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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

Discrete domain: Uniform Binomial
Poisson Negative Binomial
SUM domain: Gaussian Gaussian-GCauchy
Student Sum-Stable
Log-Normal Non-central Student
MAX domain: Gumbel (EV 0) Frechet (EV 1)
Weibull (EV 2) EV
Actual Distributions
POT domain: Exponential (GP 0) Pareto (GP 1)
Beta (GP 2) GP

Multivariate Mode

SUM domain: Bivariate Gaussian
MAX domain: Gumbel-McFadden (EV) Marshall-Olkin (EV)
Huesler-Reiss (EV)
POT domain: Gumbel-McFadden (GP) Marshall-Olkin (GP)
Huesler-Reiss (GP)

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