Estimation problem

An estimation problem is a term from mathematical statistics that understands estimation theory as a statistical decision problem and summarizes all information relevant for the estimation. This includes which values the data assume, which probability measures are taken into account, which properties of the probability measures considered are to be estimated, and how great the damage is caused by an estimation error.

definition

An estimation problem is a quadruple

{\ displaystyle ({\ mathcal {M}}, E, g, L)}

consisting of

A statistical model ${\ displaystyle {\ mathcal {M}} = ({\ mathcal {X}}, {\ mathcal {A}}, (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta})}$
A decision space . ${\ displaystyle E = (\ Delta, {\ mathcal {A}} _ {\ Delta})}$
A function to be appreciated

{\ displaystyle g \ colon \ Theta \ to \ Delta}

A loss function

{\ displaystyle L \ colon \ Theta \ times \ Delta \ to \ mathbb {R}}

Explanations

The estimation problem summarizes all relevant information about the estimation:

The statistical model provides information about which values the data assume (values in ). It also contains all the quantities to which probabilities are to be assigned. The system of quantities is chosen canonically . The family contains all probability measures that are considered relevant in the given situation. When examining a die roll, different probability measures come into question than for examining shoe sizes. ${\ displaystyle {\ mathcal {X}}}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta}}$

The decision room is a special measuring room and contains all the information about what you can decide for. If one wants to estimate the parameter of a Bernoulli distribution , then every estimate is a decision. In this case, all numbers between zero and one can be used for decisions. The situation is different when estimating the expected value of a normal distribution: Here every real number comes into question as an estimate and thus as a decision. The decision-making space is thus greater here. ${\ displaystyle E}$

The function , which in the parametric case is also called a parameter function, assigns a decision to each , and often, as in the following examples, there is even a functional dependence on . It indicates what is to be estimated so that it can be examined afterwards how far the estimate deviates. A typical example is the function that assigns the expected value to each . Alternatively, it could assign the appropriate median to each . ${\ displaystyle g}$ ${\ displaystyle \ vartheta \ in \ Theta}$ ${\ displaystyle P _ {\ vartheta}}$ ${\ displaystyle \ vartheta}$ ${\ displaystyle \ operatorname {E} (P _ {\ vartheta})}$ ${\ displaystyle \ vartheta}$

The loss function assigns a decision from the decision space to a real number as a function of, which indicates how great the damage is caused by the decision for , if present. It is then expanded to a risk function with which various estimators and decision rules can be compared. ${\ displaystyle e}$ ${\ displaystyle \ vartheta}$ ${\ displaystyle e}$ ${\ displaystyle \ vartheta}$

Example of expected value estimation

A typical case of an estimation problem is the estimation of the expected value of a probability distribution in independently repeated attempts. In doing so, one usually looks at the set , provided with the σ-algebra . If you don't have any further information, you first define the family of all probability measures with a finite expected value and then consider their n-fold product measures . (The notation as an indexed family seems unnatural here and has been retained for reasons of a uniform representation. In this case, one would choose and set the set of all probability measures considered .) ${\ displaystyle n}$ ${\ displaystyle {\ mathcal {X}} = \ mathbb {R} ^ {n}}$ ${\ displaystyle {\ mathcal {A}} = {\ mathcal {B}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta}}$ ${\ displaystyle \ Theta}$ ${\ displaystyle P _ {\ vartheta} = \ vartheta}$

The statistical model is thus given by

{\ displaystyle {\ mathcal {M}} = (\ mathbb {R} ^ {n}, {\ mathcal {B}} (\ mathbb {R} ^ {n}), (P _ {\ vartheta} ^ {\ otimes n}) _ {\ vartheta \ in \ Theta})}

.

The decision space is as follows

{\ displaystyle E = (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}

,

because the decision corresponds to the estimate of the expected value and this is a real number.

The function to be estimated is then

{\ displaystyle g (\ vartheta) = \ operatorname {E} (P _ {\ vartheta})}

.

It assigns each the expected value of the associated probability measure . ${\ displaystyle \ vartheta}$ ${\ displaystyle P _ {\ vartheta}}$

A possible loss function would be the Gaussian loss caused by

{\ displaystyle L (e, \ vartheta) = (eg (\ vartheta)) ^ {2}}

given is.

literature

Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 20 , doi : 10.1007 / 978-3-642-41997-3 .
Friedrich Liese, Klaus-J. Miescke: Statistical Decision Theory . Estimation, Testing, and Selection. Springer-Verlag, New York 2008, ISBN 978-0-387-73193-3 , pp. 107 , doi : 10.1007 / 978-0-387-73194-0 .