Difference between revisions of "Statistics"
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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 | + | 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.=== | ===Examples.=== | ||
| + | 1) Let $ X _ {1} \dots X _ {n} $ | ||
| + | be independent identically-distributed random variables which have a variance. The statistics | ||
| − | 1 | + | $$ |
| − | + | \overline{X}\; = | |
| − | + | \frac{1}{n} | |
| + | \sum _ { i= } 1 ^ { n } X _ {i} \ \textrm{ and } \ \ | ||
| + | s ^ {2} = | ||
| + | \frac{1}{(} | ||
| + | n- 1) \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} | ||
| + | $$ | ||
| − | are then unbiased estimators for the mathematical expectation | + | 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]]) | ||
| − | + | $$ | |
| + | X _ {(} 1) \leq \dots \leq X _ {(} n) , | ||
| + | $$ | ||
| − | constructed from the observations | + | constructed from the observations $ X _ {1} \dots X _ {n} $, |
| + | are statistics. | ||
| − | 3) Let the random variables | + | 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 | ||
| − | + | $$ | |
| + | I _ {n} ( \lambda ) = | ||
| + | \frac{1}{2 \pi n } | ||
| + | \left | \sum _ { k= } 1 ^ { n } X _ {k} e ^ | ||
| + | {- ik \lambda } \right | ^ {2} ,\ \ | ||
| + | \lambda \in [- \pi , \pi ], | ||
| + | $$ | ||
| − | called the [[Periodogram|periodogram]], is an asymptotically-unbiased estimator for | + | called the [[Periodogram|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|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. | ||
Revision as of 08:23, 6 June 2020
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=14854