Point estimator

In mathematical statistics, a point estimator is an estimation function that assigns a value to each sample that is intended to estimate a certain property of the underlying probability measure . In most applications, the quantity of interest is a parameter of the probability distribution of the observations (such as the mean of a normal distribution ) ${\displaystyle \mu }$ ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$

In addition to area estimators, point estimators are the central object of investigation in estimation theory and, in a more general sense, a decision function that assigns the observations present to an estimated value of the variable of interest.

A point estimate is a function of the random observations, a point estimate is the calculated value of the point estimate for existing observations.

definition

A statistical model and a decision space are given . So for each is also contained in the σ-algebra . Then it is called a measurable function ${\displaystyle (X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle e\in E}$ ${\displaystyle \{e\}}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle T\colon (X,{\mathcal {A}})\to (E,{\mathcal {E}})}$

a point estimator. So it is always for everyone . ${\displaystyle M\in {\mathcal {E}}}$ ${\displaystyle T^{-1}(M)\in {\mathcal {A}}}$

Most of the time, the decision room is chosen. ${\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}})}$

example

A binomial model is given, i.e. a statistical model with and and as a set of probability measures the binomial distributions for . ${\displaystyle X=\{0,1,\dots ,n\}}$ ${\displaystyle {\mathcal {A}}={\mathcal {P}}(X)}$ ${\displaystyle \operatorname {Bin} _{n,\vartheta }}$ ${\displaystyle \vartheta \in [0,1]}$

This model formalizes, for example, how often "heads" were tossed after n-times tossing a coin. The obvious question now is to estimate the probability with which the coin shows “heads” on the basis of the available data. The appropriate decision space is the basic set , since the probability must lie in this interval, provided with Borel's σ-algebra , which contains all point sets. ${\displaystyle [0,1]}$ ${\displaystyle {\mathcal {B}}([0,1])}$

A possible point estimator would then be, for example

${\displaystyle M:(X,{\mathcal {A}})\to ([0,1],{\mathcal {B}}([0,1]))}$

defined by

${\displaystyle M(x)={\frac {x}{n}}}$.

How good and useful such point estimators are still has to be examined separately. Because it would be the same

${\displaystyle M^{*}(x)={\frac {1}{2}}}$

a possible point estimator. However, regardless of the number of tosses that show "heads", it delivers that the coin is fair, which is apparently nonsensical.

Use and construction

Point estimators are used in particular to estimate:

• Expected values
• Variances
• Standard deviations

Classic and proven methods for constructing point estimators are

• the maximum likelihood method ,
• the least squares method or
• the moment method .

→ Main article : Estimation method (statistics)

Quality criteria for point estimators

There are various quality criteria for point estimators. The four most common are sufficiency, efficiency, expectation and consistency.

• Sufficiency guarantees that the point estimator uses all of the data information relevant to the estimation.
• Faithful to expectation (undistorted, unadulterated): A point estimator is faithful to expectations if it correctly indicates the actual value of the variable of interest on average. In this sense, the estimate has no systematic error.
• Consistency : Consistency clearly means that for a growing number of observations, the point estimate tends to approximate the actual value of the quantity of interest.
• Efficiency : A point estimator is efficient if its spread is minimal compared to other point estimators. In this sense, an efficient estimator has no unnecessary dispersion.

Web links

• MS Nikulin: Point estimator . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

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