Пусть
x
→
=
(
x
1
,
…
,
x
n
)
{\displaystyle {\vec {x}}=\left(x_{1},\ldots ,x_{n}\right)}
выборка из распределения , зависящего от параметра
θ
∈
Θ
{\displaystyle \theta \in \Theta }
θ
^
≡
θ
^
(
x
→
)
{\displaystyle {\hat {\theta }}\equiv {\hat {\theta }}\left({\vec {x}}\right)}
E
[
θ
^
]
=
θ
,
∀
θ
∈
Θ
{\displaystyle \mathbb {E} \left[{\hat {\theta }}\right]=\theta ,\quad \forall \theta \in \Theta }
где
В противном случае оценка называется смещённой, и случайная величина
E
θ
^
−
θ
{\displaystyle \mathbb {E} {\hat {\theta }}-\theta }
смеще́нием .
Выборочное среднее
X
¯
=
1
n
∑
i
=
1
n
X
i
{\displaystyle {\bar {X}}={\frac {1}{n}}\sum \limits _{i=1}^{n}X_{i}}
X
i
{\displaystyle X_{i}}
E
X
i
=
μ
<
∞
{\displaystyle \mathbb {E} X_{i}=\mu <\infty }
∀
i
∈
N
{\displaystyle \forall i\in \mathbb {N} }
E
X
¯
=
μ
{\displaystyle \mathbb {E} {\bar {X}}=\mu }
Пусть независимые случайные величины
X
i
{\displaystyle X_{i}}
дисперсию
D
X
i
=
σ
2
{\displaystyle \mathrm {D} X_{i}=\sigma ^{2}}
S
n
2
=
1
n
∑
i
=
1
n
(
X
i
−
X
¯
)
2
{\displaystyle S_{n}^{2}={\frac {1}{n}}\sum \limits _{i=1}^{n}\left(X_{i}-{\bar {X}}\right)^{2}}
выборочная дисперсия ,и
S
2
=
1
n
−
1
∑
i
=
1
n
(
X
i
−
X
¯
)
2
{\displaystyle S^{2}={\frac {1}{n-1}}\sum \limits _{i=1}^{n}\left(X_{i}-{\bar {X}}\right)^{2}}
исправленная выборочная дисперсия .Тогда
S
n
2
{\displaystyle S_{n}^{2}}
S
2
{\displaystyle S^{2}}
σ
2
{\displaystyle \sigma ^{2}}
S
n
2
{\displaystyle S_{n}^{2}}
Пусть
μ
{\displaystyle \mu }
X
¯
{\displaystyle {\overline {X}}}
E
[
S
n
2
]
=
E
[
1
n
∑
i
=
1
n
(
X
i
−
X
¯
)
2
]
.
{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}{\bigg ]}.}
Добавив и отняв
μ
{\displaystyle \mu }
E
[
S
n
2
]
=
E
[
1
n
∑
i
=
1
n
(
(
X
i
−
μ
)
−
(
X
¯
−
μ
)
)
2
]
.
{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\big (}(X_{i}-\mu )-({\overline {X}}-\mu ){\big )}^{2}{\bigg ]}.}
Возведём в квадрат и получим:
E
[
S
n
2
]
=
E
[
1
n
∑
i
=
1
n
(
X
i
−
μ
)
2
−
2
(
X
¯
−
μ
)
1
n
∑
i
=
1
n
(
X
i
−
μ
)
+
n
n
(
X
¯
−
μ
)
2
]
.
{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu ){\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {n}{n}}({\overline {X}}-\mu )^{2}{\bigg ]}.}
Заметив, что
1
n
∑
i
=
1
n
(
X
i
−
μ
)
=
X
¯
−
1
n
(
n
μ
)
{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )={\overline {X}}-{\frac {1}{n}}(n\mu )}
E
[
S
n
2
]
=
E
[
1
n
∑
i
=
1
n
(
X
i
−
μ
)
2
−
(
X
¯
−
μ
)
2
]
.
{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-({\overline {X}}-\mu )^{2}{\bigg ]}.}
Учитывая, что
E
[
a
+
b
]
=
E
[
a
]
+
E
[
b
]
{\displaystyle \operatorname {E} [a+b]=\operatorname {E} [a]+\operatorname {E} [b]}
E
[
1
n
∑
i
=
1
n
(
X
i
−
μ
)
2
]
=
σ
2
{\displaystyle \operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}=\sigma ^{2}}
дисперсия ;
E
[
(
X
¯
−
μ
)
2
]
=
1
n
σ
2
{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}={\frac {1}{n}}\sigma ^{2}}
E
[
(
X
¯
−
μ
)
2
]
=
E
[
(
1
n
∑
i
=
1
n
(
X
i
−
μ
)
)
2
]
=
E
[
1
n
2
∑
i
=
1
n
(
X
i
−
μ
)
2
+
2
n
2
∑
i
=
1
,
j
=
1
,
i
<
j
n
(
X
i
−
μ
)
(
X
j
−
μ
)
]
{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}=\operatorname {E} {\big [}{\big (}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu ){\big )}^{2}{\big ]}=\operatorname {E} {\big [}{\frac {1}{n^{2}}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}+{\frac {2}{n^{2}}}\sum _{i=1,j=1,i<j}^{n}(X_{i}-\mu )(X_{j}-\mu ){\big ]}}
X
i
{\displaystyle X_{i}}
X
j
{\displaystyle X_{j}}
E
[
(
X
i
−
μ
)
]
=
0
{\displaystyle \operatorname {E} {\big [}(X_{i}-\mu ){\big ]}=0}
E
[
∑
i
=
1
,
j
=
1
,
i
<
j
n
(
X
i
−
μ
)
(
X
j
−
μ
)
]
=
∑
i
=
1
,
j
=
1
,
i
<
j
n
E
[
(
X
i
−
μ
)
]
E
[
(
X
j
−
μ
)
]
=
0
{\displaystyle \operatorname {E} {\big [}\sum _{i=1,j=1,i<j}^{n}(X_{i}-\mu )(X_{j}-\mu ){\big ]}=\sum _{i=1,j=1,i<j}^{n}\operatorname {E} {\big [}(X_{i}-\mu ){\big ]}\operatorname {E} {\big [}(X_{j}-\mu ){\big ]}=0}
получим:
E
[
S
n
2
]
=
σ
2
−
E
[
(
X
¯
−
μ
)
2
]
=
=
σ
2
−
1
n
σ
2
=
=
n
−
1
n
σ
2
<
σ
2
.
{\displaystyle {\begin{aligned}\operatorname {E} [S_{n}^{2}]&=\sigma ^{2}-\operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}=\\&=\sigma ^{2}-{\frac {1}{n}}\sigma ^{2}=\\&={\frac {n-1}{n}}\sigma ^{2}<\sigma ^{2}.\end{aligned}}}
править
M. G. Kendall. "The advanced theory of statistics (vol. I). Distribution theory (2nd edition)". Charles Griffin & Company Limited, 1945.
M. G. Kendall and A. Stuart. "The advanced theory of statistics (vol. II). Inference and relationship (2nd edition)". Charles Griffin & Company Limited, 1967.
A. Papoulis. Probability, random variables, and stochastic processes (3rd edition). McGrow-Hill Inc., 1991.
G. Saporta. "Probabilités, analyse des données et statistiques". Éditions Technip, Paris, 1990.
J. F. Kenney and E. S. Keeping. Mathematics of Statistics. Part I & II. D. Van Nostrand Company, Inc., 1961, 1959.
I. V. Blagouchine and E. Moreau: "Unbiased Adaptive Estimations of the Fourth-Order Cumulant for Real Random Zero-Mean Signal", IEEE Transactions on Signal Processing , vol. 57, no. 9, pp. 3330–3346, September 2009. An Illuminating Counterexample 
![{\displaystyle \mathbb {E} \left[{\hat {\theta }}\right]=\theta ,\quad \forall \theta \in \Theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdc344c9ff9be373c1b7769945026f1887e7652)
![{\displaystyle \mathbb {E} \left[X\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da88481bc3637feba360a17031ef10deed2c4f2)
![{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}{\bigg ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50da850b727d47c9fe8b12ec37bea937a78ba124)
![{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\big (}(X_{i}-\mu )-({\overline {X}}-\mu ){\big )}^{2}{\bigg ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b153364de0d94302ae870f48cc2b34b550de330)
![{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu ){\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {n}{n}}({\overline {X}}-\mu )^{2}{\bigg ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/882aba5f44a55098075770ebb439aa3e322a8dbf)
![{\displaystyle \operatorname {E} [S_{n}^{2}]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-({\overline {X}}-\mu )^{2}{\bigg ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/514273e8c6fbe946eaf54fd37b4e372b453bdc02)
![{\displaystyle \operatorname {E} [a+b]=\operatorname {E} [a]+\operatorname {E} [b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd848b476e4ab95f1a3d70dd609a18db78651c1)
![{\displaystyle \operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}=\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e365745787a66276a60c63bb86184f817e2c31af)
![{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}={\frac {1}{n}}\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d386a26eb18f6db56ca2f12646c2484c0e35cea2)
![{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}=\operatorname {E} {\big [}{\big (}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu ){\big )}^{2}{\big ]}=\operatorname {E} {\big [}{\frac {1}{n^{2}}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}+{\frac {2}{n^{2}}}\sum _{i=1,j=1,i<j}^{n}(X_{i}-\mu )(X_{j}-\mu ){\big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f36ad26f1dec804bc60178e0d70d510c18de0ab)
![{\displaystyle \operatorname {E} {\big [}(X_{i}-\mu ){\big ]}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d190ba6fc6a755677fbdb0e1252523ff4382f3)
![{\displaystyle \operatorname {E} {\big [}\sum _{i=1,j=1,i<j}^{n}(X_{i}-\mu )(X_{j}-\mu ){\big ]}=\sum _{i=1,j=1,i<j}^{n}\operatorname {E} {\big [}(X_{i}-\mu ){\big ]}\operatorname {E} {\big [}(X_{j}-\mu ){\big ]}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f3ff1318c86888a29f6f8483e803a251a35186)
![{\displaystyle {\begin{aligned}\operatorname {E} [S_{n}^{2}]&=\sigma ^{2}-\operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}=\\&=\sigma ^{2}-{\frac {1}{n}}\sigma ^{2}=\\&={\frac {n-1}{n}}\sigma ^{2}<\sigma ^{2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0baf68f30f0b093fbb5a695eabf281da2b8513)