Graphics Reference
In-Depth Information
Figure
.
.
Two-dimensional projection of the density evolution depicted in Fig.
.
. From the Ruin
Probabilities Toolbox
Quantile Lines
6.4.3
hefunction x
p
t
iscalledasample p-quantilelineifforeach t
t
, T
, x
p
t
isthe
(
)
[
]
(
)
sample p-quantile, i.e.,if it satisfies F
n
,whereF
n
is the empirical
distribution function (edf).Recall that fora sample of observations
(
x
p
−)
p
F
n
(
x
p
)
x
,...,x
n
,the
edf is defined as
n
#
F
n
(
x
)=
i
x
i
x
(
.
)
in other words, it is a piecewise constant function with jumps of size
n at points x
i
Burnecki et al. (
).
Quantile lines are a very helpful tool in the analysis of stochastic processes. For
example,theycanprovideasimplejustification ofthestationarity ofaprocess(orthe
lack of it); (see Janicki and Weron,
). In Figs.
.
,
.
, and
.
, they visualize
the evolution of the risk process.
Quantilelinescanbealsoausefultoolforcomparingtwoprocesses;seeFig.
.
.It
depicts quantile lines and two sample trajectories of the risk process and its diffusion
approximation; consult Burnecki et al. (
) for a discussion of different approxi-
mations in the context of ruin probability. he parameters of the risk process are the
same as in Fig.
.
. We can see that the quantile lines of the risk process and its ap-
proximation coincide. his justifies the use of the Brownian approximation for these
parameters of the risk process.
