Approximate confidence range

In mathematical statistics, a special class of confidence ranges is referred to as approximate confidence ranges . In contrast to conventional confidence ranges, they do not always maintain their confidence level , but only when considering an ever-increasing sample. Asymptotic properties of statistics such as asymptotic normality and the limit theorems of stochastics are used to construct approximate confidence ranges, which greatly expands the range of application.

If the range is an interval, one also speaks of an approximate confidence interval . The range estimators which provide approximate confidence ranges are correspondingly called approximate range estimators .

definition

Framework

For were measuring rooms and families of probability measures on . ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle (X_ {n}, {\ mathcal {A}} _ {n})}$ ${\ displaystyle (P _ {\ vartheta, n}) _ {\ vartheta \ in \ Theta}}$ ${\ displaystyle (X_ {n}, {\ mathcal {A}} _ {n})}$

In most cases, the measurement rooms and families of probability distributions are successively larger product models .

Be another measuring room as well ${\ displaystyle (\ Gamma, {\ mathcal {A}} _ {\ Gamma})}$

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

the function to be estimated and be a sequence of range estimators , where ${\ displaystyle (C_ {n}) _ {n \ in \ mathbb {N}}}$

{\ displaystyle C_ {n} \ colon X_ {n} \ to {\ mathcal {A}} _ {\ Gamma}}

.

formulation

Under the above framework conditions, the sequence of range estimators is called an approximate range estimator for the confidence level if ${\ displaystyle (C_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle g}$ ${\ displaystyle 1- \ alpha}$

{\ displaystyle \ liminf _ {n \ to \ infty} P _ {\ vartheta} ^ {n} (\ {x \ in X ^ {n} \ mid g (\ vartheta) \ in C_ {n} (x) \ }) \ geq 1- \ alpha}

for all

{\ displaystyle \ vartheta \ in \ Theta}

applies. Here the Limes inferior . ${\ displaystyle \ liminf}$

example

Typical examples of approximate confidence intervals can be found in the binomial model . A detailed description can be found in the article Confidence Interval for the Success Probability of the Binomial Distribution . Are exemplary Bernoulli-distributed for everyone and is ${\ displaystyle X_ {i}}$ ${\ displaystyle i \ in \ mathbb {N}}$

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

the sample mean , so is

{\ displaystyle C_ {n} (X) = \ left [{\ overline {X}} _ {n} -u_ {1- \ alpha / 2} {\ sqrt {\ frac {{\ overline {X}} _ {n} (1 - {\ overline {X}} _ {n})} {n}}}, {\ overline {X}} _ {n} + u_ {1- \ alpha / 2} {\ sqrt { \ frac {{\ overline {X}} _ {n} (1 - {\ overline {X}} _ {n})} {n}}} \ right]}

a possible approximate confidence interval for the probability of success of the binomial distribution at the confidence level . ${\ displaystyle 1- \ alpha}$

swell

Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 235-238 , doi : 10.1515 / 9783110215274 .
Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 230-240 , 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 .

Individual evidence

^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 230 , doi : 10.1007 / 978-3-642-41997-3 .

[Rüschendorf230-1] Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 230 , doi : 10.1007 / 978-3-642-41997-3 .