Power Exponential distribution for fitting a GAMLSS
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.
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)
mu.link |
Defines the |
sigma.link |
Defines the |
nu.link |
Defines the |
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 |
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))
returns a gamlss.family object which can be used to fit a Power Exponential distribution in the gamlss() function.
mu is the mean and sigma is the standard deviation of the Power Exponential distribution
Mikis Stasinopoulos, Bob Rigby
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.
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")
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.