Tukey lambda distribution

  Tukey(λ)

  λ ∈ ℝ — shape parameter

  x ∈ [ − 1 /λ,  1 /λ ]    if    λ > 0 ,
  x ∈ ℝ           if    λ ≤ 0 .

  ${\displaystyle \left(\ Q(\ p\ ;\lambda \ ),\ {\frac {1}{\ q(\ p\ ;\lambda \ )\ }}\ \right)\quad ~{\mathsf {for\ any}}~\quad p\;:\;0\leq \ p\ \leq \ 1}$

  ${\displaystyle {\Bigl (}\ Q(p;\lambda ),\ p\ {\Bigr )}~~{\mathsf {for\ any}}~~p\;:\;0\leq \ p\ \leq \ 1~}$   (general case)
  ${\displaystyle {\frac {1}{\ e^{-x}+1\ }}\quad ~{\mathsf {if}}~\quad \lambda \ =\ 0\quad }$ (special case exact solution)

  ${\displaystyle 0\quad ~{\mathsf {if}}~\quad \lambda >-1\ }$

  0

  0

  ${\displaystyle {\frac {2}{\ \lambda ^{2}\ }}\left(\ {\frac {1}{\ 1+2\ \lambda \ }}-{\frac {\ \Gamma \,\!(\lambda +1)^{2}\ }{\ \Gamma \,\!(\ 2\ \lambda +2\ )\ }}\right)\quad ~{\mathsf {if}}~\quad \lambda >-{\tfrac {\ 1\ }{2}}\ }$
  ${\displaystyle {\frac {\ \pi ^{2}\ }{3}}\qquad \qquad ~{\mathsf {if}}~\quad \lambda \ =\ 0}$

  ${\displaystyle 0\qquad \qquad ~{\mathsf {if}}~\quad \lambda >-{\tfrac {\ 1\ }{3}}\ }$

  ${\displaystyle ~{\frac {\ (2\ \lambda +1)^{2}\cdot g_{2}^{2}\cdot {\big (}\ 3\ g_{2}^{2}-4\ g_{1}\ g_{3}+g_{4}\ {\big )}\ }{\ (\ 8\ \lambda +2\ )\cdot g_{4}\cdot {\big (}\ g_{1}^{2}-g_{2}\ {\big )}^{2}}}\ -\ 3\quad {\mathsf {if}}~~\lambda >0\ ,}$

  ${\displaystyle {\frac {\ 6\ }{5}}\qquad \qquad ~{\mathsf {if}}~\quad \lambda \ =\ 0\ ;}$

  ${\displaystyle {\mathsf {where}}\quad g_{k}\ \equiv \ \Gamma \,\!(\ k\,\lambda +1\ )\quad {\mathsf {and}}\quad \lambda \ >-{\tfrac {\ 1\ }{4}}~.}$

 ${\displaystyle h(\lambda )=\int _{0}^{1}\ln {\bigl (}\ q(p;\lambda )\ {\bigr )}\ \operatorname {d} p~}$^[1]

  ${\displaystyle \phi (t;\lambda )=\int _{0}^{1}\exp {\bigl (}\ i\ t\ Q(p;\lambda )\ {\bigr )}\ \operatorname {d} p~}$^[2]