tsal {tsallisqexp}

The Tsallis Distribution

Description

Density function, distribution function, quantile function, random generation.

Usage

dtsal(x, shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale),
log=FALSE)

ptsal(x, shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale),
lower.tail=TRUE, log.p=FALSE)

qtsal(p,  shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale),
lower.tail=TRUE, log.p=FALSE)

rtsal(n, shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale))

tsal.mean(shape, scale, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale))

Arguments

x

vector of quantiles.

q

vector of quantiles or a shape parameter.

p

vector of probabilities.

n

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

shape

shape parameter.

scale, kappa

scale parameters.

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].

Details

The Tsallis distribution is defined by the following density

f(x) = \frac{1}{ \kappa}(1-(1-q)x/\kappa)^{1/(1-q)}

for all x. It is convenient to introduce a re-parameterization shape = -1/(1-q), scale = shape*\kappa which makes the relationship to the Pareto clearer, and eases estimation. If we have both shape/scale and q/kappa parameters, the latter over-ride.

Value

dtsal gives the density, ptsal gives the distribution function, qtsal gives the quantile function, and rtsal generates random deviates. tsal.mean computes the expected value.

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

Author(s)

Cosma Shalizi (original R code), Christophe Dutang (R packaging)

References

Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions, http://bactra.org/research/tsallis-MLE/ and https://arxiv.org/abs/math/0701854.

Examples

#####
# (1) density function
x <- seq(0, 5, length=24)

cbind(x, dtsal(x, 1/2, 1/4))

#####
# (2) distribution function

cbind(x, ptsal(x, 1/2, 1/4))