Help for package MWright

Type:

Package

Title:

Mainardi-Wright Family of Distributions

Version:

0.3.2

Author:

Dexter Cahoy

Maintainer:

Dexter Cahoy <cahoyd@uhd.edu>

Description:

Implements random number generation, plotting, and estimation algorithms for the two-parameter one-sided and two-sided M-Wright (Mainardi-Wright) family. The M-Wright distributions naturally generalize the widely used one-sided (Airy and half-normal or half-Gaussian) and symmetric (Airy and Gaussian or normal) models. These are widely studied in time-fractional differential equations. References: Cahoy and Minkabo (2017) <doi:10.3233/MAS-170388>; Cahoy (2012) <doi:10.1007/s00180-011-0269-x>; Cahoy (2012) <doi:10.1080/03610926.2010.543299>; Cahoy (2011); Mainardi, Mura, and Pagnini (2010) <doi:10.1155/2010/104505>.

License:

GPL (≥ 3)

Encoding:

UTF-8

LazyData:

true

Imports:

stats, cubature

RoxygenNote:

6.1.1

NeedsCompilation:

Packaged:

2019-08-07 22:19:44 UTC; cahoyd

Repository:

CRAN

Date/Publication:

2019-08-07 23:00:05 UTC

MWright Package

Description

Contains random number generation, plotting, and estimation algorithms for the two-parameter one-sided and two-sided M-Wright (Mainardi-Wright) family. The M-Wright distributions naturally generalize widely used one-sided (Airy and half-normal or half-normal) and symmetric (Airy and Gaussian or normal) models. These are widely studied in time-fractional diffusion processes.

Details

References:

Cahoy and Minkabo (2017)<doi:10.3233/MAS-170388>

Cahoy (2012) <doi:10.1007/s00180-011-0269-x>

Cahoy (2012) <doi:10.1080/03610926.2010.543299>

Cahoy (2011)

Mainardi, Mura, and Pagnini (2010) <doi:10.1155/2010/104505>

Author(s)

Dexter Cahoy cahoyd@uhd.edu

One-sided M-Wright distribution

Description

Plots the density function.

Usage

dmwright1(ah, sh, m, max)

Arguments

ah

point estimate for shape parameter alpha.

sh

point estimate for scale parameter s.

m

number of data points (pairs) to use for plotting.

max

maximum x-axis value to use for plotting.

Value

numeric matrix

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


xy=dmwright1(0.45, 2.5, 1000, 10)
plot(xy[,1], xy[,2],  lwd = 2, type="l",ylab="", xlab="x")

mwright1_sided <- rmwright1(1000, 0.45, 2.5)
hist(mwright1_sided, br=30, prob=TRUE)
lines(xy[,1], xy[,2],  lwd=2 )

Two-sided M-Wright distribution

Description

Plots the density function.

Usage

dmwright2(ah, sh, m, max)

Arguments

ah

point estimate for shape parameter alpha.

sh

point estimate for scale parameter s.

m

number of data points (pairs) to use for plotting.

max

maximum x-axis value to use for plotting.

Value

numeric matrix

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


xy=dmwright2(0.45, 2.5, 1000, 10)
plot(xy[,1], xy[,2],  lwd = 2, type="l",ylab="", xlab="x")

mwright2_sided <- rmwright2(1000, 0.45, 2.5)
hist(mwright2_sided, br=30, prob=TRUE)
lines(xy[,1], xy[,2],  lwd=2 )

Interval estimation for one-sided M-Wright distribution

Description

Confidence intervals for the model parameters.

Usage

int_est1(x, lev)

Arguments

x

numeric vector

lev

confidence level.

Value

matrix

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


mwright_1sided <- rmwright1(1000, 0.7, 0.4)
int_est1(mwright_1sided ,0.95)

Interval estimation for two-sided M-Wright distribution

Description

Confidence intervals for the model parameters.

Usage

int_est2(x, lev)

Arguments

x

numeric vector

lev

confidence level.

Value

matrix

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


mwright_2sided <- rmwright2(1000, 0.7, 0.4)
int_est2(mwright_2sided ,0.95)

Distribution function for one-sided M-Wright distribution

Description

Calculates a left-tail probability.

Usage

pmwright1(alp, sc, upper)

Arguments

alp

point estimate for shape parameter alpha.

sc

point estimate for scale parameter s.

upper

non-negative upper quantile

Value

numeric

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


pmwright1(runif(1), runif(1,0,10),Inf  )

pmwright1(runif(1), runif(1,0,10), 0.5 )

Distribution function for two-sided M-Wright distribution

Description

Calculates a left-tail probability.

Usage

pmwright2(alp, sc, upper)

Arguments

alp

point estimate for shape parameter alpha.

sc

point estimate for scale parameter s.

upper

upper quantile

Value

numeric

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


pmwright2(runif(1), runif(1,0,10),  Inf )

pmwright2(runif(1), runif(1,0,10), 0.5 )

Point estimation for one-sided M-Wright distribution

Description

This provides point estimates for the shape and scale parameters.

Usage

point_est1(x)

Arguments

x

numeric vector.

Value

numeric vector

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples

x <- rmwright1(1000, 0.7, 0.4)
point_est1(x)

Point estimates for two-sided M-Wright distribution

Description

This provides point estimates for the shape and scale parameters.

Usage

point_est2(x)

Arguments

x

numeric vector.

Value

numeric vector

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples

x <- rmwright2(1000, 0.7, 0.4)
point_est2(x)

Random number generation for one-sided M-Wright distribution

Description

Generates random numbers.

Usage

rmwright1(n, nu, sc)

Arguments

n

sample size.

nu

a number between 0 and 1.

sc

a non-negative scale value.

Value

a vector of one-sided M-Wright distributed random numbers

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoy (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


mwright_1sided <- rmwright1(1000, 0.7, 0.4)
hist(mwright_1sided, br=30)

Random number generation for two-sided M-Wright distribution

Description

Generates random numbers.

Usage

rmwright2(n, nu, sc)

Arguments

n

sample size.

nu

a number between 0 and 1.

sc

a non-negative scale value.

Value

a vector of two-sided M-Wright distributed random numbers.

References

Cahoy and Minkabo (2017). Inference for three-parameter M-Wright distributions with applications. Model Assisted Statistics and Applications, 12(2), 115-125. https://doi.org/10.3233/MAS-170388

Cahoy (2012). Moment estimators for the two-parameter M-Wright distribution. Computational Statistics, 27(3), 487-497. https://doi.org/10.1007/s00180-011-0269-x

Cahoy (2012). Estimation and simulation for the M-Wright function. Communications in Statistics-Theory and Methods, 41(8), 1466-1477. https://doi.org/10.1080/03610926.2010.543299

Cahoyd (2011). On the parameterization of the M-Wright function. Far East Journal of Theoretical Statistics, 34(2), 155-164. http://www.pphmj.com/abstract/5767.htm

Mainardi, Mura, and Pagnini (2010). The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ., Volume 2010. https://doi.org/10.1155/2010/104505

Examples


mwright_2sided <- rmwright2(1000, 0.7, 0.4)
hist(mwright_2sided, br=30)