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
1 2 3 4 5 |
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 |
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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