Pivot statistics

A pivot statistic , also known as a pivot size , or a pivot for short , is a special function in mathematical statistics . These are statistics with certain invariance properties that are used to construct confidence ranges . The name is derived from the French pivot (German anchor, here in the sense of pivot and pivot point) and is based on the invariance properties.

definition

A statistical model is given ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

A pivot is a random variable as a function of the sample variable and , the distribution of which is independent of . ${\displaystyle T(X,\vartheta )}$ ${\displaystyle X}$ ${\displaystyle \vartheta }$ ${\displaystyle \vartheta }$

A pivot statistic is strictly formally defined as follows: A decision space and a function to be estimated are given ${\displaystyle (\Gamma ,{\mathcal {\mathcal {A}}}_{\Gamma })}$

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

Mostly is . Then it is called a measurable figure ${\displaystyle \Gamma \subset \mathbb {R} }$

${\displaystyle T\colon {\mathcal {X}}\times \Gamma \to \Gamma }$

a pivot statistic for if it fulfills the following properties: ${\displaystyle g}$

• For all and all the amount is in included. ${\displaystyle B\in {\mathcal {A}}_{\Gamma }}$ ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle \{x\in {\mathcal {X}}\mid T(x,g(\vartheta ))\in B\}}$ ${\displaystyle {\mathcal {A}}}$
• There is a probability distribution on , so for all always applies. ${\displaystyle Q}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$ ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle P_{\vartheta }\circ T(\cdot ,g(\vartheta ))^{-1}=Q}$

example

A fixed one is given and the normal distribution with expected value and variance is assumed . Let be n times the product measure . ${\displaystyle \sigma _{0}>0}$ ${\displaystyle {\mathcal {N}}(\mu ,\sigma _{0}^{2})}$ ${\displaystyle \mu }$ ${\displaystyle \sigma _{0}^{2}}$ ${\displaystyle {\mathcal {N}}^{n}(\mu ,\sigma _{0}^{2})}$

The product model with a fixed variance and an unknown expected value is considered as a statistical model . ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),({\mathcal {N}}^{n}(\mu ,\sigma _{0}^{2}))_{\mu \in \mathbb {R} })}$

Then a pivot statistic is given by

${\displaystyle T(X,\mu )={\frac {{\sqrt {n}}({\overline {X}}-\mu )}{\sigma _{0}}}}$.

Here is

${\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$

the sample mean . The fact that it is a pivot follows directly from the calculation rules for normally distributed random variables (see invariance of the normal distribution versus convolution ), because it is . By normalizing with the standard deviation , one obtains that there is always standard normal distribution , i.e. for all . ${\displaystyle {\overline {X}}-\mu \sim {\mathcal {N}}(0,\sigma _{0}^{2}/n)}$ ${\displaystyle \sigma _{0}/{\sqrt {n}}}$ ${\displaystyle T}$ ${\displaystyle T(\cdot ,\mu )\sim {\mathcal {N}}(0,1)}$ ${\displaystyle \mu \in \mathbb {R} }$

Construction of confidence areas using pivots

If a pivot statistic exists and a set is given, then through ${\displaystyle T}$ ${\displaystyle B\in {\mathcal {A}}_{\Gamma }}$

${\displaystyle C_{B}(x)=\{\gamma \in \Gamma \mid T(x,\gamma )\in B\}}$

an area estimator is defined. Based on the definition of the range estimator, then

${\displaystyle \{x\in {\mathcal {X}}\mid g(\vartheta )\in C_{B}(x)\}=\{x\in {\mathcal {X}}\mid T(x,g(\vartheta ))\in B\}}$

and thus

${\displaystyle P_{\vartheta }(\{x\in {\mathcal {X}}\mid g(\vartheta )\in C_{B}(x)\})=P_{\vartheta }(\{x\in {\mathcal {X}}\mid T(x,g(\vartheta ))\in B\})=Q(B)}$

for everyone due to the pivot property of . The range estimator thus provides a confidence range for the confidence level . The choice of the set thus determines the confidence level and geometry of the confidence range. ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle T}$ ${\displaystyle C_{B}}$ ${\displaystyle Q(B)}$ ${\displaystyle B}$

Example for the construction of confidence ranges

Under the same conditions as in the example above, a confidence range is to be determined for the mean value for the confidence level . There is a set must first be chosen so that ${\displaystyle 1-\alpha }$ ${\displaystyle Q={\mathcal {N}}(0,1)}$ ${\displaystyle B}$

${\displaystyle Q(B)={\mathcal {N}}(0,1)(B)=1-\alpha }$.

The choice of depends largely on the application. One-sided confidence intervals are common ${\displaystyle B}$

${\displaystyle B_{1}=[a_{1},+\infty )}$ or ${\displaystyle B_{2}=(-\infty ,a_{2}]}$

or two-sided confidence intervals

${\displaystyle B_{3}=[-a_{3},a_{3}]}$.

You must now choose that for is. For this one selects the appropriate - quantiles of the standard normal distribution and obtains as well as and . ${\displaystyle a_{1},a_{2},a_{3}}$ ${\displaystyle {\mathcal {N}}(0,1)(B_{i})=1-\alpha }$ ${\displaystyle i=1,2,3}$ ${\displaystyle p}$ ${\displaystyle u_{p}}$ ${\displaystyle a_{1}=u_{\alpha }}$ ${\displaystyle a_{2}=u_{1-\alpha }}$ ${\displaystyle a_{3}=u_{1-\alpha /2}}$

This results for the range estimator with the quantity ${\displaystyle B_{1}}$

${\displaystyle {\begin{aligned}C_{B_{1}}(X)&=\{\mu \in \mathbb {R} \mid T(X,\mu )\in [u_{\alpha },+\infty )\}\\&=\left\{\mu \in \mathbb {R} \;\mid \;{\frac {{\sqrt {n}}({\overline {X}}-\mu )}{\sigma _{0}}}\geq u_{\alpha }\right\}\\&=\left\{\mu \in \mathbb {R} \mid \mu \geq {\overline {X}}-{\frac {\sigma _{0}u_{1-\alpha }}{\sqrt {n}}}\right\}\end{aligned}}}$,

because due to the symmetry of the standard normal distribution applies. ${\displaystyle u_{1-\alpha }=-u_{\alpha }}$

The one-sided confidence interval for the confidence level for the expected value is thus obtained ${\displaystyle 1-\alpha }$

${\displaystyle C_{B_{1}}(X)=\left[{\overline {X}}-{\frac {\sigma _{0}u_{1-\alpha }}{\sqrt {n}}},+\infty \right)}$.

By proceeding in the same way with the quantities and , the second one-sided confidence interval is obtained ${\displaystyle B_{2}}$ ${\displaystyle B_{3}}$

${\displaystyle C_{B_{2}}(X)=\left(-\infty ,{\overline {X}}+{\frac {\sigma _{0}u_{1-\alpha }}{\sqrt {n}}}\right]}$

and as a two-sided confidence interval

${\displaystyle C_{B_{3}}(X)=\left[{\overline {X}}-{\frac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2};{\overline {X}}+{\frac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2}\right]}$.

Related concepts

The approximate pivot statistics are closely related to the pivot statistics . They serve to construct approximate confidence ranges and are based on limit value considerations.

Individual evidence

1. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 234 , doi : 10.1515 / 9783110215274 .
2. ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 142 , doi : 10.1007 / 978-3-642-17261-8 .
3. ^ ^{A b} Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 231 , doi : 10.1007 / 978-3-642-41997-3 .