Difference between revisions of "Statistics"
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$$ | $$ | ||
{\mathsf P} ^ {T} \{ B \} = {\mathsf P} \{ T( X) \in B \} = \ | {\mathsf P} ^ {T} \{ B \} = {\mathsf P} \{ T( X) \in B \} = \ | ||
− | {\mathsf P} \{ X \in T ^ {-} | + | {\mathsf P} \{ X \in T ^ {- 1 }( B) \} = |
$$ | $$ | ||
$$ | $$ | ||
= \ | = \ | ||
− | {\mathsf P} ^ {X} \{ T ^ {-} | + | {\mathsf P} ^ {X} \{ T ^ {- 1} ( B) \} \ \ |
(\forall B \in {\mathcal A}). | (\forall B \in {\mathcal A}). | ||
$$ | $$ | ||
Line 41: | Line 41: | ||
\overline{X}\; = | \overline{X}\; = | ||
\frac{1}{n} | \frac{1}{n} | ||
− | \sum _ { i= } | + | \sum _ {i=1} ^ { n } X _ {i} \ \textrm{ and } \ \ |
s ^ {2} = | s ^ {2} = | ||
\frac{1}{(} | \frac{1}{(} | ||
− | n- 1) \sum _ { i= } | + | n- 1) \sum _{i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} |
$$ | $$ | ||
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I _ {n} ( \lambda ) = | I _ {n} ( \lambda ) = | ||
\frac{1}{2 \pi n } | \frac{1}{2 \pi n } | ||
− | \left | \sum _ { k= } | + | \left | \sum _{k=1} ^ { n } X _ {k} e ^ |
{- ik \lambda } \right | ^ {2} ,\ \ | {- ik \lambda } \right | ^ {2} ,\ \ | ||
\lambda \in [- \pi , \pi ], | \lambda \in [- \pi , \pi ], | ||
Line 85: | Line 85: | ||
====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) |
Statistics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistics&oldid=48822