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Completeness, Ancillarity, and Basu’s Theorem

stat210a.berkeley.edu · 2,010 words · saved by 1 readers

Check that Laplace median problem is on HW 2 or other, check that minimality for uniform scale is in previous lecture. As we have seen, for a given statistical problem we may have many different sufficient statistics, some of which reduce the data more than others. We usually want to look for one that is minimal sufficient, meaning that it strips away as much irrelevant information as possible and only retains the information that is relevant to estimating the parameter. In some cases the minimal sufficient statistic has an additional property called completeness. The definition of completeness is initially counterintuitive, but it has a number of useful implications we will explore throughout the semester. A statistic 𝑇 ( 𝑋 ) is complete for a family of distributions 𝑃 = { 𝑃 𝜃 : 𝜃 ∈ Θ } if no nontrivial function of 𝑇 can have expectation zero for every distribution in the family: 𝐸 𝜃 𝑓 ( 𝑇 ( 𝑋 ) ) = 0 ∀ 𝜃 ∈ Θ ⟹ 𝑓 ( 𝑇 ) = 𝑃 -a.s. 0 ::: callout-note The name for co

\[ \newcommand{\cB}{\mathcal{B}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cX}{\mathcal{X}} \newcommand{\EE}{\mathbb{E}} \newcommand{\PP}{\mathbb{P}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\td}{\,\textrm{d}} \newcommand{\simiid}{\stackrel{\textrm{i.i.d.}}{\sim}} \newcommand{\simind}{\stackrel{\textrm{ind.}}{\sim}} \newcommand{\eqas}{\stackrel{\textrm{a.s.}}{=}} \newcommand{\eqPas}{\stackrel{\cP\textrm{-a.s.}}{=}} \newcommand{\eqmuas}{\stackrel{\mu\textrm{-a.s.}}{=}} \newcommand{\eqD}{\stackrel{D}{=}}…

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