| Type: | Package |
| Title: | The p-Generalized Normal Distribution |
| Version: | 2.0 |
| Date: | 2015-11-23 |
| Author: | Steve Kalke |
| Maintainer: | Steve Kalke <steve.kalke@googlemail.com> |
| Description: | Evaluation of the pdf and the cdf of the univariate, noncentral, p-generalized normal distribution. Sampling from the univariate, noncentral, p-generalized normal distribution using either the p-generalized polar method, the p-generalized rejecting polar method, the Monty Python method, the Ziggurat method or the method of Nardon and Pianca. The package also includes routines for the simulation of the bivariate, p-generalized uniform distribution and the simulation of the corresponding angular distribution. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| LazyData: | true |
| NeedsCompilation: | no |
| Packaged: | 2015-11-24 07:39:02 UTC; Steve |
| Repository: | CRAN |
| Date/Publication: | 2015-11-24 15:08:34 |
The p-Generalized Normal Distribution
Description
The pgnorm-package includes routines to evaluate (cdf,pdf) and simulate the univariate p-generalized normal distribution with form parameter p, expectation mean and standard deviation \sigma. The pdf of this distribution is given by
f(x,p,mean,\sigma)=(\sigma_p/ \sigma) \, C_p \, \exp \left( - \left( \frac{\sigma_p}{\sigma } \right)^p \frac{\left| x-mean \right|^p}{p} \right) ,
where C_p=p^{1-1/p}/2/\Gamma(1/p) and \sigma_p^2=p^{2/p} \, \Gamma(3/p)/\Gamma(1/p), which becomes
f(x,p,mean,\sigma)=C_p \, \exp \left( - \frac{\left| x \right|^p}{p} \right),
if \sigma=\sigma_p and mean=0. The random number generation can be realized with one of five different simulation methods including the p-generalized polar method, the p-generalized rejecting polar method, the Monty Python method, the Ziggurat method and the method of Nardon and Pianca. Additionally to the simulation of the p-generalized normal distribution, the related p-generalized uniform distribution on the p-generalized unit circle and the corresponding angular distribution can be simulated by using the functions "rpgunif" and "rpgangular", respectively.
Details
| Package: | pgnorm |
| Type: | Package |
| Version: | 2.0 |
| Date: | 2015-11-23 |
| License: | GPL (>= 2) |
| LazyLoad: | yes |
Author(s)
Steve Kalke <steve.kalke@googlemail.com>
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm(10,3)Dataset 1 of the Monty Python method
Description
The dataset contains tail algorithm constants for sampling from the tail of the p-generalized normal distribution in context of a simulation of the p-generalized normal distribution with the Monty Python method.
Usage
data(datasetpgnmp1)Examples
data(datasetpgnmp1)Dataset 2 of the Monty Python method
Description
The dataset contains optimal rectangle widths in context of a simulation of the p-generalized normal distribution with the Monty Python method.
Usage
data(datasetpgnmp2)Examples
data(datasetpgnmp2)Dataset of the Ziggurat method
Description
The dataset contains tail algorithm constants for sampling from the tail of the p-generalized normal distribution in context of a simulation of the p-generalized normal distribution with the Ziggurat method.
Usage
data(datasetpgnzig)Examples
data(datasetpgnzig)A function to evaluate the p-generalized normal density
Description
The function evaluates the density f(x,p,mean,sigma) of the univariate p-generalized normal distribution according to
f(x,p,mean,\sigma)=(\sigma_p/ \sigma) \, C_p \, \exp \left( - \left( \frac{\sigma_p}{\sigma } \right)^p \frac{\left| x-mean \right|^p}{p} \right) ,
where C_p=p^{1-1/p}/2/\Gamma(1/p) and \sigma_p^2=p^{2/p} \, \Gamma(3/p)/ \Gamma(1/p).
Usage
dpgnorm(y, p, mean, sigma)Arguments
y |
The real argument of the function. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
mean |
A real number expressing the expectation of the distribution. The default is 0. |
sigma |
A positive number expressing the standard deviation of the distribution. The default is |
Value
A real number.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-dpgnorm(0,3,1,2)A function to evaluate the p-generalized normal cdf
Description
The function evaluates the cdf of the univariate p-generalized normal distribution according to the density
f(x,p,mean,\sigma)=(\sigma_p/ \sigma) \, C_p \, \exp \left( - \left( \frac{\sigma_p}{\sigma } \right)^p \frac{\left| x-mean \right|^p}{p} \right) ,
where C_p=p^{1-1/p}/2/\Gamma(1/p) and \sigma_p^2=p^{2/p} \, \Gamma(3/p)/\Gamma(1/p) .
Usage
ppgnorm(y, p, mean, sigma)Arguments
y |
A real number, the argument of the function. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
mean |
A real number expressing the expectation of the distribution. The default is 0. |
sigma |
A positive number expressing the standard deviation of the distribution. The default is |
Value
A real number.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-ppgnorm(2,p=3) A random number generator for the angular distribution
Description
The function simulates the univariate angular distribution corresponding to the p-generalized uniform distribution on the p-generalized unit circle.
Usage
rpgangular(n,p)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgangular(10000,3)A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate p-generalized normal distribution by using one of the following methods: the p-generalized polar method (pgenpolar), the p-generalized rejecting polar method (pgenpolarrej), the Monty Python method (montypython), the Ziggurat method (ziggurat) and the method of Nardon and Pianca (nardonpianca).
Usage
rpgnorm(n, p, mean, sigma, method)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. In case of the Monty Python method and the Ziggurat method, p can be chosen from |
mean |
A real number expressing the expectation of the distribution. The default is 0. |
sigma |
A positive number expressing the standard deviation of the distribution. The default is |
method |
A string expressing the method to be used for the simulation ("pgenpolar", "pgenpolarrej", "montypython", "ziggurat" or "nardonpianca"). The default is "nardonpianca". |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm(10000,3,method="pgenpolar")A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate, central, p-generalized normal distribution by using the Monty Python method.
Usage
rpgnorm_montypython(n,p)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. In case of the Monty Python method, p can be chosen from |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm_montypython(10000,3)A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate, central, p-generalized normal distribution by using the method of Nardon and Pianca.
Usage
rpgnorm_nardonpianca(n,p)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm_nardonpianca(10000,3)A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate, central, p-generalized normal distribution by using the p-generalized polar method.
Usage
rpgnorm_pgenpolar(n,p)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm_pgenpolar(10000,3)A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate, central, p-generalized normal distribution by using the p-generalized rejecting polar method.
Usage
rpgnorm_pgenpolarrej(n,p)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm_pgenpolarrej(10000,3)A random number generator for the p-generalized normal distribution
Description
The function simulates the univariate, central, p-generalized normal distribution by using the Ziggurat method.
Usage
rpgnorm_ziggurat(n,p,x)Arguments
n |
The natural number of random variables to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. In case of the Ziggurat method, p can be chosen from |
x |
(optional) A real vector containing the |
Value
An n-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgnorm_ziggurat(10000,3)A random number generator for the p-generalized uniform distribution
Description
The function simulates the bivariate, p-generalized uniform distribution on the p-generalized unit circle.
Usage
rpgunif(n,p)Arguments
n |
The natural number of random vectors to be simulated. |
p |
A positive number expressing the form parameter of the distribution. The default is 2. |
Value
A real n \times 2 matrix.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-rpgunif(10000,3)A function for setting up the Ziggurat.
Description
The function approximates the rightmost x-coordinates of the first n-1 rectangles defining the Ziggurat in case of the central, p-generalized normal distribution.
Usage
zigsetup(p, n, tol)Arguments
p |
A positive number expressing the form parameter of the distribution. The default is 2. In case of the Ziggurat method, p can be chosen from |
n |
The number of rectangles that build up the Ziggurat. The default is |
tol |
A positive number expressing the approximation accuracy of the function. The default is |
Value
An (n-1)-dimensional, real vector.
Author(s)
Steve Kalke
References
S. Kalke and W.-D. Richter (2013)."Simulation of the p-generalized Gaussian distribution." Journal of Statistical Computation and Simulation. Volume 83. Issue 4.
Examples
y<-zigsetup(3,20,10^(-6))