Statistics (function)

A statistic is a special mathematical function in the subfield of mathematics of the same name, statistics . Statistics do not differ structurally from estimators such as point estimators , but are distinguished from them because of the fundamentally different tasks for which they are used.

definition

A statistical model and a measuring room are given . Then it is called a measurable function ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (E,{\mathcal {E}})}$

${\displaystyle S:(X,{\mathcal {A}})\to (E,{\mathcal {E}})}$

a statistic. Measurability means that for all of the σ-algebra the archetypes are contained in the σ-algebra . ${\displaystyle M}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle S^{-1}(M)}$ ${\displaystyle {\mathcal {A}}}$

Comment on the definition

The measurability of a function is guaranteed, for example, if it is continuous from the one mapped in the and the corresponding Borel σ-algebras are selected as σ-algebras . Usually these σ-algebras are chosen by default. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle {\mathcal {B}}}$

The measurability is necessary in order to be able to define the distribution of the statistics analogous to the procedure for random variables . This means that you can also use expressions like

${\displaystyle P_{\vartheta }(S\in M){\text{ für }}M\in {\mathcal {E}}}$

want to investigate. This is then defined as the image dimension via

${\displaystyle P_{\vartheta }(S\in M)=P_{\vartheta }(S^{-1}(M))}$.

The measurability guarantees here that the right side is well defined.

Demarcation

Mathematically, the terms "measurable function", "random variable", "(point) estimator" and "statistics" agree. The central point of their common definition is that it enables the construction of distributions and image dimensions.

An important point when distinguishing between estimator and statistics is the use and interpretation of the function. Statistics organize and structure existing information (such as order statistics ) or are aids for the construction of procedures (such as test statistics ).

In contrast to this, the estimation functions evaluate existing data, try to guess a value as well as possible and are subject to certain quality criteria.

The distinction is sometimes difficult. Consider a statistical model that formalizes the tossing of a possibly asymmetrical coin n times, that is

${\displaystyle X=\{0,1\}^{n},\;{\mathcal {A}}={\mathcal {P}}(X)}$   and   . ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }=\{\operatorname {Ber} _{\vartheta }^{n}\,|\,\vartheta \in [0,1]\}}$

Ber stands for the Bernoulli distribution . Then the sample mean is

${\displaystyle M\colon X\to [0,1]}$   defined by   ${\displaystyle M(x)={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$

a point estimator for the parameter . The function ${\displaystyle \vartheta }$

${\displaystyle S\colon X\to \mathbb {R} }$   defined by   ${\displaystyle M(x)=\sum _{i=1}^{n}x_{i}}$

differs from the point estimator only by the prefactor , but can be used as a statistic that reduces the observation depth. It reduces the complete description of the experiment with the order of the throws to the information how often the desired side was thrown. ${\displaystyle {\tfrac {1}{n}}}$

literature

• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .
• Tatjana Lange , Karl Mosler : Statistics compact. Basic knowledge for economists and engineers. Springer textbook . Springer Gabler, Berlin, Heidelberg 2017, ISBN 978-3-662-53466-3 .