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Applied Mathematical Sciences, Vol. 8, 2014, no. 162, 8085 - 8097
HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410802
Construction Actuarial Model for Aggregate Loss
under Exponentiated Inverted Weibull
Distribution
Osama Hanafy Mahmoud
Department of Mathematics, Statistics and Insurance
Sadat Academy for Management Sciences, Egypt
Copyright © 2014 Osama Hanafy Mahmoud. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
The problem of planning reinsurance policies in property and causality
insurance companies and how to estimate the loss reserve play an important role
for the results of the insurance company. Also, they have past or present data for
number of claims and its amounts which want to use in prediction future claim
frequency and claim severity. Many statistical distribution are fitting to find an
appropriate distribution to represent our data. In this paper, we introduce a
statistical distribution known as Exponentiated Inverted Weibull (EIW)
distribution to represent the claim amount and its characteristics for applying it in
actuarial studies. Second, we test the tail weight of the distribution and maximum
likelihood estimation for its parameters. we present our aggregate loss model
under collective risk theory when the claim frequency distribution is Poisson or
Negative Binomial distribution. Also, we present how to calculate the reinsurance
pure premium in case of stop loss reinsurance. Finally, The simulation numerical
example is given to represent our results.
Keywords: Exponentiated Inverted Weibull Distribution, Tail Weight of
Distribution, Maximum Likelihood Estimation, Aggregate Loss Model
8086 Osama Hanafy Mahmoud
1. Introduction
Insurance companies need to investigate claims experience and apply
mathematical techniques for many purposes such as ratemaking, reserving,
reinsurance arrangements and solvency.
Many papers have been presented to aggregate losses as: Heckman and
Meyers (1983) discussed aggregate loss distributions from the perspective of
collective risk theory for severity and count distributions. They include examples
for calculating the pure premium for a policy with an aggregate limit,
calculating the pure premium of an aggregate stop-loss policy for group life
insurance; and calculating the insurance charge for a multi-line retrospective
rating plan, including a line which is itself subject to an aggregate limit. Venter
(1983), Distribution functions are introduced based on power transformations of
beta and gamma distributions, and properties of these distributions are discussed.
The gamma, beta, F, Pareto, Burr, Weibull and loglogistic distributions are special
cases. The transformed gamma is used to model aggregate distributions by
matching moments. The transformed beta is used to account for parameter
uncertainty in this model.
Robertson (1992), Provided an application of the fast Fourier transform to
the computation of aggregate loss distributions from arbitrary frequency and
severity distributions. Papush el al (2001), addressed the question what type of
Normal , Lognormal and Gamma distributions is the most appropriate to use to
approximate aggregate loss distribution. Vilar et al. (2008), described a
nonparametric approach to make inference for aggregate loss models in the
insurance framework by assuming that an insurance company provides a historical
sample of claims given by claim occurrence times and claim sizes.
Bortoluzzo et al. (2009), aimed estimating claim size in the auto insurance by
using zero adjusted Inverse Gaussian distribution. Shevchenko (2010) reviewed
numerical algorithms that can be successfully used to calculate the aggregate loss
distributions. In particular Monte Carlo, Panjer recursion and Fourier
transformation methods are presented and compared. Also, several closed-form
approximations based on moment matching and asymptotic result for heavy-tailed
distributions are reviewed.
One of the most significant goals of any insurance risk activity is to achieve a
satisfactory model for the probability distribution of the total claim amount. In
this paper, we introduce a statistical distribution known as Exponentiated Inverted
Weibull (EIW) distribution to represent the claim amount and its characteristics
for applying it in actuarial studies.
This paper is organized as follows: Section 2 we presentsThe Model claim
severity under Exponentiated Inverted Weibull Distribution and test the tail weight
Construction actuarial model for aggregate loss 8087
of a distribution. In Section 3 we discuss the problem of estimating the parameters
of distribution by using maximum likelihood method. Section 4 we present our
aggregate loss model under collective risk theory when the claim frequency
distribution is Poisson or Negative Binomial distribution. Section 5, how to
calculate the reinsurance pure premium in case of stop loss reinsurance. Finally,
The simulation numerical example is given to represent our results.
2. The Model under Exponentiated Inverted Weibull Distribution
Recently many studies in probability distributions and its applications presented
the Exponentiated Inverted Weibull distribution as:
Flaih et al (2012), Considered the standard exponentiated inverted weibull
distribution (EIW) that generalizes the standard inverted weibull distribution (IW),
the new distribution has two shape parameters. The moments, median, survival
function, hazard function, maximum likelihood estimators, least-squares
estimators, fisher information matrix and asymptotic confidence intervals have
been discussed. A real data set is analyzed and it is observed that the (EIW)
distribution can provide a better fitting than (IW) distribution. Aljouharah Aljuaid,
(2013), presented Bayes and classical estimators have been obtained for two
parameters exponentiated inverted Weibull distribution when sample is available
from complete and type II censoring scheme. Hassan (2013), dealt with the
optimal designing of failure step- stress partially accelerated life tests with two
stress levels under type-I censoring. The lifetime of the test items is assumed to
follow exponentiated inverted Weibull distribution. Hassan et al. (2014),
presented estimation of population parameters for the exponentiated inverted
Weibull distribution based on grouped data with equi and unequi-spaced grouping.
Several alternative estimation schemes, such as, the method of maximum
likelihood, least squares, minimum chi-square, and modified minimum chi-square
are considered.
If our claims amount of insurance portfolio nxxxx ,,,, 321 follow the
Exponentiated Inverted Weibull (EIW) distribution with parameters and in
the following form for the probability density function (pdf):
0,, )()( )1( xxexf x (1)
Therefore, its cumulative probability function (cpf) can be written in the form:
0,, )()( xexF x
(2)
The (right-) tail of a distribution is the portion of the distribution corresponding to
8088 Osama Hanafy Mahmoud
large values of the random variable. A distribution is said to be a heavy-tailed
distribution if it significantly puts more probability on larger values of the random
variable. We also say that the distribution has a larger tail weight. In contrast, a
distribution that puts less and less probability for larger values of the random
variable is said to be light-tailed distribution.
To test the tail weight of a distribution, we can use the Existence of Moments
method as follows:
A distribution )(xf is said to be light-tailed if 1)( rxE for all 0r
and the distribution )(xf is said to be heavy-tailed if either )( rxE does not
exist for all 0r or the moments exist only up to a certain value of a positive
integer r , Finan (2014).
The rth moments of the exponentiated inverted weibull distribution is given as
follows:
0,, )()(0
)1(
xdxxexxE xrr
This can be written as:
rx r
E(x
r
r
0,, )1() (3)
Proof: The pdf of the EIW distribution is:
0,, )()( )1( xxexf x
The rth moments function can be written in the form:
0
)()( dxxfxxE rr
dxxexxE xrr )()( )1(
0
By taking transformation xH
We can write the rth moments function as:
dHHH
eH
xE
Hr
r
1
11
)1(
0
)(
By simplification of the above equation ,we can get
dHeHxE H
rr
r
0
11
)(
This integral known as gamma function , therefore the rth moments function is:
r
xE
r
r 1)(
Construction actuarial model for aggregate loss 8089
From the above equation, we can find to obtain the rth moment must the value of
greater than r to be exist.
Since the moments are not finite for all positive r; the exponentiated inverted
weibull distribution is heavy-tailed.
From the above equation, we can find the mean and the variance of EIW
distribution as follows:
By putting r=1
rx E(x
0,, )1
1()
1
(4)
And the second moment by putting r =2 in the form:
rx E(x
0,, )2
1()
2
2 (5)
Thus the variance is:
22
22
)1
1()2
1(
))(()()
xExEV(x
3. Maximum Likelihood Estimation (MLE) for parameters
Suppose that we have postulated a probability model, such as the Exponented
Inverted Weibull distribution, to describe a given loss amount distribution. The
next step in our procedure should be to estimate values for the parameters of the
model.
We use the maximum likelihood method (MLE) for estimating the
unknown parameters and of Exponented Inverted Weibull distribution, as
follows:-
N
1i
)( ) , ( ixfL
Then the likelihood function is as follows,
N
i
i
xNN
N
i
i
x
xe
xeL
N
ii
1
)1()(
1
)1(
)( )(
)(),(
1
(6)
by taking the natural logarithm for the likelihood function , we get
8090 Osama Hanafy Mahmoud
N
i
i
N
i
i xxNNL11
ln)1(lnln),(ln (7)
So, we need to estimate the two parameters and . The first derivatives
for the natural logarithm of the likelihood function with respect to and , are
given by
N
i
ixNL
1
),(ln
(8)
N
i
i
N
i
ii xxxNL
11
ln)ln(),(ln
(9)
The maximum likelihood estimators of and could be obtained by
equating the equations (8) and (9) by zero, and solving them simultaneously using
an iterative technique. We obtain the approximate variance covariance matrix by
replacing expected values by their maximum likelihood estimators and inverting
the Fisher – information matrix, defined by:
2
22
2
2
2
lnln
lnln
LL
LL
I
Where, the second derivatives of the natural logarithm of likelihood function
defined in equation (6) are given as follows:
22
2 ),(ln
NL
(10)
N
i
ii xxL
1
2
)ln(),(ln
(11)
N
i
ii xxNL
1
2
22
2
))(ln(),(ln
(12)
The MLE ˆ and ˆ have an asymptotic variance covariance matrix
obtained by inverting the Fisher – information matrix.
4. Aggregate loss model under collective Risk Theory
Suppose that portfolio has N claims in the past period of time in our experience
and each unit has ix is the claim size which is independent identical distributed
Construction actuarial model for aggregate loss 8091
exponented inverted Weibull with parameters and its p.d.f in the equation
(1) and cumulative probability function in the equation (2) .
Then the aggregate losses is S where the sum of claim amounts as:
NxxxxS 321
Suppose also that the individual loss amounts ix are independent on the annual
loss frequency N.
Then it follows that:
The probability density function (pdf) of aggregate losses is
0
* )()()(k
k
xs sfkNprSf (13)
Where )(* sf k
x is called the k th fold convolution of Nxxxx 321
the k th fold convolutions are often extremely difficult to compute in practice and
therefore one encounters difficulties dealing with the probability distribution of S:
An alternative approach is to use various approximation techniques. We consider
a technique known as the Panjer recursive formula.
The mean and variance of aggregate loss distribution can get as:
)()()()()(
)()()(
2NVxExVNEsV
xENEsE
(14)
The pricing problem usually reduces to finding moment of S . A common pricing
formula is )( )( svksEprice
Where the price is the expected payout plus a risk loading which k times the
variance of the payout for some k.
The expected payout )(sE is also known as the pure premium and it can be
shown to be )()( xENE .
Estimation the Mean and the Variance of Aggregate losses Distribution:
We will consider Poisson and Negative Binomial distributions for the frequency
distribution of losses as follows:
I. Poisson Distribution
Suppose that the annual frequency of losses from a portfolio follows a Poisson
distribution with parameter .
In this case the mean of the loss distribution is:
8092 Osama Hanafy Mahmoud
sE )1
1()(
1
(15)
sV )2
1()(
2
(16)
II. Negative Binomial Distribution
Suppose that the annual frequency of losses from a portfolio follows a Negative
Binomial distribution with parameters pr and .
In this case the mean of the loss distribution is:
p
prsE )
11(
1)(
1
(17)
p
p
p
prsV
22
)1
1(1
)2
1(1
)(
(18)
5. Stop Loss Reinsurance
Gauger and Hosking (2008) and Finan (2014) presented the stop loss reinsurance
as:
When a deductible D is applied to the aggregate loss S over a definite period, then
the insurance payment will be
DSDS
DSDSSDSMaxDS
,
,0^0,
the reinsurer will pay the insurer an amount equal to DS
. The insurer's
retained loss is thus DS^ .
The main problem is how to calculate the reinsurance pure premium?.
the reinsurance pure premium is payment as the stop-loss re insurance. Its
expected cost is called the net stop-loss premium and can be computed as:
dxxfdxdxxFDSE S
DD
S )()(1
(19)
6. Numerical Results In this section, we will present a numerical investigation of the maximum
likelihood estimation for the parameters of and .
We need to estimate the two parameters and by using the maximum
likelihood method. So, we will need to solve the three non-linear equations of
Construction actuarial model for aggregate loss 8093
logarithm likelihood function (8) and (9) simultaneously using Newton-Raphson
method .The iterative technique, can be applied as follows:
mmmm ACxx 1
1
where
and
2
22
2
2
2
lnln
lnln
mm
mmmm
LL
LL
C
Assuming initial values for each of and , the Newton-Raphson iterative
procedure is continued until either the number of iterations will be ( 200 ) or
when |Xm – Xm+1 | < 5 10-5 .
In the following table, the estimates of unknown parameters, the relative bias
which is the absolute difference between the estimated parameter and its true
value divided by its true value.
-ˆ
Bais Re0
0
ltive
And the mean square error (MSE) which is the mean square of the difference
between the estimated parameter are presented for all the estimated parameters
considering different initial points of the parameters.
ˆM
2
0
NSE
where N is the number of experiments carry out .
m
mm
m
mm
m
mm L
L
Axx
ln
ln
,ˆ
ˆ,
ˆ
ˆ
1
11
8094 Osama Hanafy Mahmoud
Table (1)
Estimators for parameters of EIW distribution, Relative Bias and MSE
MSE Relative Bias Estimator Parameters 0
0
0.02393 0.003084 50.1547 5 50
0.007265 0.01676 5.085238
0.140869 0.006217 60.37536 5 60
0.00727 0.016772 5.085291
0.140825 0.006215 60.37527 6 60
0.010474 0.016771 6.102346
0.3848482 0.008784 70.62036 6 70
0.0104615 0.0167612 6.102282
0.8045189 0.011087 80.89695 7 80
0.0142606 0.01677358 7.119418
Table (1), shows The Estimators of the parameters and of the model,
Relative Bias, and MSE. We can notice that the absolute value of the difference
between the true value of the parameter and its estimator is small value converges
to zero, so these estimators are said to be consistent estimators.
Estimate the mean and the variance of EIW distribution
By substituting the estimated values of the parameters ˆ and ˆ in equations (4)
and (5), we can get the mean and the variance of EIW distribution as shown in
Table (2):
Table (2)
The estimated mean and variance of EIW distribution
)( xE )( 2xE )( xV
50.1547 5.085238 2.26264 5.60568 0.486147
60.37536 5.085291 2.34661 6.029741 0.5229219
60.37527 6.102346 2.013053 4.382492 0.3301104
70.62036 6.102282 2.065445 4.613583 0.3475178
80.89695 7.119418 1.881189 3.798702 0.2598307
Construction actuarial model for aggregate loss 8095
From Table (2), we can notice there is direct relationship between the value of and the value of the mean and the variance of distribution. Also, there is inverse
relationship between the value of and the value of the mean and the variance
of distribution.
Estimation the mean and the variance of Aggregate losses distribution:
When the annual frequency of losses from a portfolio follows a Poisson
distribution with parameter 8 by substituting in equations (15) and (16), or
the Negative Binomial distribution with parameters 6.0 and 10 pr by
substituting in equations (17) and (18) in table (3) as follows:
Table (3)
Estimation the mean and the variance of Aggregate losses distribution
Concluding remarks:
In this study, we address the exponentaited inverted weubil distribution issue and
its empirical application of aggregate losses. By testing the tail of the EIW
distribution, we find it has heavy tail. The maximum likelihood method was
applied for estimating the parameters of distribution. Under collective risk model,
we estimate the mean and the variance of the aggregate losses distribution where
the frequency distribution for claim counts is Poisson or Negative Binomial. If we
identify the aggregate losses distribution, we can depend on it to ratemaking,
arrangement for stop of loss reinsurance and estimate the needed loss reserve.
Poisson distribution Negative Binomial
distribution
)( sE )( sV )( sE )( sV
50.1547 5.085238 18.1011 44.84544 33.93958 199.2747
60.37536 5.085291 18.77309 48.23793 35.19992 214.3496
60.37527 6.102346 16.10442 35.05993 30.19579 156.916
70.62036 6.102282 16.52356 36.90866 30.98168 165.1902
80.89695 7.119418 15.04951 30.38961 28.21783 136.6051
8096 Osama Hanafy Mahmoud
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Construction actuarial model for aggregate loss 8097
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Received: October 17, 2014; Published: November 18, 2014