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
a (non-randomized) range estimator if for each the amount
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 .
example
Let the measurement room be given and the product model as a statistical model
- ,
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
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.
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
generated σ-algebra . Then the function is a measurable function and thus a non-randomized decision function .
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 :
- For each is a probability measure on .
- For each there is a measurable function.
is then the likelihood of opting for a crowd if occurs .
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 .