Package 'pgnorm'

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 <[email protected]>

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)

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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 $\sigma_p$.

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 $\sigma_p$.

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 $(1, \infty) \cup \{ 0.25, 0.45, 0.5, 0.6, 0.75 \}$.
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 $\sigma_p=p^{1/p} \, \sqrt{\Gamma(3/p)/ \Gamma(1/p)}$, the natural standard deviation of the $p$-generalized normal distribution.
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 $(1, \infty) \cup \{ 0.25, 0.45, 0.5, 0.6, 0.75 \}$.

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 $(1, \infty) \cup \{ 0.25, 0.45, 0.5, 0.6, 0.75 \}$.
x	(optional) A real vector containing the $2^8-1$ rightmost endpoints of the $2^8$ ziggurat-rectangles.

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 $(1, \infty) \cup \{ 0.25, 0.45, 0.5, 0.6, 0.75 \}$.
n	The number of rectangles that build up the Ziggurat. The default is $2^8$.
tol	A positive number expressing the approximation accuracy of the function. The default is $10^{-9}$.

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))

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