Generalized log-logistic distribution
The Generalized log-logistic distribution (GLL) has three parameters
and
.
Generalized log-logisticParameters |
location (real)
scale (real)
shape (real) |
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Support |
![{\displaystyle x\geqslant \mu -\sigma /\xi \,\;(\xi \geqslant 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed09c9000c75066664416df8d1d5d7069c862ad4)
![{\displaystyle x\leqslant \mu -\sigma /\xi \,\;(\xi <0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7640e1354efd975861bdc1ca8e517fd1c4ae0dc8) |
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PDF |
![{\displaystyle {\frac {(1+\xi z)^{-(1/\xi +1)}}{\sigma \left(1+(1+\xi z)^{-1/\xi }\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0a6c2c6053b69a27ebfba1f776ac3d566461d6)
where ![{\displaystyle z=(x-\mu )/\sigma \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8244b89656eac44fc7b134d8447038fd7e17d6c6) |
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CDF |
![{\displaystyle \left(1+(1+\xi z)^{-1/\xi }\right)^{-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df6ac77f0f190d6d8c796ea816ce43a157f1b68)
where ![{\displaystyle z=(x-\mu )/\sigma \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8244b89656eac44fc7b134d8447038fd7e17d6c6) |
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Mean |
![{\displaystyle \mu +{\frac {\sigma }{\xi }}(\alpha \csc(\alpha )-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6265b5df9b7e36b4bb9afc35fb4c1d024be3b7)
where ![{\displaystyle \alpha =\pi \xi \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a04798813006cf1e5d0076964b9b9ec7e7da31) |
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Median |
![{\displaystyle \mu \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340) |
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Mode |
![{\displaystyle \mu +{\frac {\sigma }{\xi }}\left[\left({\frac {1-\xi }{1+\xi }}\right)^{\xi }-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3234781adb448370f3a7d4073c460b630bdab66) |
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Variance |
![{\displaystyle {\frac {\sigma ^{2}}{\xi ^{2}}}[2\alpha \csc(2\alpha )-(\alpha \csc(\alpha ))^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7608a00d698835b4f898f584eaf8983af7c6bd6c)
where ![{\displaystyle \alpha =\pi \xi \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a04798813006cf1e5d0076964b9b9ec7e7da31) |
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The cumulative distribution function is
![{\displaystyle F_{(\xi ,\mu ,\sigma )}(x)=\left(1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3414be99e4c1d2247beeee9fd5e74ac5abfd754b)
for
, where
is the location parameter,
the scale parameter and
the shape parameter. Note that some references give the "shape parameter" as
.
The probability density function is
![{\displaystyle {\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0909a60aab0523829a4e3880207129fe3feb54a7)
again, for