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Tukey g-and-h Parametric Distribution

Source: R/tukeyGH.R
tukeyGH.Rd

The Tukey g-and-h (TGH) distribution is a flexible family of parametric distributions that can accommodate a wide range of skewness and kurtosis values, making it useful for modeling non-normal data.

Usage

tukeyGH(
  x,
  type = "d",
  location = 0,
  scale = 1,
  g = 0,
  h = 0,
  log = FALSE,
  log.p = FALSE,
  n = NULL
)

Arguments

x

A numeric vector or a single value, depending on the function being called.

type

A character string specifying the function to be called (d for density, p for cumulative distribution, q for quantile, or r for random number generation).

location

The location parameter of the TGH distribution.

scale

The scale parameter of the TGH distribution.

g

The skewness parameter of the TGH distribution.

h

The kurtosis parameter of the TGH distribution.

log

If TRUE, the log-density is returned for the density function. Default is FALSE.

log.p

If TRUE, the log-probability is returned for the cumulative distribution and quantile functions. Default is FALSE.

n

For random number generation, the number of random values to be generated. Default is NULL.

Value

A numeric vector.

Details

The Tukey g-and-h distribution, introduced by John W. Tukey in 1977, is defined by two transformation functions, g and h, which are applied to a standard normal distribution. The g transformation controls the skewness of the distribution, while the h transformation controls the kurtosis (or heaviness of the tails). The TGH distribution is given by:

$$T_{g,h}(Z) = \frac{1}{g} (e^{g Z} -1) e^{\frac{1}{2} h Z^2}$$

Where \(Z\) is a random variable with standard normal distribution. The parameters \(g\) and \(h\) stand for the bias and elongation of the tails, respectively, of Tukey’s \(g\)-and-\(h\) distribution.

References

  • Tukey, J.W., (1977). Modern techniques in data analysis. NSF‐sponsored regional research conference at Southeastern Massachusetts University, North Dartmouth, MA.

  • Martinez, J., Iglewicz, B., (1984). Some properties of the Tukey g and h family of distributions. Communications in Statistics: Theory and Methods, 13(3):353–369.

  • Hoaglin, D.C., (1985). Summarizing shape numerically: The g-and-h distributions. In: Hoaglin, D.C., Mosteller, F., Tukey, J.W., (eds), Data Analysis for Tables, Trends, and Shapes. New York:Wiley.

Author

Christian L. Goueguel

Examples

x <- seq(-5, 5, length.out = 100)
y1 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = 0, h = 0)
y2 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = 0.5, h = 0)
y3 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = -0.5, h = 0)
y4 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = 0, h = 0.1)
y5 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = 0.5, h = 0.1)
y6 <- tukeyGH(x, type = "d", location = 0, scale = 1, g = -0.5, h = 0.1)
plot(x, y1, type = "b", col = "red", pch = 16,
main = "Tukey g-and-h distribution", ylim = c(0, 0.6))
lines(x, y2, type = "b", col = "blue", pch = 15)
lines(x, y3, type = "b", col = "green", pch = 17)
lines(x, y4, type = "b", col = "black", pch = 6)
lines(x, y5, type = "b", col = "gold", pch = 18)
lines(x, y6, type = "b", col = "orange", pch = 8)
legend(
"topright",
legend = c(
"g = 0, h = 0", "g = 0.5, h = 0", "g = -0.5, h = 0",
"g = 0, h = 0.1", "g = 0.5, h = 0.1", "g = -0.5, h = 0.1"),
col = c("red", "blue", "green", "black", "gold", "orange"),
lty = 1
)