lmfor (version 1.6)

logit-logistic: The Four-parameter Logit-logistic Distribution

Description

Density, distribution function, quantile function and random generation for the four-parameter logit-logistic distribution.

Usage

dll(x, mu, sigma, xi=0, lambda=1, log = FALSE)
pll(q, mu, sigma, xi=0, lambda=1, lower.tail=TRUE, log.p=FALSE)
qll(p, mu, sigma, xi=0, lambda=1, lower.tail=TRUE, log.p=FALSE)
rll(n, mu, sigma, xi=0, lambda=1)

Arguments

x, q

vector of quantiles

p

vector of probabilitiies

n

number of observations. If length(n) > 1, the length is taken to be the number required.

mu,sigma,xi,lambda

parameters of the distribution, xi (minimum) defaults to 0 and lambda (max-min) to 1.

log, log.p

logical; if TRUE, probabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise \(P[X > x]\).

Value

dll gives the density, pll gives the distribution function, qll gives the quantile function, and rll generates random deviates.

Invalid arguments will result in return value NaN.

The length of the result is determined by n for rll, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Details

The logit-logistic cdf and pdf are $$F(d|\xi,\lambda,\mu,\sigma) = \frac{1}{1+e^{(\frac{\mu}{\sigma})} (\frac{d-\xi}{\xi+\lambda-d})^{-\frac{1}{\sigma}}}$$

$$f(d|\xi,\lambda,\mu,\sigma) = \frac{\lambda}{\sigma}\frac{1}{(d-\xi)(\xi+\lambda-d)} \frac{1}{e^{-\frac{\mu}{\sigma}}(\frac{d-\xi}{\xi+\lambda-d})^{\frac{1}{\sigma}}+e^{\frac{\mu}{\sigma}}(\frac{d-\xi}{\xi+\lambda-d})^{-\frac{1}{\sigma}}+2}$$ Parameter \(\xi\) is the minimum, \(\lambda>0\) the width of range (max-min), \(\mu\) controls the skewness and \(\sigma\) the curtosis.

References

Mingliang Wang and Keith Rennolls, 2005. Tree diameter distribution modelling: introducing the logit-logistic distribution. Canadian Journal of Forest Research, 35(6): 1305-1313, 10.1139/x05-057.

Mehtatalo, Lauri and Lappi, Juha 2020. Biometry for Forestry and Environmental Data: with examples in R. New York: Chapman and Hall/CRC. 426 p. 10.1201/9780429173462

Examples

# NOT RUN {
data(spati)
d<-spati$d[spati$plot==22]
hist(d,freq=FALSE)
d0<-seq(0,60,0.1)
lines(d0,dll(d0,0.630,0.573,3.561,35.2))
# }