Area estimator

A range estimator is a specific estimator in mathematical statistics . In contrast to a point estimator , range estimators are set-valued images , i.e. they assign a set to each outcome of a statistical experiment and not a single value. These sets are mostly ellipses, spheres or intervals. In the latter case, one speaks of an interval estimator .

Range estimates form the mathematical basis for determining confidence ranges . These are the quantities for which a given coverage probability is guaranteed.

As with decision functions, a distinction is made between randomized and non- randomized range estimators.

Non-randomized area estimators

A measurement room and a statistical model are given . Then a figure is called ${\displaystyle (M,{\mathcal {A}}_{M})}$ ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

${\displaystyle C\colon X\to {\mathcal {A}}_{M}}$

a (non-randomized) range estimator if for each the amount ${\displaystyle m\in M}$

${\displaystyle A(m)=\{x\in X\,|\,m\in C(x)\}}$

is contained in σ-algebra . is called the acceptance range of and contains all elements of the basic set which, when they occur, cover the value . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A(m)}$ ${\displaystyle m}$ ${\displaystyle m}$

example

Let the measurement room be given and the product model as a statistical model ${\displaystyle ([0,1],{\mathcal {B}}([0,1]))}$

${\displaystyle (\{0,1\}^{100},{\mathcal {P}}(\{0,1\})^{\otimes 100},\operatorname {Ber} _{\vartheta }^{\otimes 100})}$,

that models the 100-fold coin toss. denotes the Bernoulli distribution . A typical interval estimator would then be a mapping that assigns an interval around the arithmetic mean to each outcome of the experiment. If this is called and is , then the function would be ${\displaystyle \operatorname {Ber} _{\vartheta }}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle \varepsilon >0}$

${\displaystyle C(x)=[{\bar {x}}-\varepsilon ,{\bar {x}}+\varepsilon ]}$

an area estimator.

Strictly speaking, one would still have to cut the interval in order to guarantee, even for larger ones, that it is always a subset of the basic set of the measuring area. ${\displaystyle [0,1]}$ ${\displaystyle \varepsilon }$

Classification as decision-making functions

Range estimators can be represented as set-valued decision functions in the general framework of a statistical decision problem. To do this, one chooses the σ-algebra as the basic set of the decision space . The elements of the basic set of the decision space are then sets. The σ-algebra on the basic set of the decision space is defined via that of the auxiliary sets ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}_{M}}$

${\displaystyle T_{m}:=\{K\in {\mathcal {A}}_{M}\,|\,m\in K\}}$

generated σ-algebra . Then the function is a measurable function and thus a non-randomized decision function . ${\displaystyle \Sigma :=\sigma (T_{m};m\in M)}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {A}}-\Sigma }$

Randomized range estimators

Using this construction, a randomized range estimator can then also be defined: This is a Markov kernel from to , that is, for : ${\displaystyle \delta }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \delta \colon X\times \Sigma \to [0,1]}$

• For each is a probability measure on . ${\displaystyle x\in X}$ ${\displaystyle \delta (x,\;\cdot \;)}$ ${\displaystyle (\Omega ,\Sigma )}$
• For each there is a measurable function. ${\displaystyle S\in \Sigma }$ ${\displaystyle \delta (\;\cdot \;,S)}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \delta (x,K)}$is then the likelihood of opting for a crowd if occurs . ${\displaystyle x}$ ${\displaystyle K\in {\mathcal {A}}_{M}=\Omega }$

construction

Common methods for constructing range estimates include: a. Pivot statistics and approximate pivot statistics .

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .