# 一般化双曲型分布

ナビゲーションに移動 検索に移動

## 一次元一般化双曲型分布

### 確率密度関数

{\displaystyle {\begin{aligned}gh(x;\lambda ,\alpha ,\beta ,\delta ,\mu )=&a(\lambda ,\alpha ,\beta ,\delta ,\mu )(\delta ^{2}+(x-\mu )^{2})^{(\lambda -{\frac {1}{2}})/2}\\&\times K_{\lambda -1/2}(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}})\exp {(\beta (x-\mu ))}\end{aligned}}}

ここで、

${\displaystyle a(\lambda ,\alpha ,\beta ,\delta ,\mu )={\frac {(\alpha ^{2}-\beta ^{2})^{\lambda /2}}{{\sqrt {2\pi }}\alpha ^{\lambda -1/2}\delta ^{\lambda }K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}}$

${\displaystyle K_{\lambda }(x)}$ は、第3種の変形ベッセル関数
${\displaystyle \mu }$ 位置(location)パラメータ (実数)
${\displaystyle \lambda }$ (実数)
${\displaystyle \alpha }$ (実数)
${\displaystyle \beta }$ 歪度(skewness)/非対称性(asymmetry)パラメータ (実数)
${\displaystyle \delta }$ 尺度(scale)パラメータ (実数)
${\displaystyle x\in (-\infty ;+\infty )}$
λ>0 のとき、${\displaystyle \delta \geq 0,\;|\beta |<\alpha }$
λ=0 のとき、${\displaystyle \delta >0,\;|\beta |<\alpha }$
λ<0 のとき、${\displaystyle \delta >0,\;|\beta |\leq \alpha }$

### モーメント

{\displaystyle {\begin{aligned}&\zeta _{u}&=&\delta {\sqrt {\alpha ^{2}-(\beta +u)^{2}}}\\&\zeta &=&\zeta _{u=0}\end{aligned}}}

とする。

#### 期待値

{\displaystyle {\begin{aligned}E(X)&=\mu +{\frac {\delta \beta }{\sqrt {\alpha ^{2}-\beta ^{2}}}}{\frac {K_{\lambda +1}(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}\\[0.5em]&=\mu +{\frac {\delta ^{2}\beta }{\zeta }}{\frac {K_{\lambda +1}(\zeta )}{K_{\lambda }(\zeta )}}\end{aligned}}}

#### 分散

{\displaystyle {\begin{aligned}Var(X)&={\begin{matrix}{\frac {\delta }{\sqrt {\alpha ^{2}-\beta ^{2}}}}{\frac {K_{\lambda +1}(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}+{\frac {\delta ^{2}\beta ^{2}}{(\alpha ^{2}-\beta ^{2})}}\left[{\frac {K_{\lambda +2}(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}-\left({\frac {K_{\lambda +1}(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}\right)^{2}\right]\end{matrix}}\\[0.5em]&={\begin{matrix}{\frac {\delta ^{2}}{\zeta }}{\frac {K_{\lambda +1}(\zeta )}{K_{\lambda }(\zeta )}}+{\frac {\delta ^{4}\beta ^{2}}{\zeta ^{2}}}\left[{\frac {K_{\lambda +2}(\zeta )}{K_{\lambda }(\zeta )}}-\left({\frac {K_{\lambda +1}(\zeta )}{K_{\lambda }(\zeta )}}\right)^{2}\right]\end{matrix}}\end{aligned}}}

### モーメント母関数

モーメント母関数は以下の式で与えられる。

{\displaystyle {\begin{aligned}M_{GH}(u)&=\exp {(u\mu )}\left({\frac {\alpha ^{2}-\beta ^{2}}{(\alpha ^{2}-(\beta +u)^{2})}}\right)^{\lambda /2}{\frac {K_{\lambda }(\delta {\sqrt {\alpha ^{2}-(\beta +u)^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}\\[0.5em]&=\exp {(u\mu )}\left({\frac {\zeta }{\zeta _{u}}}\right)^{\lambda }{\frac {K_{\lambda }(\zeta _{u})}{K_{\lambda }(\zeta )}}\end{aligned}}}

### 特性関数

${\displaystyle \varphi (u)=\exp {(iu\mu )}\left({\frac {\alpha ^{2}-\beta ^{2}}{(\alpha ^{2}-(\beta +iu)^{2})}}\right)^{\lambda /2}{\frac {K_{\lambda }(\delta {\sqrt {\alpha ^{2}-(\beta +iu)^{2}}})}{K_{\lambda }(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}}$

## 特別なケース

### λ=1 の場合

{\displaystyle {\begin{aligned}gh(x;1,\alpha ,\beta ,\delta ,\mu )&=\mathrm {hyp} (x;\alpha ,\beta ,\delta ,\mu )\\&={\frac {\sqrt {\alpha ^{2}-\beta ^{2}}}{2\delta \alpha K_{1}(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}})}}\exp {(-\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}+\beta (x-\mu ))}\end{aligned}}}

### λ=-1/2 の場合

{\displaystyle {\begin{aligned}gh(x;-1/2,\alpha ,\beta ,\delta ,\mu )&=\mathrm {nig} (x;\alpha ,\beta ,\delta ,\mu )\\&={\frac {\alpha \delta }{\pi }}\exp {(\delta {\sqrt {\alpha ^{2}-\beta ^{2}}}+\beta (x-\mu ))}{\frac {K_{1}(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}})}{\sqrt {\delta ^{2}+(x-\mu )^{2}}}}\end{aligned}}}

### λ=-ν/2、α→|β| の場合

{\displaystyle {\begin{aligned}gh(x;&\lambda ={\frac {-\nu }{2}},\alpha \to |\beta |,\beta ,\delta ,\mu )\\&={\frac {\delta ^{\nu }|\beta |^{(\nu +1)/2}K_{(v+1)/2}\left({\sqrt {(\delta ^{2}+(x-\mu )^{2})\beta ^{2}}}\right)\exp {(\beta (x-\mu ))}}{2^{(v-1)/2}\Gamma \left({\frac {\nu }{2}}\right){\sqrt {\pi }}\left({\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)^{(\nu +1)/2}}}\end{aligned}}}

### λ=-ν/2、α=β=0、 ${\displaystyle \delta ={\sqrt {\nu }}}$ の場合

{\displaystyle {\begin{aligned}gh(x;&\lambda ={\frac {-\nu }{2}},\alpha =0,\beta =0,\delta ={\sqrt {\nu }},\mu )\\&={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\pi }}\delta \Gamma \left({\frac {\nu }{2}}\right)}}\left[1+{\frac {(x-\mu )^{2}}{\delta ^{2}}}\right]^{-{\frac {\nu +1}{2}}}\\&={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\pi \nu }}\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {(x-\mu )^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\end{aligned}}}

## 参考文献

(英語)

Thanh Tam, Dec 09, 2009.

(日本語)

## 脚注

a b  ${\displaystyle K_{-1/2}(x)=K_{1/2}(x)={\sqrt {\frac {\pi }{2}}}x^{-1/2}\exp {(-x)}}$