TukeyLambda: Tukey lambda distribution

Description

Quantile function, and random generation for the Tukey lambda distribution.

Usage

qtlambda(p, lambda, lower.tail = TRUE, log.p = FALSE)
rtlambda(n, lambda)

Arguments

p: vector of probabilities.
lambda: shape parameter.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
log.p: logical; if TRUE, probabilities p are given as log(p).
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Tukey lambda distribution is a continuous probability distribution defined in terms of its quantile function. It is typically used to identify other distributions.

Quantile function:

$$F^{-1}(p) = \left\{\begin{array}{ll} \frac{1}{\lambda} [p^\lambda - (1-p)^\lambda] & \lambda \ne 0 \\ \log(\frac{p}{1-p}) & \lambda = 0 \end{array}\right. $$

References

Joiner, B.L., & Rosenblatt, J.R. (1971). Some properties of the range in samples from Tukey's symmetric lambda distributions. Journal of the American Statistical Association, 66(334), 394-399.

Hastings Jr, C., Mosteller, F., Tukey, J.W., & Winsor, C.P. (1947). Low moments for small samples: a comparative study of order statistics. The Annals of Mathematical Statistics, 413-426.

Examples

Run this code


pp = seq(0, 1, by = 0.001)
partmp <- par(mfrow = c(2,3))
plot(qtlambda(pp, -1), pp, type = "l", main = "lambda = -1 (Cauchy)")
plot(qtlambda(pp, 0), pp, type = "l", main = "lambda = 0 (logistic)")
plot(qtlambda(pp, 0.14), pp, type = "l", main = "lambda = 0.14 (normal)")
plot(qtlambda(pp, 0.5), pp, type = "l", main = "lambda = 0.5 (concave)")
plot(qtlambda(pp, 1), pp, type = "l", main = "lambda = 1 (uniform)")
plot(qtlambda(pp, 2), pp, type = "l", main = "lambda = 2 (uniform)")

hist(rtlambda(1e5, -1), freq = FALSE, main = "lambda = -1 (Cauchy)")
hist(rtlambda(1e5, 0), freq = FALSE, main = "lambda = 0 (logistic)")
hist(rtlambda(1e5, 0.14), freq = FALSE, main = "lambda = 0.14 (normal)")
hist(rtlambda(1e5, 0.5), freq = FALSE, main = "lambda = 0.5 (concave)")
hist(rtlambda(1e5, 1), freq = FALSE, main = "lambda = 1 (uniform)")
hist(rtlambda(1e5, 2), freq = FALSE, main = "lambda = 2 (uniform)")
par(partmp)

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