Tukey-Lambda分布¶
有一个形状参数 \(\lambda\) 。支持是 \(x\in\mathbb{{R}}\) 。
\BEGIN{eqnarray [}} f\left(x;\lambda\right) & = & F^{{\prime}}\left(x;\lambda\right)=\frac{{1}}{{G^{{\prime}}\left(F\left(x;\lambda\right);\lambda\right)}}=\frac{{1}}{{F^{{\lambda-1}}\left(x;\lambda\right)+\left[1-F\left(x;\lambda\right)\right]^{{\lambda-1}}}}\\ F\left(x;\lambda\right) & = & G^{{-1}}\left(x;\lambda\right)\\ G\left(p;\lambda\right) & = & \frac{{p^{{\lambda}}-\left(1-p\right)^{{\lambda}}}}{{\lambda}}\end{{eqnarray] }
\BEGIN{eqnarray *}} \mu & = & 0\\ \mu_{{2}} & = & \int_{{0}}^{{1}}G^{{2}}\left(p;\lambda\right)dp\\ & = & \frac{{2\Gamma\left(\lambda+\frac{{3}}{{2}}\right)-\lambda4^{{-\lambda}}\sqrt{{\pi}}\Gamma\left(\lambda\right)\left(1-2\lambda\right)}}{{\lambda^{{2}}\left(1+2\lambda\right)\Gamma\left(\lambda+\frac{{3}}{{2}}\right)}}\\ \gamma_{{1}} & = & 0\\ \gamma_{{2}} & = & \frac{{\mu_{{4}}}}{{\mu_{{2}}^{{2}}}}-3\\ \mu_{{4}} & = & \frac{{3\Gamma\left(\lambda\right)\Gamma\left(\lambda+\frac{{1}}{{2}}\right)2^{{-2\lambda}}}}{{\lambda^{{3}}\Gamma\left(2\lambda+\frac{{3}}{{2}}\right)}}+\frac{{2}}{{\lambda^{{4}}\left(1+4\lambda\right)}}\\ & & -\frac{{2\sqrt{{3}}\Gamma\left(\lambda\right)2^{{-6\lambda}}3^{{3\lambda}}\Gamma\left(\lambda+\frac{{1}}{{3}}\right)\Gamma\left(\lambda+\frac{{2}}{{3}}\right)}}{{\lambda^{{3}}\Gamma\left(2\lambda+\frac{{3}}{{2}}\right)\Gamma\left(\lambda+\frac{{1}}{{2}}\right)}}.\end{{eqnarray* }
请注意， \(\lim_{{\lambda\rightarrow0}}G\left(p;\lambda\right)=\log\left(p/\left(1-p\right)\right)\)
\BEGIN{eqnarray [}} h\left[X\right] & = & \int_{{0}}^{{1}}\log\left[G^{{\prime}}\left(p\right)\right]dp\\ & = & \int_{{0}}^{{1}}\log\left[p^{{\lambda-1}}+\left(1-p\right)^{{\lambda-1}}\right]dp.\end{{eqnarray] }