SMNGdistribution function

SMNG and logSMNG Distributions

Density function, distribution function, quantile function and random generator for the SMNG distribution and the logSMNG. It requires the specification of a five prameters vector: mu, delta, gamma, lambda and beta.

dSMNG(
  x,
  mu = 0,
  delta,
  gamma,
  lambda,
  beta = 0,
  inf_sum = FALSE,
  rel_tol = 1e-05
)

pSMNG(q, mu, delta, gamma, lambda, beta, rel_tol = 1e-05)

qSMNG(p, mu, delta, gamma, lambda, beta, rel_tol = 1e-05)

rSMNG(n, mu, delta, gamma, lambda, beta)

dlSMNG(x, mu = 0, delta, gamma, lambda, beta, inf_sum = FALSE, rel_tol = 1e-05)

plSMNG(q, mu, delta, gamma, lambda, beta, rel_tol = 1e-05)

qlSMNG(p, mu, delta, gamma, lambda, beta, rel_tol = 1e-05)

rlSMNG(n, mu, delta, gamma, lambda, beta)

Arguments

• x, q: Vector of quantiles.
• mu: Location parameter, default set to 0.
• delta: Concentration parameter, must be positive.
• gamma: Tail parameter, must be positive.
• lambda: Shape parameter.
• beta: Skewness parameter, default set to 0 (symmetric case).
• inf_sum: Logical: if FALSE (default) the integral representation of the SMNG density is used, otherwise the infinite sum is employed.
• rel_tol: Level of relative tolerance required for the integrate procedure or for the infinite sum convergence check. Default set to 1e-5.
• p: Vector of probabilities.
• n: Sample size.

Returns

dSMNG and dlSMNG provide the values of the density function at a quantile x for, respectively a SMNG distribution and a log-SMNG.

pSMNG and plSMNG provide the cumulative distribution function at a quantile q.

qSMNG and qlSMNG provide the quantile corresponding to a probability level p.

rSMNG and rlSMNG generate n independent samples from the desired distribution.

Details

The SMNG distribution is a normal scale-mean mixture distribution with a GIG as mixing distribution. The density can be expressed as an infinite sum of Bessel K functions and it is characterized by 5 parameters.

Moreover, if X is SMNG distributed, then $Z=exp(X)$ is distributed as a log-SMNG distribution.

Examples

### Plots of density and cumulative functions of the SMNG distribution
x<-seq(-10,10,length.out = 500)

plot(x,dSMNG(x = x,mu = 0,delta = 1,gamma = 1,lambda = 1,beta= 2),
    type="l",ylab="f(x)")
lines(x,dSMNG(x = x,mu = 0,delta = 1,gamma = 1,lambda = 1,beta= -2),col=2)
title("SMNG density function")

plot(x,pSMNG(q = x,mu = 0,delta = 1,gamma = 1,lambda = 1,beta= 2),
    type="l",ylab="F(x)")
lines(x,pSMNG(q = x,mu = 0,delta = 1,gamma = 1,lambda = 1,beta= -2),col=2)
title("SMNG cumulative function")

### Plots of density and cumulative functions of the logSMNG distribution
x<-seq(0,20,length.out = 500)

plot(x,dlSMNG(x = x,mu = 0,delta = 1,gamma = 1,lambda = 2,beta = 1),
    type="l",ylab="f(x)",ylim = c(0,1.5))
lines(x,dlSMNG(x = x,mu = 0,delta = 1,gamma = 1,lambda = 2,beta = -1),col=2)
title("logSMNG density function")

plot(x,plSMNG(q = x,mu = 0,delta = 1,gamma = 1,lambda = 2,beta = 1),
    type="l",ylab="F(x)",ylim = c(0,1))
lines(x,plSMNG(q = x,mu = 0,delta = 1,gamma = 1,lambda = 2,beta = -1),col=2)
title("logSMNG cumulative function")