SMNGdistribution {BayesLN} R Documentation

## SMNG and logSMNG Distributions

### Description

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`.

### Usage

```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.

### 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.

### Value

`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.

### 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")

```

[Package BayesLN version 0.2.2 Index]