Completeness (statistics)

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 ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle f\in {\mathcal {L}}(X,{\mathcal {A}},{\mathcal {P}}){\text{ mit }}\operatorname {E} _{P}(f)=0\quad \mathrm {f{\ddot {u}}r\;alle\;} P\in {\mathcal {P}}}$

applies

${\displaystyle f=0\quad {\mathcal {P}}-{\text{fast sicher}}}$.

Here referred to the space of all -integrable functions (see lp space ). ${\displaystyle {\mathcal {L}}(X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {P}}}$

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 . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}|_{\mathcal {T}}}$ ${\displaystyle (X,{\mathcal {T}})}$ ${\displaystyle {\mathcal {P}}|_{\mathcal {T}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {T}}}$

For statistics

A statistic

${\displaystyle T:(X,{\mathcal {A}})\to (Y,{\mathcal {A}}^{*})}$

is called complete if the σ-algebra it generates is complete or is complete. ${\displaystyle \sigma (T)}$ ${\displaystyle {\mathcal {P}}|_{\sigma (T)}}$

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: ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \vartheta }$ ${\displaystyle {\mathcal {P}}=(P_{\vartheta })_{\vartheta \in \Theta }}$

A statistic is called complete if it can be integrated with for all ${\displaystyle T}$ ${\displaystyle g}$

${\displaystyle \operatorname {E} _{\vartheta }(g(T))=0\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta }$

always follows that

${\displaystyle P_{\vartheta }(g(T)=0)=1\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta }$.

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 ${\displaystyle {\mathcal {L}}(X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle f,g\in {\mathcal {L}}(X,{\mathcal {A}},{\mathcal {P}})}$

${\displaystyle \int f\mathrm {d} P=\int g\mathrm {d} P\quad \mathrm {f{\ddot {u}}r\;alle\;} P\in {\mathcal {P}}}$

follows

${\displaystyle f=g\quad {\mathcal {P}}-{\text{fast sicher}}}$

The distribution class is big enough to be able to distinguish all functions . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {L}}(X,{\mathcal {A}},{\mathcal {P}})}$

Counterexamples

Let be independently and identically distributed random variables with expected value and bounded variance. ${\displaystyle X_{1},X_{2}}$ ${\displaystyle \vartheta }$

Then is . So the function is an unbiased estimator of 0 and the integrand is not the null function . ${\displaystyle \operatorname {E} [X_{1}-X_{2}]=\operatorname {E} [g(X_{1},X_{2})]=0}$ ${\displaystyle g}$

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. ${\displaystyle B(X,{\mathcal {A}})}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle B(X,{\mathcal {A}})\cap L(X,{\mathcal {A}},{\mathcal {P}})}$

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. ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}\cap L(X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$

Limited L-completeness

The two terms above can also be combined: A distribution class is called restricted- complete ${\displaystyle {\mathcal {L}}}$ if it is complete for . The definitions of bounded- complete σ-algebra and bounded- complete statistics follow as above. ${\displaystyle {\mathcal {L}}\cap B(X,{\mathcal {A}})\cap L(X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$

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.