Tukey lambda distribution
Quantile function, and random generation for the Tukey lambda distribution.
qtlambda(p, lambda, lower.tail = TRUE, log.p = FALSE) rtlambda(n, lambda)
p |
vector of probabilities. |
lambda |
shape parameter. |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
log.p |
logical; if TRUE, probabilities p are given as log(p). |
n |
number of observations. If |
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) = [if λ != 0:] (p^λ - (1-p)^λ)/λ [if λ = 0:] log(p/(1-p))
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.
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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