ここでは、母数が1つ・推定量が不偏である場合から始めて、いくつかのかなり一般的な場合へと拡張していく。どのバージョンでもある種の正規性の仮定をおくが、それはほとんどの「普通のふるまいをする」確率分布については成り立つものである。この条件については後述する。
クラメール・ラオの限界を、母数が複数の場合にも拡張しよう。母数ベクトル を
θ
=
(
θ
1
,
θ
2
,
…
,
θ
d
)
T
∈
R
d
{\displaystyle {\boldsymbol {\theta }}=\left(\theta _{1},\theta _{2},\dots ,\theta _{d}\right)^{T}\in \mathbb {R} ^{d}}
とし（右肩の T は転置を表す（以下同じ））、それによって決まる確率密度関数
f
(
x
;
θ
)
{\displaystyle f(x;{\boldsymbol {\theta }})}
f
{\displaystyle f}
フィッシャー情報行列 は、
d
×
d
{\displaystyle d\times d}
I
m
,
k
{\displaystyle I_{m,k}}
I
m
,
k
=
E
[
∂
∂
θ
m
ln
f
(
x
;
θ
)
∂
∂
θ
k
ln
f
(
x
;
θ
)
]
=
−
E
[
∂
2
∂
θ
m
∂
θ
k
ln
f
(
x
;
θ
)
]
{\displaystyle {\begin{aligned}I_{m,k}&=\operatorname {E} \left[{\frac {\partial }{\partial \theta _{m}}}\ln f\left(x;{\boldsymbol {\theta }}\right){\frac {\partial }{\partial \theta _{k}}}\ln f\left(x;{\boldsymbol {\theta }}\right)\right]\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta _{m}\partial \theta _{k}}}\ln f\left(x;{\boldsymbol {\theta }}\right)\right]\end{aligned}}}
で定まる行列のことである。
T
(
X
)
{\displaystyle {\boldsymbol {T}}(X)}
T
(
X
)
=
(
T
1
(
X
)
,
…
,
T
d
(
X
)
)
T
{\displaystyle {\boldsymbol {T}}(X)=(T_{1}(X),\ldots ,T_{d}(X))^{T}}
E
[
T
(
X
)
]
{\displaystyle \operatorname {E} [{\boldsymbol {T}}(X)]}
ψ
(
θ
)
{\displaystyle {\boldsymbol {\psi }}({\boldsymbol {\theta }})}
このとき、
T
(
X
)
{\displaystyle {\boldsymbol {T}}(X)}
分散共分散行列 に対するクラメール・ラオの限界は、
Cov
(
T
(
X
)
)
≥
∂
ψ
(
θ
)
∂
θ
(
[
I
(
θ
)
]
−
1
∂
ψ
(
θ
)
∂
θ
)
T
{\displaystyle \operatorname {Cov} \left({\boldsymbol {T}}(X)\right)\geq {\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}\left([I\left({\boldsymbol {\theta }}\right)]^{-1}{\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}\right)^{T}}
となる。ここで、
行列に対する不等式
A
≥
B
{\displaystyle A\geq B}
A
−
B
{\displaystyle A-B}
非負定値 であるということである。
∂
ψ
(
θ
)
/
∂
θ
{\displaystyle \partial {\boldsymbol {\psi }}({\boldsymbol {\theta }})/\partial {\boldsymbol {\theta }}}
ヤコビ行列 （
i
j
{\displaystyle ij}
∂
ψ
i
(
θ
)
/
∂
θ
j
{\displaystyle \partial \psi _{i}({\boldsymbol {\theta }})/\partial \theta _{j}}
T
(
X
)
{\displaystyle {\boldsymbol {T}}(X)}
θ
{\displaystyle {\boldsymbol {\theta }}}
ψ
(
θ
)
=
θ
{\displaystyle {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)={\boldsymbol {\theta }}}
Cov
(
T
(
X
)
)
≥
I
(
θ
)
−
1
{\displaystyle \operatorname {Cov} \left({\boldsymbol {T}}(X)\right)\geq I\left({\boldsymbol {\theta }}\right)^{-1}}
のようになる。フィッシャー情報行列の逆行列 を計算するのが面倒な場合は、単に対応する対角成分の逆数をとることで、（より緩いかもしれないが）1つの下限が得られる[ 8]
Var
(
T
m
(
X
)
)
=
[
Cov
(
T
(
X
)
)
]
m
m
≥
[
I
(
θ
)
−
1
]
m
m
≥
(
[
I
(
θ
)
]
m
m
)
−
1
{\displaystyle \operatorname {Var} (T_{m}(X))=\left[\operatorname {Cov} \left({\boldsymbol {T}}(X)\right)\right]_{mm}\geq \left[I\left({\boldsymbol {\theta }}\right)^{-1}\right]_{mm}\geq \left(\left[I\left({\boldsymbol {\theta }}\right)\right]_{mm}\right)^{-1}}
f
(
x
;
θ
)
{\displaystyle f(x;\theta )}
θ
{\displaystyle \theta }
I
(
θ
)
=
E
[
(
∂
∂
θ
ln
f
(
X
;
θ
)
)
2
]
=
∫
R
f
(
x
;
θ
)
1
(
f
(
x
;
θ
)
)
2
(
∂
f
(
x
;
θ
)
∂
θ
)
2
d
x
=
−
∫
R
f
(
x
;
θ
)
f
(
x
;
θ
)
∂
f
(
x
;
θ
)
∂
θ
−
(
∂
f
(
x
;
θ
)
∂
θ
)
2
(
f
(
x
;
θ
)
)
2
d
x
=
−
∫
R
f
(
x
;
θ
)
(
∂
2
∂
θ
2
ln
f
(
x
;
θ
)
)
d
x
=
−
E
[
∂
2
∂
θ
2
ln
f
(
X
;
θ
)
]
{\displaystyle {\begin{aligned}I(\theta )&=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta }}\ln f(X;\theta )\right)^{2}\right]\\&=\int _{\mathbb {R} }f(x;\theta ){\frac {1}{\left(f(x;\theta )\right)^{2}}}\left({\frac {\partial f(x;\theta )}{\partial \theta }}\right)^{2}\,dx\\&=-\int _{\mathbb {R} }f(x;\theta ){\frac {f(x;\theta ){\frac {\partial f(x;\theta )}{\partial \theta }}-\left({\frac {\partial f(x;\theta )}{\partial \theta }}\right)^{2}}{\left(f(x;\theta )\right)^{2}}}\,dx\\&=-\int _{\mathbb {R} }f(x;\theta )\left({\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(x;\theta )\right)\,dx\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]\end{aligned}}}
（3番目の等号の箇所で
∫
R
∂
f
(
x
;
θ
)
∂
θ
d
x
=
∂
∂
θ
∫
R
f
(
x
;
θ
)
d
x
=
∂
∂
θ
(
1
)
=
0
{\displaystyle \int _{\mathbb {R} }{\frac {\partial f(x;\theta )}{\partial \theta }}\,dx={\frac {\partial }{\partial \theta }}\int _{\mathbb {R} }f(x;\theta )\,dx={\frac {\partial }{\partial \theta }}(1)=0}
であることを用いた）
と変形でき、クラメール・ラオの不等式は次のようにも書ける。
Var
(
θ
^
)
≥
1
I
(
θ
)
=
1
−
E
[
∂
2
∂
θ
2
ln
f
(
X
;
θ
)
]
{\displaystyle \operatorname {Var} ({\hat {\theta }})\geq {\frac {1}{I(\theta )}}={\frac {1}{-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]}}}
こちらの公式の方が下限を評価するのにより有用な場合がある。
母数が1つの場合のクラメール・ラオの不等式を一般的に証明する。
X
{\displaystyle X}
f
(
x
;
θ
)
{\displaystyle f(x;\theta )}
確率変数 とし、
T
=
t
(
X
)
{\displaystyle T=t(X)}
X
{\displaystyle X}
θ
{\displaystyle \theta }
ψ
(
θ
)
{\displaystyle \psi (\theta )}
E
[
T
]
=
ψ
(
θ
)
{\displaystyle \operatorname {E} \left[T\right]=\psi (\theta )}
目標は、任意の
θ
{\displaystyle \theta }
Var
(
t
(
X
)
)
≥
[
ψ
′
(
θ
)
]
2
I
(
θ
)
{\displaystyle \operatorname {Var} (t(X))\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}}
を示すことである。
V
{\displaystyle V}
V
=
∂
∂
θ
ln
f
(
X
;
θ
)
=
1
f
(
X
;
θ
)
∂
∂
θ
f
(
X
;
θ
)
{\displaystyle V={\frac {\partial }{\partial \theta }}\ln f(X;\theta )={\frac {1}{f(X;\theta )}}{\frac {\partial }{\partial \theta }}f(X;\theta )}
ここで連鎖律 を使った。
V
{\displaystyle V}
E
[
V
]
=
∫
R
f
(
x
;
θ
)
[
1
f
(
x
;
θ
)
∂
∂
θ
f
(
x
;
θ
)
]
d
x
=
∂
∂
θ
∫
R
f
(
x
;
θ
)
d
x
=
∂
∂
θ
(
1
)
=
0
{\displaystyle {\begin{aligned}\operatorname {E} \left[V\right]&=\int _{\mathbb {R} }f(x;\theta )\left[{\frac {1}{f(x;\theta )}}{\frac {\partial }{\partial \theta }}f(x;\theta )\right]\,dx\\&={\frac {\partial }{\partial \theta }}\int _{\mathbb {R} }f(x;\theta )\,dx={\frac {\partial }{\partial \theta }}(1)=0\end{aligned}}}
ここで積分と偏微分の順序が交換可能であること（正規性の条件の2番目）を使った。
V
{\displaystyle V}
T
{\displaystyle T}
共分散
Cov
(
V
,
T
)
{\displaystyle \operatorname {Cov} (V,T)}
E
[
V
]
=
0
{\displaystyle \operatorname {E} \left[V\right]=0}
Cov
(
V
,
T
)
=
E
[
V
T
]
{\displaystyle \operatorname {Cov} (V,T)=\operatorname {E} \left[VT\right]}
Cov
(
V
,
T
)
=
E
[
T
⋅
{
1
f
(
X
;
θ
)
∂
∂
θ
f
(
X
;
θ
)
}
]
=
∫
R
t
(
x
)
[
1
f
(
x
;
θ
)
∂
∂
θ
f
(
x
;
θ
)
]
f
(
x
;
θ
)
d
x
=
∂
∂
θ
[
∫
R
t
(
x
)
f
(
x
;
θ
)
d
x
]
=
∂
∂
θ
E
[
T
]
=
ψ
′
(
θ
)
{\displaystyle {\begin{aligned}\operatorname {Cov} (V,T)&=\operatorname {E} \left[T\cdot \left\{{\frac {1}{f(X;\theta )}}{\frac {\partial }{\partial \theta }}f(X;\theta )\right\}\right]\\[6pt]&=\int _{\mathbb {R} }t(x)\left[{\frac {1}{f(x;\theta )}}{\frac {\partial }{\partial \theta }}f(x;\theta )\right]f(x;\theta )\,dx\\[6pt]&={\frac {\partial }{\partial \theta }}\left[\int _{\mathbb {R} }t(x)f(x;\theta )\,dx\right]={\frac {\partial }{\partial \theta }}\operatorname {E} \left[T\right]=\psi ^{\prime }(\theta )\end{aligned}}}
ここで再び、積分と微分が交換可能であるという条件（正規性の条件の2番目）を使った。
コーシー・シュワルツの不等式 から、
Var
(
T
)
Var
(
V
)
≥
|
Cov
(
V
,
T
)
|
=
|
ψ
′
(
θ
)
|
{\displaystyle {\sqrt {\operatorname {Var} (T)\operatorname {Var} (V)}}\geq \left|\operatorname {Cov} (V,T)\right|=\left|\psi ^{\prime }(\theta )\right|}
よって
Var
(
T
)
≥
[
ψ
′
(
θ
)
]
2
Var
(
V
)
=
[
ψ
′
(
θ
)
]
2
I
(
θ
)
{\displaystyle \operatorname {Var} (T)\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{\operatorname {Var} (V)}}={\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}}
これが示したかったことである。
確率変数列
X
1
,
X
2
,
⋯
,
X
n
{\displaystyle X_{1},X_{2},\cdots ,X_{n}}
θ
{\displaystyle \theta }
X
:=
(
X
1
,
X
2
,
⋯
,
X
n
)
{\displaystyle {\boldsymbol {X}}:=(X_{1},X_{2},\cdots ,X_{n})}
尤度関数は、結合確率密度関数
f
n
(
x
1
,
x
2
,
⋯
,
x
n
;
θ
)
=
f
n
(
x
;
θ
)
{\displaystyle f_{n}(x_{1},x_{2},\cdots ,x_{n};\theta )=f_{n}({\boldsymbol {x}};\theta )}
x
{\displaystyle {\boldsymbol {x}}}
θ
{\displaystyle \theta }
スコア関数は、尤度関数の自然対数をとってから
θ
{\displaystyle \theta }
∂
∂
θ
ln
f
n
(
x
;
θ
)
{\displaystyle {\frac {\partial }{\partial \theta }}\ln f_{n}({\boldsymbol {x}};\theta )}
これらはいずれも実数値関数であるので、
I
(
θ
)
=
E
[
(
∂
∂
θ
ln
f
n
(
X
;
θ
)
)
2
]
{\displaystyle I(\theta )=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta }}\ln f_{n}({\boldsymbol {X}};\theta )\right)^{2}\right]}
となる。 本記事でここまでに述べた事柄は、次の置き換えをすれば基本的に全て同じ形式で成り立つ。
X
→
X
,
x
→
x
,
∫
R
(
⋯
)
d
x
→
∫
R
n
(
⋯
)
d
x
{\displaystyle X\to {\boldsymbol {X}},\quad x\to {\boldsymbol {x}},\quad \int _{\mathbb {R} }(\cdots )\,dx\to \int _{\mathbb {R} ^{n}}(\cdots )\,d{\boldsymbol {x}}}
特に、確率変数列
X
=
(
X
1
,
X
2
,
⋯
,
X
n
)
{\displaystyle {\boldsymbol {X}}=(X_{1},X_{2},\cdots ,X_{n})}
独立同分布 で、その確率密度関数が
f
(
x
;
θ
)
{\displaystyle f(x;\theta )}
尤度関数は
f
n
(
x
;
θ
)
=
∏
i
=
1
n
f
(
x
i
;
θ
)
{\displaystyle f_{n}({\boldsymbol {x}};\theta )=\prod _{i=1}^{n}f(x_{i};\theta )}
スコア関数は
∂
∂
θ
ln
f
n
(
x
;
θ
)
=
∑
i
=
1
n
(
∂
∂
θ
ln
f
(
x
i
;
θ
)
)
{\displaystyle {\frac {\partial }{\partial \theta }}\ln f_{n}({\boldsymbol {x}};\theta )=\sum _{i=1}^{n}\left({\frac {\partial }{\partial \theta }}\ln f(x_{i};\theta )\right)}
フィッシャー情報量は
I
(
θ
)
=
−
E
[
∂
2
∂
θ
2
ln
f
n
(
X
;
θ
)
]
=
−
E
[
∂
2
∂
θ
2
∑
i
=
1
n
{
ln
f
(
X
i
;
θ
)
}
]
=
−
∑
i
=
1
n
(
E
[
∂
2
∂
θ
2
{
ln
f
(
X
;
θ
)
}
]
)
=
−
n
E
[
∂
2
∂
θ
2
ln
f
(
X
;
θ
)
]
{\displaystyle {\begin{aligned}I(\theta )&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f_{n}({\boldsymbol {X}};\theta )\right]\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\sum _{i=1}^{n}\{\ln f(X_{i};\theta )\}\right]\\&=-\sum _{i=1}^{n}\left(\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\{\ln f(X;\theta )\}\right]\right)\\&=-n\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]\end{aligned}}}
となる。
↑ Cramér, Harald (1946). Mathematical Methods of Statistics . Princeton, NJ: Princeton Univ. Press. ISBN 0-691-08004-6 . OCLC 185436716 ↑ Rao, Calyampudi Radakrishna (1945). “Information and the accuracy attainable in the estimation of statistical parameters”. Bulletin of the Calcutta Mathematical Society 37 : 81–89. MR 0015748 . ↑ Rao, Calyampudi Radakrishna (1994). S. Das Gupta. ed. Selected Papers of C. R. Rao . New York: Wiley. ISBN 978-0-470-22091-7 . OCLC 174244259 ↑ Darmois, Georges (1945). “Sur les limites de la dispersion de certaines estimations”. Rev. Int. Inst. Statist. 13 : 9–15. ↑ Fréchet, Maurice (1943). “Sur l'extension de certaines évaluations statistiques au cas de petit échantillons”. Rev. Inst. Int. Statist. 11 : 182–205. ↑ Gart, John J. (1958). “An extension of the Cramér–Rao inequality”. Ann. Math. Stat. 29 : 367–380. ↑ Malécot, Gustave (1999). “Statistical methods and the subjective basis of scientific knowledge [ translated from Année X 1947 by Daniel Gianola] ”. Genet. Sel. Evol. 31 : 269–298. ↑ For the Bayesian case, see eqn. (11) of Bobrovsky; Mayer-Wolf; Zakai (1987). “Some classes of global Cramer–Rao bounds”. Ann. Stat. 15 (4): 1421–38. ↑ Kay, S. M. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory . Prentice Hall. p. 47. ISBN 0-13-042268-1
Bos, Adriaan van den (2007). Parameter Estimation for Scientists and Engineers . Hoboken: John Wiley & Sons . pp. 45–98. ISBN 0-470-14781-4 Kay, Steven M. (1993). Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory . Prentice Hall. ISBN 0-13-345711-7 Shao, Jun (1998). Mathematical Statistics . New York: Springer . ISBN 0-387-98674-X
![{\displaystyle I(\theta )=\operatorname {E} \left[\left({\frac {\partial \ell (X;\theta )}{\partial \theta }}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5872142e5c8ef87b2a5fadb8ad347ad05ccea1)
![{\displaystyle \operatorname {E} \left[T(X)\right]=\psi (\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04644246a16192b671fb9faa23411d3d9a970881)
![{\displaystyle \operatorname {Var} (T)\geq {\frac {[\psi '(\theta )]^{2}}{I(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9598e0faa4e0af6df574dab1b607aa3ed76e78ee)
![{\displaystyle b(\theta )=\operatorname {E} [{\hat {\theta }}]-\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da54579c77c5ff3f92c0780adf15c66349e6370f)
![{\displaystyle \operatorname {Var} ({\hat {\theta }})\geq {\frac {[1+b'(\theta )]^{2}}{I(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7308ec6dbe616c805ede8a23c6601f04562c1)
![{\displaystyle \operatorname {E} \left[({\hat {\theta }}-\theta )^{2}\right]\geq {\frac {[1+b'(\theta )]^{2}}{I(\theta )}}+b(\theta )^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f3dda36583dc8cdb1e59d9879c379d73400b93)
![{\displaystyle \operatorname {MSE} ({\hat {\theta }}):=\operatorname {E} \left[({\hat {\theta }}-\theta )^{2}\right]=\operatorname {E} \left[\left({\hat {\theta }}-\operatorname {E} [{\hat {\theta }}]\right)^{2}\right]+\left(\operatorname {E} [{\hat {\theta }}]-\theta \right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe73dd9a9f30341046936d85de3a5faa9c09834)
![{\displaystyle {\begin{aligned}I_{m,k}&=\operatorname {E} \left[{\frac {\partial }{\partial \theta _{m}}}\ln f\left(x;{\boldsymbol {\theta }}\right){\frac {\partial }{\partial \theta _{k}}}\ln f\left(x;{\boldsymbol {\theta }}\right)\right]\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta _{m}\partial \theta _{k}}}\ln f\left(x;{\boldsymbol {\theta }}\right)\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28bb94094f1c41d22aaa14590e3d8383ed92911)
![{\displaystyle \operatorname {E} [{\boldsymbol {T}}(X)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b5d13a494f822b3bd79f28275162580970dad7)
![{\displaystyle \operatorname {Cov} \left({\boldsymbol {T}}(X)\right)\geq {\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}\left([I\left({\boldsymbol {\theta }}\right)]^{-1}{\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}\right)^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef0f6d2405efc55c754739302b251a6d57b9bdb)
![{\displaystyle \operatorname {Var} (T_{m}(X))=\left[\operatorname {Cov} \left({\boldsymbol {T}}(X)\right)\right]_{mm}\geq \left[I\left({\boldsymbol {\theta }}\right)^{-1}\right]_{mm}\geq \left(\left[I\left({\boldsymbol {\theta }}\right)\right]_{mm}\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b61e88c482d5a2aaadc4af197417777b547b84c8)
![{\displaystyle {\frac {\partial }{\partial \theta }}\left[\int _{\mathbb {R} }T(x)f(x;\theta )\,dx\right]=\int _{\mathbb {R} }T(x)\left[{\frac {\partial }{\partial \theta }}f(x;\theta )\right]\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7028faea3b859ed3c5cbfb550f58aa232e813416)
![{\displaystyle {\begin{aligned}I(\theta )&=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta }}\ln f(X;\theta )\right)^{2}\right]\\&=\int _{\mathbb {R} }f(x;\theta ){\frac {1}{\left(f(x;\theta )\right)^{2}}}\left({\frac {\partial f(x;\theta )}{\partial \theta }}\right)^{2}\,dx\\&=-\int _{\mathbb {R} }f(x;\theta ){\frac {f(x;\theta ){\frac {\partial f(x;\theta )}{\partial \theta }}-\left({\frac {\partial f(x;\theta )}{\partial \theta }}\right)^{2}}{\left(f(x;\theta )\right)^{2}}}\,dx\\&=-\int _{\mathbb {R} }f(x;\theta )\left({\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(x;\theta )\right)\,dx\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b59740fcf8ec009be5beb18d93a9c29f5f8ea67)
![{\displaystyle \operatorname {Var} ({\hat {\theta }})\geq {\frac {1}{I(\theta )}}={\frac {1}{-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be536bd0edc1f8f321eb8377236a9728baf86a51)
![{\displaystyle \operatorname {E} \left[T\right]=\psi (\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9201b93f27a4f4f52f29d13db6469764140a1cca)
![{\displaystyle \operatorname {Var} (t(X))\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae7e8cb92bc1baf5a862ba8817cabd7cb4752f8)
![{\displaystyle {\begin{aligned}\operatorname {E} \left[V\right]&=\int _{\mathbb {R} }f(x;\theta )\left[{\frac {1}{f(x;\theta )}}{\frac {\partial }{\partial \theta }}f(x;\theta )\right]\,dx\\&={\frac {\partial }{\partial \theta }}\int _{\mathbb {R} }f(x;\theta )\,dx={\frac {\partial }{\partial \theta }}(1)=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27ed681b3795729741f46d0911b12084a3d15346)
![{\displaystyle \operatorname {E} \left[V\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbca38967d474fe79f8f063c629689ade1f059a)
![{\displaystyle \operatorname {Cov} (V,T)=\operatorname {E} \left[VT\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2991a81e9c8b0cace758400be84e8140f4aa71)
![{\displaystyle {\begin{aligned}\operatorname {Cov} (V,T)&=\operatorname {E} \left[T\cdot \left\{{\frac {1}{f(X;\theta )}}{\frac {\partial }{\partial \theta }}f(X;\theta )\right\}\right]\\[6pt]&=\int _{\mathbb {R} }t(x)\left[{\frac {1}{f(x;\theta )}}{\frac {\partial }{\partial \theta }}f(x;\theta )\right]f(x;\theta )\,dx\\[6pt]&={\frac {\partial }{\partial \theta }}\left[\int _{\mathbb {R} }t(x)f(x;\theta )\,dx\right]={\frac {\partial }{\partial \theta }}\operatorname {E} \left[T\right]=\psi ^{\prime }(\theta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad26ec7b45004f4b57df4727430d2c03ac54ded)
![{\displaystyle \operatorname {Var} (T)\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{\operatorname {Var} (V)}}={\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3993f230fea08cba8b549cba9defff4ed56d99)
![{\displaystyle I(\theta )=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta }}\ln f_{n}({\boldsymbol {X}};\theta )\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c127e07ee4c468b624a4a629aca4c1e51c3bc0e8)
![{\displaystyle {\begin{aligned}I(\theta )&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f_{n}({\boldsymbol {X}};\theta )\right]\\&=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\sum _{i=1}^{n}\{\ln f(X_{i};\theta )\}\right]\\&=-\sum _{i=1}^{n}\left(\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\{\ln f(X;\theta )\}\right]\right)\\&=-n\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln f(X;\theta )\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0548b4a1aaaa0cf9ec4051dd053d53cbf3acef6)
![{\displaystyle \operatorname {E} \left[T\right]=\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c06331f26aa22706c686c81e17709b7682d0fa11)
![{\displaystyle \operatorname {Var} (T)={\frac {\operatorname {Var} (X-\mu )^{2}}{n}}={\frac {1}{n}}\left[\operatorname {E} \left[(X-\mu )^{4}\right]-\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c85e0040efe1234d50e77c030aa92b8d50e6ae36)
![{\displaystyle {\begin{aligned}V&={\frac {\partial }{\partial (\sigma ^{2})}}\ln L(\sigma ^{2},X)\\&={\frac {\partial }{\partial (\sigma ^{2})}}\ln \left[{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(X-\mu )^{2}/{2\sigma ^{2}}}\right]={\frac {(X-\mu )^{2}}{2(\sigma ^{2})^{2}}}-{\frac {1}{2\sigma ^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be234ea1d30ebd97ca6f55d726de8aff6a051803)
![{\displaystyle {\begin{aligned}I&=-\operatorname {E} \left[{\frac {\partial V}{\partial (\sigma ^{2})}}\right]=-\operatorname {E} \left[-{\frac {(X-\mu )^{2}}{(\sigma ^{2})^{3}}}+{\frac {1}{2(\sigma ^{2})^{2}}}\right]\\&={\frac {\sigma ^{2}}{(\sigma ^{2})^{3}}}-{\frac {1}{2(\sigma ^{2})^{2}}}={\frac {1}{2(\sigma ^{2})^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d264dad707e10f18a2f6e18036d0c4b61be0048d)
![{\displaystyle -b(\sigma ^{2})=\sigma ^{2}-\operatorname {E} [T_{b}]=\left(1-{\frac {n}{n+2}}\right)\sigma ^{2}={\frac {2\sigma ^{2}}{n+2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a82e9dd4800e95a5b5267221f0bcaa9482876d30)
