R: Power Exponential distribution for fitting a GAMLSS

PE {gamlss.dist} | R Documentation

## Power Exponential distribution for fitting a GAMLSS

### Description

The functions define the Power Exponential distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dPE`, `pPE`, `qPE` and `rPE` define the density, distribution function, quantile function and random generation for the specific parameterization of the power exponential distribution showing below. The functions `dPE2`, `pPE2`, `qPE2` and `rPE2` define the density, distribution function, quantile function and random generation of a standard parameterization of the power exponential distribution.

### Usage

    PE(mu.link = "identity", sigma.link = "log", nu.link = "log")
    dPE(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
    pPE(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
    qPE(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
    rPE(n, mu = 0, sigma = 1, nu = 2)
    PE2(mu.link = "identity", sigma.link = "log", nu.link = "log")
    dPE2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
    pPE2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
    qPE2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
    rPE2(n, mu = 0, sigma = 1, nu = 2)

### Arguments

`mu.link` | Defines the `mu.link`, with "identity" link as the default for the mu parameter
`sigma.link` | Defines the `sigma.link`, with "log" link as the default for the sigma parameter
`nu.link` | Defines the `nu.link`, with "log" link as the default for the nu parameter
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale parameter values
`nu` | vector of kurtosis parameter
`log, log.p` | logical; if TRUE, probabilities p are given as log(p).
`lower.tail` | logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required

### Details

Power Exponential distribution (PE) is defined as

f(y|mu,sigma,nu)=(1/sigma)*(nu*exp(-0.5*|z/c|^nu))/(c*2^(1+1/nu)*Gamma(1/nu))

where c=[2^(-2/nu)Gamma(1/nu)/Gamma(3/nu)]^0.5, for y=(-Inf,+Inf), μ=(-Inf,+Inf), σ>0 and ν>0. This parametrization was used by Nelson (1991) and ensures mu is the mean and sigma is the standard deviation of y (for all parameter values of mu, sigma and nu within the rages above)

Thw Power Exponential distribution (PE2) is defined as

f(y|mu,sigma,nu)=(nu *exp(-abs(z)^2))/(2*sigma*Gamma(1/nu))

### Value

returns a `gamlss.family` object which can be used to fit a Power Exponential distribution in the `gamlss()` function.

### Note

mu is the mean and sigma is the standard deviation of the Power Exponential distribution

### Author(s)

Mikis Stasinopoulos, Bob Rigby

### References

Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 57, 347-370.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

### See Also

`gamlss.family`, `BCPE`

### Examples

    PE()# gives information about the default links for the Power Exponential distribution
    # library(gamlss)
    # data(abdom)
    # h1<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE, data=abdom) # fit
    # h2<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE2, data=abdom) # fit
    # plot(h1)
    # plot(h2)
    # leptokurtotic
    plot(function(x) dPE(x, mu=10,sigma=2,nu=1), 0.0, 20,
     main = "The PE  density mu=10,sigma=2,nu=1")
    # platykurtotic
    plot(function(x) dPE(x, mu=10,sigma=2,nu=4), 0.0, 20,
     main = "The PE  density mu=10,sigma=2,nu=4")

[Package gamlss.dist version 5.3-2 Index]