cNORM (version 3.5.1)

shash: 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)

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.

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)

References

Jones, M. C., & Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96(4), 761-780. 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")