
ddd
tt
Nuu uMu  

, (5)
which is a martingale-difference. Then, the following
martingale is
,
or it can be rewritten in the form of






Ntt Mt , is a ZMM. Based on ZMM, we

obtain

   


0EMtENt t .




0Nt t Thus,



Nt
is an estimating equation for
parameter estimation of the insurance claim counting
process. Also, as a result of the parameter estimate of the
process, this can be interpreted as an estimate or,
in other words,


t is called the compensator of


Nt, and this estimate is useful for predicting the times
of occurrence of insurance claim counts [11]. We can
depict the systematic part of the process of insurance
claim counts,



Nt, related to its compensator,

t
 
,
and the associated martingale


Ntt Mt 

t 

in
Figures 2(a) and (b), respectively, based on a sample of
15 independent random times of claims occurrence in the
NHPP with a claim intensity where
2
1
31exp 32 tt 










.
3.3. A Bayesian Estimation Approach: The
Model Specification for the Parameters of
Model Claim Intensity


Nt, In order to get the estimate of the compensator of


ˆt, on the NHPP model of the non-life insurance
laim counts, the parameters of the claim intensity func- c

U. JAROENGERATIKUN ET AL.
180
(a) (b)
Figure 2. In a sample of 15 independent random times of claims occurrence with the claim intensity


 






t
2
1
32


Nt n
,,, ,
t31exp  , (a) Non-life insurance claim counting process N(t) related to its compensator Λ(t); and (b) Mar-
tingale M(t) = N(t) – Λ(t).


33
Gamma ,qcd
c d2
c2 d cd
I(1, ), (8)
tion are estimated by the BE. Given , we sup-
pose that

123
N
t are the arrival times of the
claims on the time interval
tttt


0,t with a cumulative dis-
tribution function (a general order statistics model)
  
1exp
F
tt
 

exp
n
in
t
 . (6)
Its likelihood function [4,12] is given by
 
1
;
i
lt 




, where

is a vector of
the parameters of the model claim intensity,

denotes
the set of claim arrival times, the i of both the bell-
shaped and the beta-shaped intensity function are given

t 

*,
in Equations (1) and (2), respectively. While



 

*,,pq
 of the bell-shaped intensity function and

 
of the beta-shaped intensity function are unknown, the
estimate of

can be obtained by a Bayesian analysis.
The BE approach treats all unknown parameters in Equa-
tion (6) as random variables under their prior distribution
and derives their conditional distribution upon the known
information (sample data of claim arrival times). In this
study, we apply the following prior density for

*,



 

*
11
Gamma ,ab

22
Gamma ,ab
2
a2 b

*,,pq



*Gamma
 

22
Gamma ,cd
:
1) ,  
2
  2) , (7)
where , , and are known parameters, and
1
a1 b
the following prior density for : 

11
,cd
1) ,
2) I(1, ), p
3)
where 1, 1, , , 3 and 3 are known pa-
rameters and


Gamma ,ab denotes a gamma distribu-
tion with mean ab and variance 2 ab
,1pq
. The restriction
, is imposed with the construct I(1, ). We assume
that among parameters,

, are independent. The esti-
mated parameters are based on the joint posterior density
as,

  
;hhl

   , (9)




;lh

is the joint prior density, and where

 is
the likelihood function of

. The Equation (9) has a
complicated form or no closed form. Therefore, the BE is
implemented on the basis of the Markov chain Monte
Carlo (MCMC) algorithms with the Gibbs sampler to
solve these problems, see [13-16]. In this case, we esti-
mate the parameters

with the empirical Bayes’ esti-
mator obtained by using the BUGS program.
4. Simulation Study
In this study, a simulation model is used to investigate
how the observation of the non-life insurance claim
counting process can be used to estimate its model pa-
rameter, i.e. claim intensity or in term

t 


t,
using the estimating function provided by the martingale
method with ZMM. In particular, the NHPP of the in-
surance claim counts with bell-shaped and beta-shaped
intensities, we consider the simulation study of the proc-
esses of the insurance claim counts during the claim time
interval


0,t
,,, ,
in which the observation involves the
claim arrival times,

123
N
t. The claim arrival
times can be simulated by using the mean value function
ttt t


t
,,,,EEEE
as a claim arrival time of the HPP with mean one
[5]. It implies that 123 n are independent and
exponentially distributed with mean one, where

U. JAROENGERATIKUN ET AL. 181

Et1, 2, 3,,in
 
1
t
ii i
, for all 

. So, the nth
claim arrival time,

N
tn
t
1
12 n
tE

 

, is generated by [5,8]



3n
EEE 

i
Eof
Nt
1

, (10)
where is the invertible function


t,
0.25 with *0.  1, =

0.25and *  10 ,


,

for
the model bell-shaped intensity, a0.1
and , for the model
beta-shaped intensity.
nd
pq
*  
1.251.25pq *10  
In this simulation study of the non-life insurance claim
counting process over the time interval

0,t
10.1c
, the num-
ber of observations, , is composed of 5, 10, 15
and 20. The processes are carried out with 5000 sample
paths. In each sample path, the parameter estimate of the
model claim intensity is computed using the BE method
with the prior distributions given in Equations (7) and (8)
where 1, , , , 1

Nt n
10.01b0.01a2
a52 b

,
1, 2, 2, 3
0.1d5 1dcc 5

, 3, and the esti-
mating function, such as the ZMM which is used to esti-
mate the parameter of the process (or the com-
pensator of ), i.e. fitting the compensator
estimate
1d

t
 
Nt

t


ˆ
t to . Also, the mean squared error
(MSE) is provided to measure the fitting

Nt


ˆt


to
as the following form

Nt

2d
ii
p
uNu u
S
0
1
ˆ
MSE
t
p
i


,
where
p
S denotes the number of sample paths. Notice
that the MSE of the compensator estimate


ˆt of
for the processes, as shown in Tables 1 and 2,
depends on the parameters of the model claim intensity
as in the following results, for the NHPP with a bell-
shaped intensity, the parameters of its model claim inten-
sity

Nt
*

0.1, 0.25



Nt
(a small average number of
claims over a period), the MSE of the compensator esti-
mate of increases as the observation num-
ber increases. In the same process with the parameters of
model claim intensity

ˆt

*

10, 0.25



, the MSE of
the compensator estimate of decreases

t

Nt

ˆ

Table 1. MSE of the compensator estimate

ˆ
t of


Nt
in the NHPP of the non-life insurance claim counts with
bell-shaped intensity.
*




Nt MSE
0.1 0.25 5
10
15
20
0.873166
1.663399
2.428750
3.141596
10 0.25 5
10
15
20
5.968821
5.676725
4.630684
4.880201
Table 2. MSE of the compensator estimate


ˆt of


Nt
*
in the NHPP of the non-life insurance claim counts with
beta-shaped intensity.
p
q



Nt MSE
0.1 1.25 5
10
15
20
0.877067
1.883131
2.821956
4.170855
10 1.25 5
10
15
20
0.885957
1.971573
4.469941
5.246322
while its observation number increases until the observed
15 times of claims occurrence, and then its MSE values
begin to increase as the observation number increases.
For the NHPP with a beta-shaped intensity

0.1,
*

1.25pq


*
(a small peak level for claim intensity) and

10, 1.25pq


, the MSE of the compensator
estimate


ˆt
 of


Nt

Nt
increases exponentially as their
observation number increases.
Some examples in these situations of the NHPP with
both bell-shaped and beta-shaped intensities, of non-life
insurance claim counts based on a sample of 5, 10, 15
and 20 times of claims occurrence, are illustrated in Fig-
ures 3 and 4, including the and its compensator


t
*=10
 0.25
. Figure 3 shows a sample path of the process with
a bell-shaped intensity ,

. The


Nt
and its compensator


t
*=10
 0.25
are characterized by the pa-
rameters of model claim intensity ,


,
the compensator


t

Nt
*=10


Nt
fits well with , as the ob-
servation number is 15 and 20 (slightly larger than the
claim intensity ). While the and its com-
pensator


t
*=10

1.25pq
in the process with a beta-shaped inten-
sity ,

 is shown in Figure 4, the
compensator


t

Nt fits with , as the observation
number is small.
5. Conclusions and Discussion
5.1. Conclusions
In a non-life insurance claim counting process over the
time interval


0,t, the simulation study of the NHPP
with both bell-shaped and beta-shaped intensities dem-
onstrates the fitting of the compensator estimate


ˆt

to


Nt
*
. The model fitting depends on the parameters
of model claim intensity and model specification of
claim intensity. Firstly, regarding the NHPP with the
parameters of the model bell-shaped intensity, a

has
almost no claim occurrences over a period and any

The compensator estimate is a good fit to

ˆt



Nt
*
with small MSE and small number of observations. In
the same process with the parameters of the model claim
ntensity, an average number of claims over a period i


U. JAROENGERATIKUN ET AL.
201 OJS
182
(a) (b)
(c) (d)
Figure 3. N(t) and its compensator Λ(t) in the NHPP with the parameters of a bell-shaped intensity λ* = 10, σ = 0.25 based on
a sample of (a) 5 claims; (b) 10 claims; (c) 15 claims; and (d) 20 claims.
(a) (b)
(c) (d)
Figure 4. N(t) and its compensator Λ(t) in the NHPP with the parameters of a beta-shaped intensity λ* = 10, p = q = 1.25 based
on a sample of (a) 5 claims; (b) 10 claims; (c) 15 claims; and (d) 20 claims.
Copyright © 2 SciRes.
is not less than one and any

. The MSE of the com-
pensator estimate of is small while the
number of observations is slightly larger than the value
of

ˆt


Nt
*

. Secondly, when the model beta-shaped intensity
is considered with the parameters , the compensa-
tor estimate
pq


ˆt


Nt is a good fit to with small
MSE and small number of observations. Some examples
of the situations in the simulation study are also depicted

U. JAROENGERATIKUN ET AL. 183
Figure 5. ,

Nt


ˆtNHPP , and


ˆtCRED in the NHPP
with a bell-shaped intensity of non-life insurance claim
counts.
by a sample path relating


Nt

t
and its compensator
.
5.2. Discussion
This study is a parameter estimation approach to non-life
insurance claim counting process. The parameter


t

of the claim counting process is estimated from observa-
tions using an estimating function, such as ZMM. A re-
sult of the parameter estimate

ˆt


Nt can be interpreted as an , called the compensator estimate


ˆt


Nt

t
Nt
of
. This estimate is also useful for predicting the time
of insurance claim occurrences. Using an example, we
can depict a comparison of the two approaches of non-
life insurance claim counts, including this insurance
claim counting process and the Jewell’s credibility ap-
proach. In Figure 5, the compensator NHPP in the
NHPP with a bell-shaped intensity (dashed line) is a
good fit to over time
ˆ

 

0,t; on the other hand, the
procedure of the credibility estimate CRED (+) or
the compensator CRED of on the credibility
model does not consider the behavior of insurance claim
counts over time.

ˆt

 
ˆt


Nt
REFERENCES
[1] W. S. Jewell, “Credible Means Are Exact Bayesian for
Exponential Families,” ASTIN Bulletin, Vol. 8, No. 1,
1974, pp. 77-90.
[2] R. Kaas, D. Dannenburg and M. Goovaerts, “Exact
Credibility for Weighted Observations,” ASTIN Bulletin,
Vol. 27, No. 2, 1997, pp. 287-295.
doi:10.2143/AST.27.2.542053
[3] E. Ohlsson and B. Johansson, “Exact Credibility and
Tweedie Models,” ASTIN Bulletin, Vol. 36, No. 1, 2006,
pp. 121-133. doi:10.2143/AST.36.1.2014146
[4] T. Mikosch, “Non-Life Insurance Mathematics,” 2nd
Edition, Springer-Verlag, Berlin, 2009.
doi:10.1007/978-3-540-88233-6
[5] M. Morales, “On a Surplus Process under a Periodic En-
vironment: A Simulation Approach,” North American Ac-
tuarial Journal, Vol. 8, No. 2, 2004, pp. 76-87.
[6] Y. Lu and J. Garrido, “Doubly Periodic Non-Homogen-
eous Poisson Models for Hurricane Data,” Statistical Me-
thodology, Vol. 2, No. 1, 2005, pp. 17-35.
doi:10.1016/j.stamet.2004.10.004
[7] J. Garrido and Y. Lu, “On Double Pariodic Non-Homo-
geneous Poisson Processes,” Bulletin of the Association
of Swiss Actuaries, Vol. 2, 2004, pp. 195-212.
[8] U. Jaroengeratikun, W. Bodhisuwan and A. Thong-
teeraparp, “A Statistical Analysis of Intensities Estima-
tion on the Modeling of Non-Life Insurance Claim
Counting Process,” Applied Mathematics, Vol. 3, No. 1,
2012, pp. 100-106. doi:10.4236/am.2012.31016
[9] P. Mukhopadhyay, “An Introduction to Estimating Func-
tions,” Alpha Science International Ltd., Harrow, 2004.
[10] P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding,
“Statistical Models Based on Counting Processes,” Sprin-
ger-Verlag New York, Inc., New York, 1993.
[11] P. Yip, “Estimating the Number of Error in a System
Using a Martingale Approach,” IEEE Transactions on
Reliability, Vol. 44, No. 2, 1995, pp. 322-326.
doi:10.1109/24.387389
[12] J. E. R. Cid and J. A. Achcar, “Bayesian Inference for
Nonhomogeneous Poisson Processed in Software Reli-
ability Models Assuming Nonmonotonic Intensity Func-
tions,” Computational Statistics & Data Analysis, Vol. 32,
No. 2, 1999, pp. 147-159.
doi:10.1016/S0167-9473(99)00028-6
[13] J. Gill, “Bayesian Methods: A Social and Behavioral Sci-
ences Approach,” Chapman and Hall/CRC, London,
2008.
[14] T. Hirata, H. Okamura and T. Dohi, “A Bayesian Infer-
ence Tool for NHPP-Based Software Reliability Assess-
ment,” In Y. H. Lee, T.-H. Kim, W.-C. Fang and D.
Slezak, Eds., Lecture Notes in Computer Science, Vol.
5899, Springer, Berlin, 2009, pp. 225-236.
[15] D. P. M. Scollnik, “Actuarial Modeling with MCMC and
BUGS,” North American Actuarial Journal, Vol. 5, No. 2,
2001, pp. 96-124.
[16] X. Zhao, C. Yu and H. Tong, “A Bayesian Approach to
Weibull Survival Model for Clinical Randomized Cen-
soring Trial Based on MCMC Simulation,” The 2nd In-
ternational Conference on Bioinformatics and Biomedical
Engineering, Shanghai, 16-18 May 2008, pp. 1181-1184.