Summarize Timeline Top Qs Fact Check 切断正規分布の台が
[
a
,
∞
)
{\displaystyle [a,\infty )}
μ
{\displaystyle \mu }
σ
{\displaystyle \sigma }
(
−
∞
,
a
)
{\displaystyle (-\infty ,a)}
期待値 は大きくなる。逆に、台が
(
−
∞
,
b
]
{\displaystyle (-\infty ,b]}
一方、分散 については、どのような切断を行ったとしても元となる正規分布よりも必ず小さくなる。
α
=
(
a
−
μ
)
/
σ
{\displaystyle \alpha =(a-\mu )/\sigma }
β
=
(
b
−
μ
)
/
σ
{\displaystyle \beta =(b-\mu )/\sigma }
E
(
X
|
a
<
X
<
b
)
=
μ
+
φ
(
α
)
−
φ
(
β
)
Φ
(
β
)
−
Φ
(
α
)
σ
{\displaystyle \operatorname {E} (X|a<X<b)=\mu +{\frac {\varphi (\alpha )-\varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}\sigma }
Var
(
X
|
a
<
X
<
b
)
=
σ
2
[
1
+
α
φ
(
α
)
−
β
φ
(
β
)
Φ
(
β
)
−
Φ
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α
)
−
(
φ
(
α
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−
φ
(
β
)
Φ
(
β
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−
Φ
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)
2
]
{\displaystyle \operatorname {Var} (X|a<X<b)=\sigma ^{2}\left[1+{\frac {\alpha \varphi (\alpha )-\beta \varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}-\left({\frac {\varphi (\alpha )-\varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}\right)^{2}\right]}
である。なお、
μ
∉
[
a
,
b
]
{\displaystyle \mu \notin [a,b]}
桁落ち に注意しなければならない。
E
(
X
|
X
>
a
)
=
μ
+
σ
R
(
α
)
{\displaystyle \operatorname {E} (X|X>a)=\mu +{\frac {\sigma }{R(\alpha )}}}
Var
(
X
|
X
>
a
)
=
σ
2
[
1
+
α
R
(
α
)
−
(
1
R
(
α
)
)
2
]
{\displaystyle \operatorname {Var} (X|X>a)=\sigma ^{2}\left[1+{\frac {\alpha }{R(\alpha )}}-\left({\frac {1}{R(\alpha )}}\right)^{2}\right]}
である。ここで、
R
(
x
−
μ
σ
)
=
1
−
Φ
(
x
−
μ
σ
)
φ
(
x
−
μ
σ
)
{\displaystyle R\left({\frac {x-\mu }{\sigma }}\right)={\frac {1-\Phi \left({\frac {x-\mu }{\sigma }}\right)}{\varphi \left({\frac {x-\mu }{\sigma }}\right)}}}
は、対応する正規分布のミルズ比 である。
一方、上側が有界な単一切断正規分布の場合は
E
(
X
|
X
<
b
)
=
μ
−
φ
(
β
)
Φ
(
β
)
σ
{\displaystyle \operatorname {E} (X|X<b)=\mu -{\frac {\varphi (\beta )}{\Phi (\beta )}}\sigma }
Var
(
X
|
X
<
b
)
=
σ
2
[
1
−
β
φ
(
β
)
Φ
(
β
)
−
(
φ
(
β
)
Φ
(
β
)
)
2
]
{\displaystyle \operatorname {Var} (X|X<b)=\sigma ^{2}\left[1-\beta {\frac {\varphi (\beta )}{\Phi (\beta )}}-\left({\frac {\varphi (\beta )}{\Phi (\beta )}}\right)^{2}\right]}
である。
通常の正規分布と同様に、
μ
{\displaystyle \mu }
n
{\displaystyle n}
漸化式 が存在する。具体的には
n
≥
0
{\displaystyle n\geq 0}
E
[
(
x
−
μ
σ
)
n
+
2
]
=
α
n
+
1
φ
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−
β
n
+
1
φ
(
β
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Φ
(
β
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−
Φ
(
α
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+
(
n
+
1
)
E
[
(
x
−
μ
σ
)
n
]
{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]={\frac {\alpha ^{n+1}\varphi (\alpha )-\beta ^{n+1}\varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}+(n+1)\operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n}\right]}
ξ
=
(
x
−
μ
)
/
σ
{\displaystyle \xi =(x-\mu )/\sigma }
E
[
(
x
−
μ
σ
)
n
+
2
]
=
∫
α
β
ξ
n
+
2
φ
(
ξ
)
Φ
(
β
)
−
Φ
(
α
)
d
ξ
{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]=\int _{\alpha }^{\beta }{\frac {\xi ^{n+2}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}d\xi }
が得られる。
φ
′
(
ξ
)
=
−
ξ
φ
(
ξ
)
{\displaystyle \varphi '(\xi )=-\xi \varphi (\xi )}
部分積分 より
E
[
(
x
−
μ
σ
)
n
+
2
]
=
[
−
ξ
n
+
1
φ
(
ξ
)
Φ
(
β
)
−
Φ
(
α
)
]
α
β
+
(
n
+
1
)
∫
α
β
ξ
n
φ
(
ξ
)
Φ
(
β
)
−
Φ
(
α
)
d
ξ
{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]=\left[{\frac {-\xi ^{n+1}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}\right]_{\alpha }^{\beta }+(n+1)\int _{\alpha }^{\beta }{\frac {\xi ^{n}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}d\xi }
となるので、第2項を期待値として書き直すことで示すべき式が得られる。
∂
∂
μ
log
f
(
x
;
μ
,
σ
,
a
,
b
)
=
1
σ
[
x
−
μ
σ
+
φ
(
β
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−
φ
(
α
)
Φ
(
β
)
−
Φ
(
α
)
]
=
ξ
−
E
[
ξ
]
σ
{\displaystyle {\frac {\partial }{\partial \mu }}\log f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\left[{\frac {x-\mu }{\sigma }}+{\frac {\varphi (\beta )-\varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right]={\frac {\xi -\operatorname {E} [\xi ]}{\sigma }}}
および
∂
∂
σ
log
f
(
x
;
μ
,
σ
,
a
,
b
)
=
1
σ
[
(
x
−
μ
σ
)
2
−
1
+
β
φ
(
β
)
−
α
φ
(
α
)
Φ
(
β
)
−
Φ
(
α
)
]
=
ξ
2
−
E
[
ξ
2
]
σ
{\displaystyle {\frac {\partial }{\partial \sigma }}\log f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\left[\left({\frac {x-\mu }{\sigma }}\right)^{2}-1+{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right]={\frac {\xi ^{2}-\operatorname {E} [\xi ^{2}]}{\sigma }}}
I
X
(
μ
,
σ
)
=
(
E
[
ξ
2
]
−
E
[
ξ
]
2
σ
2
E
[
ξ
3
]
−
E
[
ξ
]
E
[
ξ
2
]
σ
2
E
[
ξ
3
]
−
E
[
ξ
]
E
[
ξ
2
]
σ
2
E
[
ξ
4
]
−
E
[
ξ
2
]
2
σ
2
)
{\displaystyle {\mathcal {I}}_{X}(\mu ,\sigma )=\left({\begin{matrix}{\frac {\operatorname {E} [\xi ^{2}]-\operatorname {E} [\xi ]^{2}}{\sigma ^{2}}}&{\frac {\operatorname {E} [\xi ^{3}]-\operatorname {E} [\xi ]\operatorname {E} [\xi ^{2}]}{\sigma ^{2}}}\\{\frac {\operatorname {E} [\xi ^{3}]-\operatorname {E} [\xi ]\operatorname {E} [\xi ^{2}]}{\sigma ^{2}}}&{\frac {\operatorname {E} [\xi ^{4}]-\operatorname {E} [\xi ^{2}]^{2}}{\sigma ^{2}}}\end{matrix}}\right)}
により与えられる。
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
![{\displaystyle \sigma ^{2}\left[1-{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}-\left({\frac {\varphi (\beta )-\varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/963b158e0df5440c66d5f5e645f4f7fb94db6c48)
![{\displaystyle \ln \left(\sigma {\sqrt {2\pi e}}\left[\Phi (\beta )-\Phi (\alpha )\right]\right)-{\frac {1}{2}}{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0049cda66daad92b317ac364d77a2a1226792a84)
![{\displaystyle (-\infty ,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9d3156528d17e410760f1d0cd5034f6011a635)
![{\displaystyle \operatorname {Var} (X|a<X<b)=\sigma ^{2}\left[1+{\frac {\alpha \varphi (\alpha )-\beta \varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}-\left({\frac {\varphi (\alpha )-\varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28c428e4dc463df927f279f67efc66b8eb5d1592)
![{\displaystyle \mu \notin [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f6401781e8f34343f6d08dc3ea072524ef84e8)
![{\displaystyle \operatorname {Var} (X|X>a)=\sigma ^{2}\left[1+{\frac {\alpha }{R(\alpha )}}-\left({\frac {1}{R(\alpha )}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3841a167792bd65a55b228ddf2d09769196934bf)
![{\displaystyle \operatorname {Var} (X|X<b)=\sigma ^{2}\left[1-\beta {\frac {\varphi (\beta )}{\Phi (\beta )}}-\left({\frac {\varphi (\beta )}{\Phi (\beta )}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc63b218756fe742a1e3e262ea6036dd6f71d5d)
![{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]={\frac {\alpha ^{n+1}\varphi (\alpha )-\beta ^{n+1}\varphi (\beta )}{\Phi (\beta )-\Phi (\alpha )}}+(n+1)\operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/568663e62ce79d6acaf6e1e141cd6e10be958199)
![{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]=\int _{\alpha }^{\beta }{\frac {\xi ^{n+2}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f62b843d49e4167f0fad816dd1ac473950462a7)
![{\displaystyle \operatorname {E} \left[\left({\frac {x-\mu }{\sigma }}\right)^{n+2}\right]=\left[{\frac {-\xi ^{n+1}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}\right]_{\alpha }^{\beta }+(n+1)\int _{\alpha }^{\beta }{\frac {\xi ^{n}\varphi (\xi )}{\Phi (\beta )-\Phi (\alpha )}}d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f78e33335062361270410af5abc6fe82ef3e828e)
![{\displaystyle {\frac {\partial }{\partial \mu }}\log f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\left[{\frac {x-\mu }{\sigma }}+{\frac {\varphi (\beta )-\varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right]={\frac {\xi -\operatorname {E} [\xi ]}{\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5863abc0367bc01e613004477d9a8e008dd9ccc0)
![{\displaystyle {\frac {\partial }{\partial \sigma }}\log f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\left[\left({\frac {x-\mu }{\sigma }}\right)^{2}-1+{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right]={\frac {\xi ^{2}-\operatorname {E} [\xi ^{2}]}{\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/485e8052eff1a3356799ec44d57f0285de39f713)
![{\displaystyle {\mathcal {I}}_{X}(\mu ,\sigma )=\left({\begin{matrix}{\frac {\operatorname {E} [\xi ^{2}]-\operatorname {E} [\xi ]^{2}}{\sigma ^{2}}}&{\frac {\operatorname {E} [\xi ^{3}]-\operatorname {E} [\xi ]\operatorname {E} [\xi ^{2}]}{\sigma ^{2}}}\\{\frac {\operatorname {E} [\xi ^{3}]-\operatorname {E} [\xi ]\operatorname {E} [\xi ^{2}]}{\sigma ^{2}}}&{\frac {\operatorname {E} [\xi ^{4}]-\operatorname {E} [\xi ^{2}]^{2}}{\sigma ^{2}}}\end{matrix}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8198643dc27e6b17bb0434486f57b372fecd98)
![{\displaystyle x=\mu +\sigma \Phi ^{-1}\left(\Phi (\alpha )+y[\Phi (\beta )-\Phi (\alpha )]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9bd95037dc5dd7285d2ba1b41ded1dc15d13b4a)
