Graphics Reference
In-Depth Information
thelossandwaiting timedistributions tooneortwo probability laws.Moreover, they
will allow for a visual assessment of the goodness-of-fit.
Mean Excess Function
6.3.1
For a random claim amount variable X, the mean excess function or mean resid-
ual life function is the expected payment per claim on a policy with a fixed amount
deductible of x, where claims of less than or equal to x are completely ignored:
x
∫
−
F
(
u
)
du
e
(
x
)=
E
(
X
−
x
X
x
)=
(
.
)
−
F
(
x
)
In practice, the mean excess function e is estimated by e
n
, based on a representative
sample x
,...,x
n
:
x
i
x
x
i
e
n
(
x
)=
−
x .
(
.
)
#
i
x
i
x
Note that in a financial risk management context, switching fromthe right tail to the
let tail, e
is referred to as the expected shortfall (Weron,
).
When considering the shapes of mean excess functions, the exponential distri-
bution with the cumulative distribution function (cdf) F
(
x
)
plays
a central role. It has the memoryless property, meaning that whether or not the in-
formation X
(
x
)=
−
exp
(−
βx
)
x is given, the expected value of X
−
x is the same as if one started
at x
. he mean excess function for the exponential distri-
bution is therefore constant. One can in fact easily calculate that e
=
andcalculatedE
(
X
)
(
x
)=
β for all
x
inthiscase.
If the distribution of X has a heavier tail than the exponential distribution, we
find that the mean excess function ultimately increases, and when it has a lighter tail
e
provides important information
on the sub-exponential or super-exponential nature of the tail of the distribution at
hand. hat is why, in practice, this tool is used not only to discover the relevant class
of the claim size distribution, but also to investigate all kinds of phenomena. We will
apply it to data on both PCS loss and waiting times.
Mean excess functions of the well known and widely used distributional classes
are given by the following formulae (selected shapes are sketched in Fig.
.
, while
the empirical mean excess functions e
n
x
ultimately decreases. Hence,the shape of e
x
(
)
(
)
(
x
)
for the PCS catastrophe data are plotted
in Fig.
.
):
log-normal distribution with cdf F
(
x
)=
Φ
(
log x
−
μ
)
σ
:
ln x
−
μ
−
σ
σ
σ
exp
μ
Φ
+
−
e
(
x
)=
−
x ,
ln x
−
μ
σ
−
Φ
where Φ
(ċ)
is the standard normal (with mean
and variance
) distribution
function;