shash: Sinh-Arcsinh (shash) Distribution

In cNORM: Continuous Norming

Sinh-Arcsinh (shash) Distribution

Description

Density, distribution function, quantile function and random generation for the Sinh-Arcsinh distribution with location parameter mu, scale parameter sigma, skewness parameter epsilon, and tail weight parameter delta.

Usage

dshash(x, mu = 0, sigma = 1, epsilon = 0, delta = 1, log = FALSE)

pshash(
  q,
  mu = 0,
  sigma = 1,
  epsilon = 0,
  delta = 1,
  lower.tail = TRUE,
  log.p = FALSE
)

qshash(
  p,
  mu = 0,
  sigma = 1,
  epsilon = 0,
  delta = 1,
  lower.tail = TRUE,
  log.p = FALSE
)

rshash(n, mu = 0, sigma = 1, epsilon = 0, delta = 1)

Arguments

x, q	vector of quantiles
mu	location parameter (default: 0)
sigma	scale parameter (must be > 0, default: 1)
epsilon	skewness parameter (default: 0, symmetric distribution)
delta	tail weight parameter (must be > 0, default: 1 for normal-like tails)
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 Sinh-Arcsinh distribution (Jones & Pewsey, 2009) is defined by the transformation:

X = \mu + \sigma \cdot \sinh\left(\frac{\text{asinh}(Z) - \epsilon}{\delta}\right)

where Z \sim N(0,1) is a standard normal variable.

The four parameters control:

• mu: Location (similar to mean)

• sigma: Scale (similar to standard deviation)

• epsilon: Skewness (epsilon = 0 gives symmetry)

• delta: Tail weight (delta = 1 gives normal-like tails, delta > 1 gives heavier tails, delta < 1 gives lighter tails)

Value

dshash gives the density, pshash gives the distribution function, qshash gives the quantile function, and rshash generates random deviates.

The length of the result is determined by n for rshash, and is the maximum of the lengths of the numerical arguments for the other functions.

References

Jones, M. C., & Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96(4), 761-780. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/asp053")}

See Also

Normal for the normal distribution.

Examples

# Generate random samples
x <- rshash(1000, mu = 0, sigma = 1, epsilon = 0.5, delta = 1.2)

# Density
plot(density(x))
curve(dshash(x, mu = 0, sigma = 1, epsilon = 0.5, delta = 1.2),
      add = TRUE, col = "red")

# Cumulative probability
pshash(0, mu = 0, sigma = 1, epsilon = 0.5, delta = 1.2)

# Quantiles
qshash(c(0.025, 0.5, 0.975), mu = 0, sigma = 1, epsilon = 0.5, delta = 1.2)

# Compare with normal distribution (epsilon = 0, delta = 1)
par(mfrow = c(2, 2))
x_vals <- seq(-4, 4, length.out = 200)
plot(x_vals, dshash(x_vals), type = "l", main = "Symmetric (like normal)")
plot(x_vals, dshash(x_vals, epsilon = 1), type = "l", main = "Right skewed")
plot(x_vals, dshash(x_vals, delta = 2), type = "l", main = "Heavy tails")
plot(x_vals, dshash(x_vals, delta = 0.5), type = "l", main = "Light tails")

cNORM documentation built on Feb. 27, 2026, 1:07 a.m.