Shifted log-logistic distribution

Shifted log-logistic
Probability density function $\mu =0,\sigma =1,$ values of $\xi$ as shown in legend
Cumulative distribution function $\mu =0,\sigma =1,$ values of $\xi$ as shown in legend
Parameters	$\mu \in (-\infty ,+\infty )\,$ location (real) $\sigma \in (0,+\infty )\,$ scale (real) $\xi \in (-\infty ,+\infty )\,$ shape (real)
Support	$x\geqslant \mu -\sigma /\xi \,\;(\xi >0)$ $x\leqslant \mu -\sigma /\xi \,\;(\xi <0)$ $x\in (-\infty ,+\infty )\,\;(\xi =0)$
PDF	${\frac {(1+\xi z)^{-(1/\xi +1)}}{\sigma \left(1+(1+\xi z)^{-1/\xi }\right)^{2}}}$ where $z=(x-\mu )/\sigma \,$
CDF	$\left(1+(1+\xi z)^{-1/\xi }\right)^{-1}\,$ where $z=(x-\mu )/\sigma \,$
Mean	$\mu +{\frac {\sigma }{\xi }}(\alpha \csc(\alpha )-1)$ where $\alpha =\pi \xi \,$
Median	$\mu \,$
Mode	$\mu +{\frac {\sigma }{\xi }}\left[\left({\frac {1-\xi }{1+\xi }}\right)^{\xi }-1\right]$
Variance	${\frac {\sigma ^{2}}{\xi ^{2}}}[2\alpha \csc(2\alpha )-(\alpha \csc(\alpha ))^{2}]$ where $\alpha =\pi \xi \,$

The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution.^[1]^[2] It has also been called the generalized logistic distribution,^[3] but this conflicts with other uses of the term: see generalized logistic distribution.

Definition

The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter $\delta$ . Thus if $X$ has a log-logistic distribution then $X+\delta$ has a shifted log-logistic distribution. So $Y$ has a shifted log-logistic distribution if $\log(Y-\delta )$ has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic.

The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation.

In this parameterisation, the cumulative distribution function (CDF) of the shifted log-logistic distribution is

F(x;\mu ,\sigma ,\xi )={\frac {1}{1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }}}

for $1+\xi (x-\mu )/\sigma \geqslant 0$ , where $\mu\in\mathbb R$ is the location parameter, $\sigma >0\,$ the scale parameter and $\xi\in\mathbb R$ the shape parameter. Note that some references use $\kappa =-\xi \,\!$ to parameterise the shape.^[3]^[4]

The probability density function (PDF) is

f(x;\mu ,\sigma ,\xi )={\frac {\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-(1/\xi +1)}}{\sigma \left[1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }\right]^{2}}},

again, for $1+\xi (x-\mu )/\sigma \geqslant 0.$

The shape parameter $\xi$ is often restricted to lie in [-1,1], when the probability density function is bounded. When $|\xi |>1$ , it has an asymptote at $x=\mu -\sigma /\xi$ . Reversing the sign of $\xi$ reflects the pdf and the cdf about $x=\mu .$ .

Related distributions

When $\mu =\sigma /\xi ,$ the shifted log-logistic reduces to the log-logistic distribution.
When $\xi$ → 0, the shifted log-logistic reduces to the logistic distribution.
The shifted log-logistic with shape parameter $\xi =1$ is the same as the generalized Pareto distribution with shape parameter $\xi =1.$

Applications

The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.^[3]^[4]^[5]

Alternate parameterization

An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter $\alpha$ , scale parameter $\beta$ and location parameter $\gamma$ , the PDF is given by ^[6]^[7]

$f(x)={\frac {\alpha }{\beta }}{\bigg (}{\frac {x-\gamma }{\beta }}{\bigg )}^{\alpha -1}{\bigg (}1+{\bigg (}{\frac {x-\gamma }{\beta }}{\bigg )}^{\alpha }{\bigg )}^{-2}$

The CDF is given by

$F(x)={\bigg (}1+{\bigg (}{\frac {\beta }{x-\gamma }}{\bigg )}^{\alpha }{\bigg )}^{-1}$

The mean is $\beta \theta \csc(\theta )+\gamma$ and the variance is $\beta ^{2}\theta [2\csc(2\theta )-\theta \csc ^{2}(\theta )]$ , where $\theta ={\frac {\pi }{\alpha }}$ .^[7]

References

↑ Venter, Gary G. (Spring 1994), "Introduction to selected papers from the variability in reserves prize program" (PDF), Casualty Actuarial Society Forum, 1: 91–101
↑ Geskus, Ronald B. (2001), "Methods for estimating the AIDS incubation time distribution when date of seroconversion is censored", Statistics in Medicine, 20 (5): 795–812, doi:10.1002/sim.700, PMID 11241577
1 2 3 Hosking, Jonathan R. M.; Wallis, James R (1997), Regional Frequency Analysis: An Approach Based on L-Moments, Cambridge University Press, ISBN 0-521-43045-3
1 2 Robson, A.; Reed, D. (1999), Flood Estimation Handbook, 3: "Statistical Procedures for Flood Frequency Estimation", Wallingford, UK: Institute of Hydrology, ISBN 0-948540-89-3
↑ Ahmad, M. I.; Sinclair, C. D.; Werritty, A. (1988), "Log-logistic flood frequency analysis", Journal of Hydrology, 98 (3–4): 205–224, doi:10.1016/0022-1694(88)90015-7
↑ "EasyFit - Log-Logistic Distribution". Retrieved 1 October 2016.
1 2 "Guide to Using - RISK7_EN.pdf" (PDF). Retrieved 1 October 2016.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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