Completeness (statistics)

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As completeness is referred to in the estimation theory , a branch of mathematical statistics , a characteristic of distribution classes , σ-algebra or measurable functions . In general, complete distribution classes are “large”, whereas complete σ-algebras are “small”.

Completeness mostly plays a role in connection with sufficiency . It provides criteria for minimal sufficiency or the existence of consistently best unbiased estimators using Lehmann-Scheffé's theorem .

definition

For distribution classes

Given a measurement space and a set of probability measures on this measurement space. Then is called complete if the set of -zero estimators is -trivial. Speak for everyone

applies

.

Here referred to the space of all -integrable functions (see lp space ).

For σ-algebras

A sub-σ-algebra of is called complete for if is completely on the measurement space . This means that the domain of definition of all probability measures is restricted to the smaller σ-algebra .

For statistics

A statistic

is called complete if the σ-algebra it generates is complete or is complete.

The distribution class is often provided with an index and then written . If the completeness is formulated in this notation, the common definition is obtained:

A statistic is called complete if it can be integrated with for all

always follows that

.

Explanation

The following definition of the completeness of a distribution class is intuitively more accessible: A distribution class is complete if and only if it is a separating family for . That is, for anyone with

follows

The distribution class is big enough to be able to distinguish all functions .

Counterexamples

Let be independently and identically distributed random variables with expected value and bounded variance.

Then is . So the function is an unbiased estimator of 0 and the integrand is not the null function .

Clarifications

Limited completeness

Let the set of restricted functions be . A distribution class is called constrained complete if it is complete for . The definitions of bounded complete σ-algebra and bounded complete statistics follow as above.

L-completeness

For a given set of functions , a distribution class is called -complete if it is complete for . The definitions of -complete σ-algebra and -complete statistics follow as above.

Limited L-completeness

The two terms above can also be combined: A distribution class is called restricted- complete if it is complete for . The definitions of bounded- complete σ-algebra and bounded- complete statistics follow as above.

application

Statistical completeness is a prerequisite for Lehmann-Scheffé's theorem , in this context the term was also introduced into statistics by EL Lehmann and H. Scheffé. The Basu's theorem provide a link between the completeness, sufficiency and distribution Freedom ago.

literature

  • Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , 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 , doi : 10.1007 / 978-3-642-17261-8 .
  • EL Lehmann, H. Scheffé: Completeness, similar regions, and unbiased estimation. I. In: Sankhyā. 10, No. 4, 1950, pp. 305-340.
  • EL Lehmann, H. Scheffé: Completeness, similar regions, and unbiased estimation. II. In: Sankhyā. 15, No. 3, 1955, pp. 219-236.
  • Helmut Pruscha: Lectures on mathematical statistics. BG Teubner, Stuttgart 2000, ISBN 3-519-02393-8 , Section II.3.