tukeyGH: Tukey g-and-h Parametric Distribution

In ChristianGoueguel/specProc: Preprocessing Tools For Laser-Induced Breakdown Spectroscopy (LIBS) Spectra

Tukey g-and-h Parametric Distribution

Description

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.

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.

Value

A numeric vector.

Author(s)

Christian L. Goueguel

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.

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
)

ChristianGoueguel/specProc documentation built on Nov. 9, 2024, 3:23 p.m.