The quantile-based Skew Logistic Distribution

Description

Density, density quantile, distribution and quantile functions and random generation for the quantile-based skew logistic distribution.

Usage

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dsl(x,parameters,inverse.eps=.Machine$double.eps,max.iterations=500)
dqsl(p,parameters)
psl(q,parameters,inverse.eps=.Machine$double.eps,max.iterations=500)
qsl(p,parameters)
rsl(n,parameters)

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

parameters

A vector of length 3, giving the parameters of the quantile-based skew logistic distribution. The 3 elements are alpha (location), beta (scale) and delta (skewing). alpha can take on any real value, beta can take on any positive value and delta must satisfy 0 <= delta <= 1. delta = 0.5 gives the logistic distribution, delta = 0 gives the reflected exponential distribution and delta = 1 gives the exponential distribution.

inverse.eps

Accuracy of calculation for the numerical determination of F(x), defaults to .Machine$double.eps

max.iterations

Maximum number of iterations in the numerical determination of F(x), defaults to 500

Details

The quantile-based skew logistic distribution is a generalisation of the logistic distribution, defined by its quantile funtion, Q(u), the inverse of the distribution function.

Q(u) = alpha + beta ( (1 - delta)*(log(u)) - delta * (log(1-u)) )

for beta >0 and 0 <= delta <= 1.

The distribution was first used by Gilchrist (2000) in the book Statistical Modelling with Quantile Functions. Full details of the properties of the distributions, including moments, L-moments and estimation via L-Moments are given in van Staden and King (2015).

The distribution is defined by its quantile function and its distribution and density functions do not exist in closed form (except for some special cases). Accordingly, the results from psl and dsl are the result of numerical solutions to the quantile function, using the Newton-Raphson method. Since the density quantile function, f(Q(u)), does exist, an additional function, dqsl, computes this.

The distribution has closed form method of L-Moment estimates (see fit.sld.lmom for details). The 4th L-Moment ratio of the the distribution is constant tau4 = 1/6 for all values of delta. The 3rd L-Moment ratio of the distribution is restricted to -1/3 <= tau3 <= 1/3, being the the 3rd L-moment ratio values of the reflected exponential and the exponential distributions respectively.

Value

dsl gives the density (based on the quantile density and a numerical solution to Q(u)=x),

dqsl gives the density quantile,

psl gives the distribution function (based on a numerical solution to Q(u)=x and dqsl

qsl gives the quantile function, and

rsl generates random deviates.

Author(s)

Robert King, robert.king@newcastle.edu.au, http://tolstoy.newcastle.edu.au/~rking/

References

Gilchrist, W.G. (2000) Statistical Modelling with Quantile Functions Chapman & Hall, print 978-1-58488-174-2, e-book 978-1-4200-3591-9.

van Staden, P.J. and King, Robert A.R. (2015) The quantile-based skew logistic distribution, Statistics and Probability Letters 96 109–116. http://dx.doi.org/10.1016/j.spl.2014.09.001

van Staden, Paul J. 2013 Modeling of generalized families of probability distribution in the quantile statistical universe. PhD thesis, University of Pretoria. http://hdl.handle.net/2263/40265

http://tolstoy.newcastle.edu.au/rking/SLD/SLD.html

Examples

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qsl(seq(0,1,0.02),c(0,1,0.123))
psl(seq(-2,2,0.2),c(0,1,.1),inverse.eps=1e-10)
rsl(21,c(3,2,0.3))