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A family of probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335501.png" />, defined on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335502.png" />, for which the unique unbiased estimator of zero in the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335503.png" />-measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335504.png" /> is the function identically equal to zero, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335505.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335506.png" />-measurable function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335507.png" /> satisfying the relation
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A family of probability measures $\{ \mathbf{P}_\theta : \theta \in \Theta \subset \mathbf{R}^k \}$, defined on a measure space $(\mathfrak{X}, \mathfrak{B})$, for which the unique unbiased estimator of zero in the class of $\mathfrak{B}$-measurable functions on $\mathfrak{X}$ is the function identically equal to zero, that is, if $f({\cdot})$ is any $\mathfrak{B}$-measurable function defined on $\mathfrak{X}$ satisfying the relation
 
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\begin{equation}\label{eq:a1}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335508.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\int_{\mathfrak{X}} f(x) \,\mathrm{d}\mathbf{P}_\theta = 0 \ \ \text{for all}\ \theta\in\Theta\,,
 
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\end{equation}
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d0335509.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d03355010.png" />-almost-everywhere, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d03355011.png" />. For example, a family of exponential distributions is complete. If the relation (*) is satisfied under the further assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d03355012.png" /> is bounded, then the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033550/d03355013.png" /> is said to be boundedly complete. Boundedly-complete families of distributions of sufficient statistics play a major role in mathematical statistics, in particular in the problem of constructing similar tests (cf. [[Similar test|Similar test]]) with a [[Neyman structure|Neyman structure]].
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then $f(x)=0$ $\mathbf{P}_\theta$-almost-everywhere, for all $\theta\in\Theta$. For example, a family of [[exponential distribution]]s is complete. If the relation \eqref{eq:a1} is satisfied under the further assumption that $f$ is bounded, then the family $\{ \mathbf{P}_\theta : \theta \in \Theta \}$ is said to be boundedly complete. Boundedly-complete families of distributions of sufficient statistics play a major role in mathematical statistics, in particular in the problem of constructing [[similar test]]s with a [[Neyman structure]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Zacks,  "The theory of statistical inference" , Wiley  (1971)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Zacks,  "The theory of statistical inference" , Wiley  (1971)</TD></TR>
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</table>
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Revision as of 20:07, 19 November 2017

A family of probability measures $\{ \mathbf{P}_\theta : \theta \in \Theta \subset \mathbf{R}^k \}$, defined on a measure space $(\mathfrak{X}, \mathfrak{B})$, for which the unique unbiased estimator of zero in the class of $\mathfrak{B}$-measurable functions on $\mathfrak{X}$ is the function identically equal to zero, that is, if $f({\cdot})$ is any $\mathfrak{B}$-measurable function defined on $\mathfrak{X}$ satisfying the relation \begin{equation}\label{eq:a1} \int_{\mathfrak{X}} f(x) \,\mathrm{d}\mathbf{P}_\theta = 0 \ \ \text{for all}\ \theta\in\Theta\,, \end{equation} then $f(x)=0$ $\mathbf{P}_\theta$-almost-everywhere, for all $\theta\in\Theta$. For example, a family of exponential distributions is complete. If the relation \eqref{eq:a1} is satisfied under the further assumption that $f$ is bounded, then the family $\{ \mathbf{P}_\theta : \theta \in \Theta \}$ is said to be boundedly complete. Boundedly-complete families of distributions of sufficient statistics play a major role in mathematical statistics, in particular in the problem of constructing similar tests with a Neyman structure.

References

[1] Yu.V. Linnik, "Statistical problems with nuisance parameters" , Amer. Math. Soc. (1968) (Translated from Russian)
[2] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)


Comments

References

[a1] S. Zacks, "The theory of statistical inference" , Wiley (1971)
How to Cite This Entry:
Distributions, complete family of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distributions,_complete_family_of&oldid=42338
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article