flâneur

Sufficiency

stat210a.berkeley.edu · 1,859 words · saved by 1 readers

Sufficiency is a central concept in statistics that allows us to focus on the essential aspects of the data set while ignoring details that are irrelevant to the inference problem. If 𝑋 ∼ 𝑃 𝜃 represents the entire data set, drawn from a model 𝑃 = { 𝑃 𝜃 : 𝜃 ∈ Θ } , then this lecture will concern the idea of a sufficient statistic 𝑇 ( 𝑋 ) that carries all of the information in the data that can help us learn about 𝜃 . A statistic 𝑇 ( 𝑋 ) is any random variable which is a function of the data 𝑋 , and which does not depend on the unknown parameter 𝜃 . We say the statistic 𝑇 ( 𝑋 ) is sufficient for the model 𝑃 if 𝑃 𝜃 ( 𝑋 ∣ 𝑇 ) does not depend on 𝜃 . This lecture will be devoted to interpreting this definition and giving examples. Example (Independent Bernoulli sequence): We introduced the binomial example from Lecture 2 by telling a story about an investigator who flips a biased coin 𝑛 times and records the total number of heads, which has a binomia

\[ \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}{=}}…

saved by

related reading