Substitution principle (statistics)

The substitution principle is a method of estimation theory , a branch of mathematical statistics , to obtain estimation functions . An important special case of the substitution principle is the moment method . An estimator obtained through the substitution principle is called a plug-in estimator or substitution estimator .

formulation

A set of probability distributions on the real numbers is given. Let the random variables be independently identically distributed according to one and be . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle X_{1},X_{2},\dots ,X_{N}}$ ${\displaystyle P_{0}\in {\mathcal {P}}}$ ${\displaystyle X=(X_{1},X_{2},\dots ,X_{N})}$

A functional should be appreciated

${\displaystyle q\colon {\mathcal {P}}\to \mathbb {R} }$

from the shape

${\displaystyle q(P)=\int _{\mathbb {R} }f(x)\mathrm {d} P}$.

Then

${\displaystyle {\hat {q}}(X)={\frac {1}{N}}\sum _{i=1}^{N}f(X_{i})}$

a possible estimator for ${\displaystyle q}$

Example: moment method

An example of the substitution principle is the moment method . If the th moment is to be estimated, the functional to be estimated is of the form ${\displaystyle n}$

${\displaystyle q(P)=\int _{\mathbb {R} }x^{n}\mathrm {d} P}$,

so it is . The substitution principle thus provides the estimator ${\displaystyle f(x)=x^{n}}$

${\displaystyle {\hat {q}}(X)={\frac {1}{N}}\sum _{i=1}^{N}X_{i}^{n}}$.

The moment method therefore delivers the same procedure for a function to be estimated . ${\displaystyle f=f(x,x^{2},\dots ,x^{n})}$

General version

The above version can be more generalized, which also makes the naming clearer. Again a set of probability distributions as well as independently identically distributed random variables according to one and be is given . A functional one is to be appreciated ${\displaystyle {\mathcal {P}}}$ ${\displaystyle X_{1},X_{2},\dots ,X_{N}}$ ${\displaystyle P_{0}\in {\mathcal {P}}}$ ${\displaystyle X=(X_{1},X_{2},\dots ,X_{N})}$

${\displaystyle q\colon {\mathcal {P}}\to \mathbb {R} }$.

Instead of estimating the functional directly, an estimator is used first

${\displaystyle {\hat {P}}_{N}=p_{N}(X_{1},X_{2},\dots ,X_{N})}$

for used. Here is an appropriately chosen measurable function . The probability measure is now substituted by the corresponding estimate using and the function obtained in this way is used as an estimate function. In the special case above, for example, the probability distribution on the real numbers is substituted by the empirical distribution . ${\displaystyle P}$ ${\displaystyle p_{N}}$ ${\displaystyle P}$ ${\displaystyle {\hat {P}}_{N}}$ ${\displaystyle q({\hat {P}}_{N})}$

swell

• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 72-77 , doi : 10.1007 / 978-3-642-17261-8 .
• Hein Putter, Willem R. van Zweet: Resampling: consistency of substitution estimators. In: Project euclid. 2002, accessed February 28, 2017 .