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}}} Литература и некоторые ссылки
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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)