Graphics Reference
In-Depth Information
α
:
Pareto distribution with cdf F
(
x
)=
−
λ
(
λ
+
x
)
λ
+
x
e
(
x
)=
,
α
;
α
−
x
τ
α
:
Burr distribution with cdf F
(
x
)=
−
λ
(
λ
+
)
λ
τ
Γ
τ
τ
−
α
α
−
Γ
+
λ
e
(
x
)=
ċ
Γ
(
α
)
λ
+
x
τ
x
τ
τ
, α
τ
,
B
x ,
ċ
−
+
−
−
x
τ
λ
+
where Γ
(ċ)
is the standard gamma function and B
(ċ
,
ċ
,
ċ)
is the beta function;
βx
τ
Weibull distribution with cdf F
(
x
)=
−
exp
(−
)
:
Γ
(
+
τ
)
τ
, βx
τ
βx
τ
e
(
x
)=
−
Γ
+
exp
(
)−
x ,
β
τ
where Γ
istheincompletegammafunction;
gamma distribution with cdf F
(ċ
,
ċ)
∫
x
α
−
exp
(
x
)=
β
(
βs
)
(−
βs
)
Γ
(
α
)
ds:
α
β
−
F
(
x, α
+
, β
)
e
(
x
)=
ċ
−
x ,
F
x, α, β
−
(
)
where F
is the gamma cdf;
mixture of two exponential distributions with cdf F
(
x, α, β
)
(
x
)=
a
−
exp
(−
β
x
) +
(
−
a
)
−
exp
(−
β
x
)
:
a
−
a
β
exp
β
x
β
exp
β
x
(−
)+
(−
)
e
(
x
)=
.
a exp
(−
β
x
)+(
−
a
)
exp
(−
β
x
)
A comparison of Figs.
.
and
.
suggests that log-normal, Pareto, and Burr dis-
tributions should provide a good fit for the loss amounts. he maximum likelihood
estimates of the parameters of these three distributions are as follows: μ
=
.
(Pareto),and α
and σ
=
.
(log-normal), α
=
.
and λ
=
.
=
.
,
and τ
λ
.
(Burr). Unfortunately, the parameters of the Burr dis-
tribution imply that the first moment is infinite, which contradicts the assumption
that the randomsequence ofclaimamounts
=
.
=
hasafinitemean. hisassumption
seems natural in the insurance world, since any premium formula usually includes
the expected value of X
k
. herefore, we exclude the Burr distribution from the anal-
ysis of claim severities.
he classification of waiting time data is not this straightforward. If we discard
large observations, then the log-normal and Burr laws should yield a good fit. How-
ever,ifall ofthewaiting times aretaken into account, then the empirical mean excess
function e
n
X
k
approximates a straight line (although one that oscillates somewhat),
which suggests that the exponential law could be a reasonable alternative.
(
x
)
