Shifted log-logistic distribution

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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 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 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]

References

↑ Venter, Gary G. (Spring 1994), "Introduction to selected papers from the variability in reserves prize program", 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
↑ 3.0 3.1 3.2 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
↑ 4.0 4.1 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

Probability distributions

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete 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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