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![]() | Hydrology Menu | ![]() | ![]() | Finance Menu |
Activate the actuary menu. The option PML concerns
the tarification with respect to the probable maximum loss.
The next four options provide a first insight in the behavior
of reserve processes within a finite time horizon. Note that
the StatPascal program initres.xpl must be utilized for computing
an estimate of the T-year initial reserve. Actuary Menu
Reserve Process Path
Ruin Time Data
Ruin Probability
PML
Reserve Process Path
Let S(t) = X[1] + ... + X[N(t)] be the total claims process
based on claim sizes X[i] and a claim number
process N(t). With the option Reserve Process Path one
can generate the path of a reserve process
U(t) = s + E(S(t)) + b(t) - S(t), t >= 0,
where
Recollect that claim sizes X[i] and their arrival
times T[i] may be jointly generated and visualized
by Data... Generate Counting/Point
Process... Marked Poisson Process....
Exponential Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
an Exponential (GP 0) distribution
with location and scale
parameters mu and sigma. One must specify these parameters in
the Reserve Process Path dialog box.
Pareto Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a Pareto (GP 1) distribution
with shape, location and scale
parameters alpha, mu and sigma. One must specify
these parameters in
the Reserve Process Path dialog box.
GP Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a generalized Pareto (GP) distribution
with shape, location
and scale parameters gamma, mu and sigma. One must
specify these parameters in
the Reserve Process Path dialog
box.
Ruin Time Data
Ruin occurs, when the reserve process path U(t) becomes negative.
Ruin times are simulated within the framework as given
in Reserve Process Path. Specify
the filename of the
data set and the Number of Simulations. The ruin times
or, if no ruin occurs, the selected time horizon
are stored as Xtremes Univariate Data.
Select one of the following submenu items to specify the claim size
distribution:
Exponential Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
an Exponential (GP 0) distribution
with location and scale
parameters mu and sigma. One must specify
these parameters in
the Ruin Time Data dialog
box.
Pareto Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a Pareto (GP 1) distribution
with shape, location and scale
parameters alpha, mu and sigma. One must specify
these parameters in
the Ruin Time Data dialog box.
GP Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a generalized Pareto (GP) distribution
with shape, location
and scale parameters gamma, mu and sigma. One must
specify these parameters in
the Ruin Time Data dialog
box.
Ruin Probability
Simulate ruin probabilities within a finite time
horizon T. The simulation runs
within the framework as given
in Reserve Process Path.
The probability of ruin up to time T is
psi(T,s) = P { tau(s) <= T }.
Applying the Ruin Probabilty option, one can calculate ruin
probabilities for initial reserves which
must be specified by smin, smax and steps.
The resulting pairs (s, psi(T,s)) are stored as
Xtremes Multivariate Data.
Exponential Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
an Exponential (GP 0) distribution
with location and scale
parameters mu and sigma. One must specify these parameters
in
the Ruin Probability dialog box.
Pareto Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a Pareto (GP 1) distribution
with shape, location and scale
parameters alpha, mu and sigma. One must specify
these parameters in
the Ruin Probability dialog box.
GP Claim Sizes
Select claim sizes X[1], X[2], X[3], ... distributed according to
a generalized Pareto (GP) distribution
with shape, location
and scale parameters gamma, mu and sigma. One must
specify these parameters in
the Ruin Probability dialog
box.
PML
The tarification of policies with respect to the
probable maximum loss (PML) is now on the agenda. First,
read carefully Section 12.3 in Statistical Analysis. Our
subsequent explanations are closely
related to those given in the Demos 9.1 and 9.2.
The option Analyze Nearest Neighbors is applicable to
a bivariate data set of type
Xtremes Multivariate Data with PMLs in the first column and
the pertaining claim sizes in the second column (e.g.,
select the columns "PML" and "Claim sizes" from im-pmlfi.dat
by applying Data... Transform Data... Select Columns).
The option Analyze Segments is applicable to a data set
of type Xtremes Multivariate Data with group numbers, lower and upper
boundaries, PMLs, claim sizes and claim degrees in the first
six columns (as given in im-pmlfi.dat).
If the given data set only consists of PMLs and claim sizes,
then the options Transform to Segments and Add
Degrees enable
the user to prepare a data set for the analysis in PML groups.
A first insight into the PML groups is gained by
applying Functional Parameter Transformation.
Transfer to Segments
A bivariate data set of type Xtremes Multivariate Data, with PMLs and
claim sizes in the two columns, is transformed to segments, i.e. to
PML groups determined by priorities p[1], ... , p[k].
These priorities must be written into an edit field on the right-hand
side of the Add Priorities dialog box. These values can also
be entered by means
of from, to, step or, respectively,
Block Size.
In the latter case, the priorities are given by PMLs.
The new data set is stored to a file with "Number of PML group", "Lower bound",
"Upper Bound", "PML" and "Claim sizes" in the first six columns.
Claim degrees can be added by
means of Add Degrees.
Add Degrees
Add claim degrees x[i,j]/z[i,j] to data sets as obtained by the
option Transfer to Segments,
where x[i,j] is a
claim size belonging to PML group i and z[i,j] is the
pertaining PML. The claim degrees are written into the sixth column.
Functional Parameter Transformation
For each of the PML groups, the priorities (boundaries), the center,
the sample means and variances of the claim sizes and the
sample means and variances of the claim degrees are stored in
a file. This option is applicable to a data set as
created by Add Degrees.
Analyze Segments
The option Analyze Segments is applicable to a data set
of type Xtremes Multivariate Data with group numbers, lower and upper
boundaries, PMLs, claim sizes and claim degrees in the first
six columns (as
already mentioned in PML).
First, choose Claim sizes or Claim degrees in
the Analyze Segments dialog box. Pressing OK, the claim
sizes or the claim degrees within each of the PML groups are
written to files seg*.dat with * denoting the group number.
Only values exceeding the threshold are dealt with. The
threshold itself (likewise, the threshold divided by the
upper group boundary) is included in the data set in the case of
claim sizes (claim degrees).
Several estimators (Hill(GP1/GP2),
MLE(GP),
Moment(GP),
MLE(GP0)) can either be applied to
claim sizes or claim degrees within each PML group.
If the Hill estimator is applied to claim degrees,
then the data are internally transformed so that the Hill estimator becomes
applicable within a beta model. Parameters of estimated beta
distributions for the original claim degrees are finally stored
in a data set.
For each of the PML groups, the group number, the center of
the group, the sample mean, the parametric mean of the estimated
GP distribution and the estimated
shape, location and scale parameters are written
to a text file with the default name para.dat.
In addition, if one of the
options Plot DF or Plot QF is activated, then parametric
and nonparametric distribution or quantile functions
are displayed in graphic windows for each of the PML groups
(for which the parametric estimation was carried out successfully).
Analyze Nearest Neighbors
The option Analyze Nearest Neighbors is applicable to
a bivariate data set of type
Xtremes Multivariate Data with PMLs in the first column and
the pertaining claim sizes in the second column (as
already mentioned in PML). This option concerns
the computations as carried out in Examples 12.3.3 and 12.3.4
in Statistical Analysis.
The estimation of the
conditional expectation E(X|z) of claim sizes given the PML
equal to z can be based on the
claim sizes y[j], j = 1, ..., k, pertaining to
the k PMLs z[j] closest to z. Then,
sigma * y[j]/k
is an empirical estimate of E(X|z).
Likewise, the estimation can be carried out within a GP model
based on the nearest neighbors. The parametric mean is used
as an estimate of E(X|z).
In the Analyze PML (Nearest Neighbors) dialog box, one
must select either the MLE(GP) or
an MLE of the scale parameter (for a fixed shape and location
parameter) and the number of neighbors. The
computations are carried out for
each PML z. Then the PML, the empirical
and parametric estimates of the conditional
expectation E(X|z) and the parameters of the estimated GP
distribution are written to a file.
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