Graphics Reference
In-Depth Information
log-normal distribution:
σ
σ
lnx
−
μ
−
lnx
−
μ
L
(
x
)=
exp
μ
+
Φ
+
x
−
Φ
;
σ
σ
Pareto distribution:
λ
α
−
α
λ
−
(
λ
+
x
)
L
(
x
)=
;
α
−
Burr distribution:
λ
τ
Γ
τ
τ
α
α
Γ
−
+
x
τ
τ
, α
τ
;
λ
L
(
x
)=
B
+
−
+
x
;
Γ
α
λ
x
τ
λ
x
τ
(
)
+
+
Weibull distribution:
Γ
(
+
τ
)
τ
, βx
α
−
βx
α
L
(
x
)=
Γ
+
+
xe
;
β
τ
gamma distribution:
α
β
F
L
(
x
)=
(
x, α
+
, β
)+
x
−
F
(
x, α, β
)
;
mixture of two exponential distributions:
a
β
−
a
L
(
x
)=
−
exp
(−
β
x
) +
−
exp
(−
β
x
)
.
β
From a curve-fitting point of view, the advantage of using the LEVFs rather than
the cdfs is that both the analytical function and the corresponding observed func-
tion L
n
, based on the observed discrete cdf, are continuous and concave, whereas
the observed claim size cdf F
n
is a discontinuous step function. Property (iii)implies
that the limited expected value function determines the corresponding cdf uniquely.
When the limited expected value functions of two distributions are close to each
other, not only are the mean values of the distributions close to each other, but the
whole distributions are too.
Since the LEVF represents the claim size distribution in the monetary dimension,
it is usually used exclusively toanalyze price data. In Fig.
.
,we depict the empirical
and analytical LEVFs for the two distributions that best fit the PCS catastrophe loss
amounts (as suggested by the mean excess function). We can see that the Pareto law
is definitely superior to the log-normal one.
Probability Plot
6.3.3
he purpose of the probability plot is to graphically assess whether the data comes
fromaspecificdistribution. Itcanprovidesomeassurancethat this assumption isnot
being violated, orit can providean early warning of aproblem with our assumptions.
