tsal.tail {tsallisqexp}

The Tsallis Distribution with a censoring parameter (tail-conditional)

Description

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

Usage

dtsal.tail(x, shape=1,scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale), xmin=0,
log=FALSE)

ptsal.tail(x, shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale), xmin=0,
lower.tail=TRUE, log.p=FALSE)

qtsal.tail(p,  shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale), xmin=0,
lower.tail=TRUE, log.p=FALSE)

rtsal.tail(n, shape=1, scale=1, q=tsal.q.from.shape(shape),
kappa=tsal.kappa.from.ss(shape,scale), xmin=0)

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.

xmin

minimum x-value.

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 with a censoring parameter is the distribution of a Tsallis distributed random variable conditionnaly on x>xmin. The density is defined as

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

for all x>xmin where C is the appropriate constant so that the integral of the density equals 1. That is C is the survival probability of the classic Tsallis distribution at x=xmin. 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.tail gives the density, ptsal.tail gives the distribution function, qtsal.tail gives the quantile function, and rtsal.tail generates random deviates.

The length of the result is determined by n for rtsal.tail, 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))