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A term used in [[Mathematical statistics|mathematical statistics]] as a name for functions of the results of observations.
 
A term used in [[Mathematical statistics|mathematical statistics]] as a name for functions of the results of observations.
  
Let a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874901.png" /> take values in the sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874902.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874903.png" />-measurable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874904.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874905.png" /> onto a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874906.png" /> is then called a statistic, and the probability distribution of the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874907.png" /> is defined by the formula
+
Let a random variable $  X $
 +
take values in the sample space $  ( \mathfrak X, {\mathcal B}, {\mathsf P}  ^ {X} ) $.  
 +
Any $  {\mathcal B} $-
 +
measurable mapping $  T( \cdot ) $
 +
from $  \mathfrak X $
 +
onto a measurable space $  ( \mathfrak Y, {\mathcal A} ) $
 +
is then called a statistic, and the probability distribution of the statistic $  T $
 +
is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874908.png" /></td> </tr></table>
+
$$
 +
{\mathsf P}  ^ {T} \{ B \}  = {\mathsf P} \{ T( X) \in B \}  = \
 +
{\mathsf P} \{ X \in T  ^ {- 1 }( B) \} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s0874909.png" /></td> </tr></table>
+
$$
 +
= \
 +
{\mathsf P}  ^ {X} \{ T  ^ {- 1} ( B) \} \ \
 +
(\forall B \in {\mathcal A}).
 +
$$
  
 
===Examples.===
 
===Examples.===
  
 +
1) Let  $  X _ {1} \dots X _ {n} $
 +
be independent identically-distributed random variables which have a variance. The statistics
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749010.png" /> be independent identically-distributed random variables which have a variance. The statistics
+
$$
 
+
\overline{X}\;  = 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749011.png" /></td> </tr></table>
+
\frac{1}{n}
 +
\sum _ {i=1} ^ { n }  X _ {i} \  \textrm{ and } \ \
 +
s  ^ {2}  =
 +
\frac{1}{(}
 +
n- 1) \sum _{i=1} ^ { n }  ( X _ {i} - \overline{X}\; )  ^ {2}
 +
$$
  
are then unbiased estimators for the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749012.png" /> and the variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749013.png" />, respectively.
+
are then unbiased estimators for the mathematical expectation $  {\mathsf E} X _ {1} $
 +
and the variance $  {\mathsf D} X _ {1} $,  
 +
respectively.
  
 
2) The terms of the [[Variational series|variational series]] (series of order statistics, cf. [[Order statistic|Order statistic]])
 
2) The terms of the [[Variational series|variational series]] (series of order statistics, cf. [[Order statistic|Order statistic]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749014.png" /></td> </tr></table>
+
$$
 +
X _ {(} 1)  \leq  \dots \leq  X _ {(} n) ,
 +
$$
  
constructed from the observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749015.png" />, are statistics.
+
constructed from the observations $  X _ {1} \dots X _ {n} $,  
 +
are statistics.
  
3) Let the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749016.png" /> form a [[Stationary stochastic process|stationary stochastic process]] with [[Spectral density|spectral density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749017.png" />. In this case the statistic
+
3) Let the random variables $  X _ {1} \dots X _ {n} $
 +
form a [[Stationary stochastic process|stationary stochastic process]] with [[Spectral density|spectral density]] $  f( \cdot ) $.  
 +
In this case the statistic
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749018.png" /></td> </tr></table>
+
$$
 +
I _ {n} ( \lambda )  =
 +
\frac{1}{2 \pi n }
 +
\left | \sum _{k=1} ^ { n }  X _ {k} e ^
 +
{- ik \lambda } \right |  ^ {2} ,\ \
 +
\lambda \in [- \pi , \pi ],
 +
$$
  
called the [[Periodogram|periodogram]], is an asymptotically-unbiased estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749019.png" />, given certain specific conditions of regularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749020.png" />, i.e.
+
called the [[Periodogram|periodogram]], is an asymptotically-unbiased estimator for $  f( \cdot ) $,  
 +
given certain specific conditions of regularity on $  f( \cdot ) $,  
 +
i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087490/s08749021.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  {\mathsf E} I _ {n} ( \lambda )  = \
 +
f( \lambda ),\ \
 +
\lambda \in [- \pi , \pi ].
 +
$$
  
 
In the theory of estimation and statistical hypotheses testing, great importance is attached to the concept of a [[Sufficient statistic|sufficient statistic]], which brings about a reduction of data without any loss of information on the (parametric) family of distributions under consideration.
 
In the theory of estimation and statistical hypotheses testing, great importance is attached to the concept of a [[Sufficient statistic|sufficient statistic]], which brings about a reduction of data without any loss of information on the (parametric) family of distributions under consideration.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Voinov,  M.S. Nikulin,  "Unbiased estimates and their applications" , Moscow  (1989)  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Voinov,  M.S. Nikulin,  "Unbiased estimates and their applications" , Moscow  (1989)  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 16:24, 6 January 2024


A term used in mathematical statistics as a name for functions of the results of observations.

Let a random variable $ X $ take values in the sample space $ ( \mathfrak X, {\mathcal B}, {\mathsf P} ^ {X} ) $. Any $ {\mathcal B} $- measurable mapping $ T( \cdot ) $ from $ \mathfrak X $ onto a measurable space $ ( \mathfrak Y, {\mathcal A} ) $ is then called a statistic, and the probability distribution of the statistic $ T $ is defined by the formula

$$ {\mathsf P} ^ {T} \{ B \} = {\mathsf P} \{ T( X) \in B \} = \ {\mathsf P} \{ X \in T ^ {- 1 }( B) \} = $$

$$ = \ {\mathsf P} ^ {X} \{ T ^ {- 1} ( B) \} \ \ (\forall B \in {\mathcal A}). $$

Examples.

1) Let $ X _ {1} \dots X _ {n} $ be independent identically-distributed random variables which have a variance. The statistics

$$ \overline{X}\; = \frac{1}{n} \sum _ {i=1} ^ { n } X _ {i} \ \textrm{ and } \ \ s ^ {2} = \frac{1}{(} n- 1) \sum _{i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} $$

are then unbiased estimators for the mathematical expectation $ {\mathsf E} X _ {1} $ and the variance $ {\mathsf D} X _ {1} $, respectively.

2) The terms of the variational series (series of order statistics, cf. Order statistic)

$$ X _ {(} 1) \leq \dots \leq X _ {(} n) , $$

constructed from the observations $ X _ {1} \dots X _ {n} $, are statistics.

3) Let the random variables $ X _ {1} \dots X _ {n} $ form a stationary stochastic process with spectral density $ f( \cdot ) $. In this case the statistic

$$ I _ {n} ( \lambda ) = \frac{1}{2 \pi n } \left | \sum _{k=1} ^ { n } X _ {k} e ^ {- ik \lambda } \right | ^ {2} ,\ \ \lambda \in [- \pi , \pi ], $$

called the periodogram, is an asymptotically-unbiased estimator for $ f( \cdot ) $, given certain specific conditions of regularity on $ f( \cdot ) $, i.e.

$$ \lim\limits _ {n \rightarrow \infty } {\mathsf E} I _ {n} ( \lambda ) = \ f( \lambda ),\ \ \lambda \in [- \pi , \pi ]. $$

In the theory of estimation and statistical hypotheses testing, great importance is attached to the concept of a sufficient statistic, which brings about a reduction of data without any loss of information on the (parametric) family of distributions under consideration.

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1988)
[2] V.G. Voinov, M.S. Nikulin, "Unbiased estimates and their applications" , Moscow (1989) (In Russian)
How to Cite This Entry:
Statistics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistics&oldid=14854
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article