stat609-36 - Lecture 36 Completeness Denition...

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beamer-tu-logLecture 36: CompletenessDefinition 6.2.16 (ancillary statistics)A statisticV(X)is ancillary iff its distribution does not depend on anyunknown quantity. A statisticV(X)is first-order ancillary iffE[V(X)]does not depend on any unknown quantity.A trivial ancillary statistic isV(X)a constant.The following examples show that there exist many nontrivial ancillarystatistics (non-constant ancillary statistics).Examples 6.2.18 and 6.2.19 (location-scale families)IfX1,...,Xnis a random sample from a location family with locationparameterμR, then, for any pair(i,j), 1i,jn,XiXjisancillary, becauseXiXj= (Xiμ)(Xjμ)and the distributionof(Xiμ,Xjμ)does not depend on any unknown parameter.Similarly,X(i)X(j)is ancillary, whereX(1),...,X(n)are the orderstatistics, and the sample varianceS2is ancillary.UW-Madison (Statistics)Stat 609 Lecture 3620141 / 11
beamer-tu-logLecture 36: CompletenessDefinition 6.2.16 (ancillary statistics)A statisticV(X)is ancillary iff its distribution does not depend on anyunknown quantity. A statisticV(X)is first-order ancillary iffE[V(X)]does not depend on any unknown quantity.A trivial ancillary statistic isV(X)a constant.The following examples show that there exist many nontrivial ancillarystatistics (non-constant ancillary statistics).Examples 6.2.18 and 6.2.19 (location-scale families)IfX1,...,Xnis a random sample from a location family with locationparameterμR, then, for any pair(i,j), 1i,jn,XiXjisancillary, becauseXiXj= (Xiμ)(Xjμ)and the distributionof(Xiμ,Xjμ)does not depend on any unknown parameter.Similarly,X(i)X(j)is ancillary, whereX(1),...,X(n)are the orderstatistics, and the sample varianceS2is ancillary.UW-Madison (Statistics)Stat 609 Lecture 3620141 / 11
beamer-tu-logNote that we do not even need to obtain the form of thedistribution ofXiXj.IfX1,...,Xnis a random sample from a scale family with scaleparameterσ>0, then by the same argument we can show that,for any pair(i,j), 1i,jn,Xi/XjandX(i)/X(j)are ancillary.IfX1,...,Xnis a random sample from a location-scale family withparametersμRandσ>0, then, for any(i,j,k), 1i,j,kn,(XiXk)/(XjXk)and(X(i)X(k))/(X(j)X(k))are ancillary.IfV(X)is a non-trivial ancillary statistic, then the the set{x:V(x) =v}does not contain any information aboutθ.IfT(X)is a statistic andV(T(X))is a non-trivial ancillary statistic,it indicates that the reduced data set byTcontains a non-trivialpart that does not contain any information aboutθand, hence, afurther simplification ofTmay still be needed.A sufficient statisticT(X)appears to be most successful inreducing the data if no nonconstant function ofT(X)is ancillary oreven first-order ancillary, which leads to the following definition.

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