BSSN {gamlssbssn}

Bimodal Skew Symmetric Normal Distribution

Description

These functions define the Bimodal Skew Symmetric Normal Distribution. This is a four parameter distribution and can be used to fit a GAMLSS model.The functions dBSSN, pBSSN, qBSSN and rBSSN define the probability distribution function, the cumulative distribution function, the inverse cumulative distribution functions and the random generation for the Bimodal Skew Symmetric Normal Distribution; respectively.

Usage

BSSN(mu.link = "identity", sigma.link = "log", nu.link = "identity",
  tau.link = "log")

dBSSN(x, mu = 0, sigma = 1, nu = 1, tau = 0.5, log = FALSE)

pBSSN(q, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE,
  log.p = FALSE, log = T)

qBSSN(p, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE,
  log.p = FALSE)

rBSSN(n, mu = 0, sigma = 1, nu = 1, tau = 0.5)

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 deafult for the sigma parameter

nu.link

Defines the nu.link, with identity link as the default for the nu parameter

tau.link

Defines the tau.link, with log link as the default for the tau parameter

x, q

Vector of quantiles

mu

Vector of location parameter values

sigma

Vector of scale parameter values

nu

Vector of nu parameter values

tau

Vector of bimodality parameter values

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

The probability density function of the BSSN distribution is given by

f_Y(y|\mu, \sigma, \nu, \tau)= c[\tau + (y-\nu)^(2)]e^{-\sigma(y-\mu)^(2)}

for -\infty < y < \infty, where c = 2\sigma^(3/2) / \gamma \sqrt\pi, \gamma= 1 + 2 \sigma \theta, \theta = \tau + \delta^{2}, \delta= \nu - \mu. -\infty <\mu <\infty and -\infty < \nu < \infty are location parameters and \sigma >0 and \tau \geq 0 denote the scale and bimodality parameters respectively.

References

Hassan, M. Y. and El-Bassiouni M. Y. (2015). Bimodal skew-symmetric normal distribution,Communications in Statistics-Theory and Methods, 45, part 5, pp 1527–1541.

Hossain, A.Rigby, R. A. Stasinopoulos D. M. and Enea, M. A flexible approach for modelling proportion response variable:LGD, 31st International workshop for Statistical Modelling Society,1, pp 127–132.

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.

Examples

op<-par(mfrow=c(3,3))
curve(dBSSN(x,  mu=1, sigma=0.1, nu=1, tau=1),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=1, sigma=0.1, nu=1, tau=5),-12, 12,ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=1, sigma=0.1, nu=1, tau=10),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=1, sigma=0.1, nu=1, tau=20),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=1, sigma=0.1, nu=0, tau=4),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=-1, sigma=0.1, nu=0, tau=3),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=1, sigma=0.1, nu=2, tau=0),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=-1, sigma=0.1, nu=-2, tau=0),-12, 12, ylab="f(x)", main="BSSN")
curve(dBSSN(x,  mu=-1, sigma=0.1, nu=-3, tau=0.8),-12, 12, ylab="f(x)", main="BSSN")
par(op)