greybox (version 2.0.2)

dgnorm: The generalized normal distribution

Description

Density, cumulative distribution, quantile functions and random number generation for the Generalised Normal distribution with the location mu, a scale and a shape parameters.

Usage

dgnorm(q, mu = 0, scale = 1, shape = 1, log = FALSE)
pgnorm(q, mu = 0, scale = 1, shape = 1, lower.tail = TRUE,
  log.p = FALSE)
qgnorm(p, mu = 0, scale = 1, shape = 1, lower.tail = TRUE,
  log.p = FALSE)
rgnorm(n, mu = 0, scale = 1, shape = 1)

Arguments

q

vector of quantiles

mu

location parameter

scale

scale parameter

shape

shape parameter

log, log.p

logical; if TRUE, probabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are \(P[X\leq x]\), otherwise \(P[X> x]\)

p

vector of probabilities

n

number of observations

Author

Maryclare Griffin and Ivan Svetunkov

Details

A generalized normal random variable \(x\) with parameters location \(\mu\), scale \(s > 0\) and shape \(\beta > 0\) has density:

$$p(x) = \beta exp{-(|x - \mu|/s)^\beta}/(2s \Gamma(1/\beta)).$$
The mean and variance of \(x\) are \(\mu\) and \(s^2 \Gamma(3/\beta)/\Gamma(1/\beta)\), respectively.

The function are based on the functions from gnorm package of Maryclare Griffin (package has been abandoned since 2018).

The quantile and cumulative functions use uniform approximation for cases shape>100. This is needed, because otherwise it is not possible to calculate the values correctly due to scale^(shape)=Inf in R.

References

• Box, G. E. P. and G. C. Tiao. "Bayesian inference in Statistical Analysis." Addison-Wesley Pub. Co., Reading, Mass (1973).

• Nadarajah, Saralees. "A generalized normal distribution." Journal of Applied Statistics 32.7 (2005): 685-694.

• Subbotin, M. T. "On the Law of Frequency of Error." Matematicheskii Sbornik 31.2 (1923): 206-301.

See Also

Distributions

Examples

# Density function values for standard normal distribution
x <- dgnorm(seq(-1, 1, length.out = 100), 0, sqrt(2), 2)
plot(x, type="l")

#CDF of standard Laplace
x <- pgnorm(c(-100:100), 0, 1, 1)
plot(x, type="l")

# Quantiles of S distribution
qgnorm(c(0.025,0.975), 0, 1, 0.5)

# Random numbers from a distribution with shape=10000 (approximately uniform)
x <- rgnorm(1000, 0, 1, 1000)
hist(x)