Approximate pivot statistics

An approximate pivot statistic is a sequence of functions in mathematical statistics that is used to construct approximate confidence ranges . It thus forms the asymptotic counterpart to pivot statistics , which are used to construct (non-approximate) confidence ranges .

definition

Framework

For were measuring rooms and families of probability measures on . Be another measuring room as well ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (X_{n},{\mathcal {A}}_{n})}$ ${\displaystyle (P_{\vartheta ,n})_{\vartheta \in \Theta }}$ ${\displaystyle (X_{n},{\mathcal {A}}_{n})}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$

${\displaystyle g\colon \Theta \to \Gamma }$

the function to be estimated.

In most cases, the measurement rooms and families of probability measures are -fold product models . Would be typical example of this , and as probability a corresponding product measure of a probability measure on . ${\displaystyle n}$ ${\displaystyle X_{n}=\mathbb {R} ^{n}}$ ${\displaystyle P_{\vartheta }^{n}}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle \mathbb {R} }$

Formalization

A series of statistics with ${\displaystyle (T_{n})_{n\in \mathbb {N} }}$

${\displaystyle T_{n}\colon X_{n}\times \Gamma \to \Gamma }$

is called an approximate pivot statistic for if: ${\displaystyle g}$

• There is a probability distribution on , so that the distribution of all to converge . So it is ${\displaystyle Q}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$ ${\displaystyle T_{n}(\cdot ,g(\vartheta ))}$ ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle Q}$

${\displaystyle P_{\vartheta }\circ T_{n}(\cdot ,g(\vartheta ))^{-1}{\stackrel {\mathcal {D}}{\rightarrow }}Q}$for and for everyone . ${\displaystyle n\to \infty }$ ${\displaystyle \vartheta \in \Theta }$

• For all levels is in included. ${\displaystyle B\in A_{\Gamma }}$ ${\displaystyle \{x\in X_{n}\mid T_{n}(x,g(\vartheta ))\in B\}}$ ${\displaystyle {\mathcal {A}}^{n}}$

The second condition guarantees that all sets can be assigned probabilities in meaningful ways by means of the probability measures , that is, the distribution of is well-defined for all . ${\displaystyle {\mathcal {A}}_{\Gamma }}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle T_{n}(\cdot ,g(\vartheta ))}$ ${\displaystyle \vartheta }$

example

Consider a Bernoulli product model, so

${\displaystyle X=\{0,1\}{\text{ und }}{\mathcal {A}}={\mathcal {P}}\{0,1\}}$

provided with the Bernoulli distribution for the parameter . ${\displaystyle \vartheta \in (0,1)}$

The -fold product model is then . The parameter of the Bernoulli distribution is to be estimated, so the function to be estimated is ${\displaystyle n}$ ${\displaystyle (X^{n},{\mathcal {A}}^{n},(\operatorname {Ber} _{\vartheta }^{n})_{\vartheta \in (0,1)})}$

${\displaystyle g(\vartheta )=\vartheta }$.

Let be the sample variable . They are independently distributed identically and it is ${\displaystyle X=(X_{1},X_{2},\dots ,X_{n})}$ ${\displaystyle X_{i}}$

${\displaystyle T_{n}(X,\vartheta )={\sqrt {n}}\left({\frac {\left({\tfrac {1}{n}}\sum _{i=1}^{n}X_{i}\right)-\vartheta }{\sqrt {\vartheta (1-\vartheta )}}}\right)}$

an approximate pivot statistic, since it converges to the standard normal distribution according to the Moivre-Laplace theorem . So it is . ${\displaystyle Q={\mathcal {N}}(0,1)}$

swell

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 233-236 , 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 , p. 144-145 , doi : 10.1007 / 978-3-642-17261-8 .