一般化双曲型分布
一般化双曲型分布の確率密度関数は以下の式で与えられる。
g h ( x ; λ , α , β , δ , μ ) = a ( λ , α , β , δ , μ ) ( δ 2 + ( x − μ ) 2 ) ( λ − 1 2 ) / 2 × K λ − 1 / 2 ( α δ 2 + ( x − μ ) 2 ) exp  ( β ( x − μ ) ) {\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、
a ( λ , α , β , δ , μ ) = ( α 2 − β 2 ) λ / 2 2 π α λ − 1 / 2 δ λ K λ ( δ α 2 − β 2 ) {\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}}})}}}
Var  ( X ) = δ α 2 − β 2 K λ + 1 ( δ α 2 − β 2 ) K λ ( δ α 2 − β 2 ) + δ 2 β 2 ( α 2 − β 2 ) [ K λ + 2 ( δ α 2 − β 2 ) K λ ( δ α 2 − β 2 ) − ( K λ + 1 ( δ α 2 − β 2 ) K λ ( δ α 2 − β 2 ) ) 2 ] = δ 2 ζ K λ + 1 ( ζ ) K λ ( ζ ) + δ 4 β 2 ζ 2 [ K λ + 2 ( ζ ) K λ ( ζ ) − ( K λ + 1 ( ζ ) K λ ( ζ ) ) 2 ] {\displaystyle {\begin{aligned}\operatorname {Var} (X)&={\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]\\&={\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{aligned}}}