Mathematical and Statistical Methods for Actuarial Sciences and Finance

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Transformation kernel estimation of insurance claim cost distributions 47

And finally, the optimal bandwidth parameter based on the asymptotic mean integrated
squared error measure forgˆ(y∗)is:


boptg =k

−^25
2

(∫

1

− 1

K(t)^2 dt

∫a

−a

g(y)dy

)^15 (∫

a

−a

{

g′′(y)

} 2

dy

)−^15

n−

(^15)


=

(

1

5

)−^25 (

3

5

(

1

4

a

(

− 40 a^2 + 48 a^4 + 15

)))^15

×

(

360 a

(

− 40 a^2 + 144 a^4 + 5

))−^15

n−

1

(^5) , (11)
The difficulty that arises when implementing the transformation kernel estimation
expressed in (9) is the selection of the value ofl. This value can be chosen subjectively
as discussed in the simulation results by Bolanc ́e et al. [3]. LetXi,i= 1 ,...,n,be
iid observations from a random variable with an unknown densityfx. The transforma-
tion kernel density estimator offxis called KIBMCE (kernel inverse beta modified
Champernowne estimator).


3 VaR estimation


In finance and insurance, the VaR represents the magnitude of extreme events and
therefore it is used as a risk measure, but VaR is a quantile. Letxbe a loss random
variable with distribution functionFx; given a probability levelp,theVaRofx
isVaR(x,p)=inf{x,Fx(x)≥p}.SinceFxis a continuous and nondecreasing
function, thenVaR(x,p)=Fx−^1 (p),wherepis a probability near 1 (0.95, 0.99,...).
One way of approximatingVaR(x,p)is based on the empirical distribution function,
but this has often been criticised because the empirical estimation is based only on
a limited number of observations, and evennpmay not be an integer number. As
an alternative to the empirical distribution approach, classical kernel estimation of
the distribution function can be useful, but this method will be very imprecise for
asymmetrical or heavy-tailed variables.
Swanepoel and Van Graan [17] propose to use a nonparametric transformation of
the data, which is equal to a classical kernel estimation of the distribution function.
We propose to use a parametric transformation based on a distribution function.
Given a transformation functionTr(x), it follows thatFx(x)=FTr(x)(Tr(x)).
So, the computation ofVaR(x,p)is based on the kernel estimation of the distribution
function of the transformed variable.
Kernel estimation of the distribution function is [1, 14]:


FˆTr(x)(Tr(x))=^1
n

∑n

i= 1

∫ Tr(x)−Tr(Xi)
b
− 1

K(t)dt. (12)

Therefore, theVaR(x,p)can be found as:

VaR(x,p)=Tr−^1 [VaR(Tr(x),p)]=Tr−^1

[

FˆTr−^1 (x)(p)

]

. (13)
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