Distribution Menu
The distribution menu contains entries to plot parametric
curves (e.g., densities, distribution functions, quantile
functions, mean and median excess functions).
The parametric options presented depend on the selected
mode and domain, so you will not see all options of the system
within a particular mode.
The following options are provided:
Univariate Mode
Multivariate Mode
Distributions Available as XPL-Functions
Gamma
Univariate Discrete Domain
Uniform Discrete Distribution
Plot a histogram of the discrete uniform distribution
with support on the integers from r to s. Specify integers
r and s with r smaller than s in the edit fields.
Mark the Close before plotting field to let the dialog
box disappear when the curve is plotted.
Binomial
Plot a histogram of the binomial distribution. Specify a positive
integer n and p in [0,1] in the edit fields.
Poisson
Plot a histogram of the Poisson distribution. The positive parameter
lambda may be specified by entering a value in the edit field.
One can use the parameter varying
mouse mode to change the parameter of the histogram in the plot window.
Negative Binomial
Plot a histogram of the negative binomial distribution which is a
mixed Poisson distribution, where the mixing is carried out with
respect to a gamma density with positive scale parameter sigma.
The following parameters must be specified:
r | parameter | nonnegative real
|
p | parameter | ( 0 , 1 )
|
Note that r is the shape parameter of the mixing gamma
distribution, where p = 1/( 1+sigma ) with sigma > 0
is the scale parameter.
Univariate SUM Domain
Gaussian
Plot a density, distribution function and quantile function of
the Gaussian distribution. The following
parameters must be specified:
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying
mouse mode to change parameters of curves in the plot window.
Gaussian-GCauchy
Plot a density, distribution function and quantile function of a mixture
of a Gaussian and a GCauchy
(generalized Cauchy) distribution. The following parameters must be
specified:
mu | location | real
|
sigma | scale (Gaussian) | positive real
|
d | contamination | [ 0 , 1 ]
|
alpha | shape (GCauchy) | positive real
|
sigma1 | scale (GCauchy) | positive real
|
The contamination parameter d determines the weight of the GCauchy
distribution. One can use the
parameter varying mouse mode to
plot curves with changed parameters.
Student
Plot a density, distribution function and quantile function of
the Student distribution. The following parameters
must be specified:
sigma | scale | positive real
|
alpha | shape | positive real
|
Xtremes uses the parameterization as given in Statistical Analysis,
page 94, with the Cauchy distribution for alpha = 1 and the Gaussian
distribution as a limiting distribution when alpha goes to infinity.
One can use the parameter varying mouse
mode to change parameters of curves in the plot window.
Sum-Stable
Plot a density, distribution function and quantile function of
a sum-stable distribution. The following parameters must be
specified:
alpha | tail-index parameter | between 0 and 2
|
beta | skewness parameter | between -1 and 1
|
mu | location parameter | real
|
alpha | scale parameter | positive real
|
Xtremes uses the continuous parameterization as described in
Statistical Analysis, Section 6.3.
Log-Normal
Plot a density, distribution function and quantile function of
the log-normal. The following parameters
must be specified:
sigma | scale | positive real
|
alpha | shape | positive real
|
Xtremes uses the parameterization as given in Statistical Analysis,
page 32.
One can use the parameter varying
mouse mode to change parameters of curves in the plot window.
Non-central Student
Univariate MAX Domain
Gumbel (EV 0)
Plot a density, distribution function and quantile function of the
Gumbel (EV 0) distribution. The following parameters must be
specified:
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying
mouse mode to change parameters of curves in the plot window.
Frechet (EV 1)
Plot a density, distribution function and quantile function of the
Frechet (EV 1) distribution. The following parameters must be
specified:
alpha | shape | positive real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying
mouse mode to change parameters of curves in the plot window.
Weibull (EV 2)
Plot a density, distribution function and quantile function of the
Weibull (EV 2) distribution. The following parameters must be
specified:
alpha | shape | negative real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying mouse mode
to change parameters of curves in the plot window.
EV
Plot a density, distribution function and quantile function of the
Extreme Value (EV) distribution in von Mises representation. The
following parameters must be specified:
gamma | shape | real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying
mouse mode to change parameters of curves in the plot window.
Actual Distributions
This option allows to visualize the m-th power F**m for certain dfs
F.
The dialog box provides the following options:
- Distribution
- Select one of the listed dfs by clicking on the radio buttons.
- m
- Enter the value for exponent.
- Parameter edit fields on the right side: alpha, gamma, sigma,
mu
- Enter values for shape, location and scale parameters in the pertaining
edit field. Note that these edit fields may differ, dependent on the
selected distribution.
Univariate GP Domain
Exponential (GP 0)
Plot a density, distribution function and quantile function of the
exponential (GP 0) distribution. The following parameters must be
specified:
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying mouse mode
to change parameters of curves in the plot window.
Pareto (GP 1)
Plot a density, distribution function and quantile function of the
Pareto (GP 1) distribution. Furthermore, the dialog box provides
options to plot the mean and median excess function of the Pareto
distribution. The following parameters must be specified:
alpha | shape | positive real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying mouse mode
to change parameters of curves in the plot window.
Beta (GP 2)
Plot a density, distribution function and quantile function of the
Beta (GP 2) distribution. Furthermore, the dialog box provides
options to plot the mean and median excess function of the Beta
distribution. The following parameters must be specified:
alpha | shape | negative real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying mouse mode
to change parameters of curves in the plot window.
GP
Plot a density, distribution function and quantile function of the
Generalized Pareto (GP) distribution in von Mises representation. The
following parameters must be specified:
gamma | shape | real
|
mu | location | real
|
sigma | scale | positive real
|
One can use the parameter varying mouse mode
to change parameters of curves in the plot window.
Multivariate SUM Domain
Bivariate Gaussian
Plot a density, distribution function or suvivor function of a
bivariate Gaussian distribution. The following parameters must be
specified:
mu1 | location (1st component) | real
|
sigma1 | scale (1st component) | positive real
|
mu2 | location (2nd component) | real
|
sigma2 | scale (2nd component) | positive real
|
rho | correlation coefficient | [ -1 , 1 ]
|
Select the pertaining radio button to let the curve being plotted
by either means of a Contour
Plot or a
three-dimensional plot. The latter option will display the function
graph in a three-dimensional coordinate system.
Mark the Close before plotting field to let the dialog box
disappear when the curve is plotted.
Multivariate MAX Domain
Gumbel-McFadden (EV)
Plot the density, distribution function or survivor function of
the bivariate Gumbel-McFadden distribution. Select the parameters
of univariate EV distributions
(gamma-parameterization) in the univariate margins.
In addition, specify:
lambda | dependence parameter | larger than 1
|
Alternatively, select the canonical dependence parameter
theta in [0,1] instead of lambda.
Select the pertaining radio button to let the curve being plotted
either by means of
a Contour Plot or a
three-dimensional plot. The latter option will display the
function graph in a three-dimensional coordinate system.
Mark the Close before plotting field to let the dialog box
disappear when the curve is plotted.
Marshall-Olkin (EV)
Plot the distribution or survivor function of the bivariate
Marshall-Olkin distribution. Select the parameters
of univariate EV distributions
(gamma-parameterization) in the univariate margins.
In addition, specify:
lambda | dependence parameter | [ 0 , 1 ]
|
Recall that the dependence parameter lambda is equal to
the canonical dependence parameter theta.
Select the pertaining radio button to let the curve being plotted
either by means of
a Contour Plot or a
three-dimensional plot. The latter option will display the function
graph in a three-dimensional coordinate system.
Huesler-Reiss (EV)
Plot density and distribution function of the Huesler-Reiss
distribution. Select the parameters
of univariate EV distributions
(gamma-parameterization) in the univariate margins.
In addition, specify:
lambda | dependence parameter | positive real
|
Alternatively, select the canonical dependence parameter
theta in [ 0 , 1 ] instead of lambda.
Select the pertaining radio button to let the curve being plotted
by either a Contour Plot or a
three-dimensional plot. The latter option will display the function
graph in a three-dimensional coordinate system.
Bivariate POT Domain
Recall that multivariate GP distributions W and multivariate
EV distributions are connected by the formula
W = log G, for log G > -1 ,
see Statistical Analysis, Chapter 10.
Gumbel-McFadden (GP)
Plot the bivariate GP distributions which are related to the
bivariate Gumbel-McFadden
distributions in the EV model.
The univariate margins are
GP distributions in the
gamma-parameterization.
Marshall-Olkin (GP)
Plot the bivariate GP distributions which are related to the
bivariate Marshall-Olkin
distributions in the EV model.
The univariate margins are
GP distributions in the
gamma-parameterization.
Huesler-Reiss (GP)
Plot the bivariate GP distributions which are related to the
bivariate Huesler-Reiss
distributions in the EV model.
The univariate margins are
GP distributions in the
gamma-parameterization.
Distributions as SP-Functions
Gamma Distributions
The standard gamma distribution with positive parameter r is
introduced in Statistical Analysis on the pages
112 and 132 by means of the densities. In the special case of
a positive integer r (see page 132), the gamma distribution
is the distribution of a sum of r iid standard exponential
random variables.
For sufficiently large r, the gamma df may be replaced by a
normal df or, to get a higher accuracy, by an expansion of
length 2 (see also Feller, Vol. II, page 538).
Up to r < 1000, the gamma df is computed exactly. Otherwise,
the gamma df at y is replaced by
Phi(z) + (1-z)**2 phi(z)/(3*sqrt(r))
with
z = (y-r)/sqrt(r),
where Phi and phi are the standard normal df and density.
© 2005 Xtremes Group
· updated Jun 21, 2005