Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Methods for Identifying Minimal Sufficient Statistics

Rafael Oliveira Cavalcante, Alexandre Galvão Patriota

Abstract

Let $f_θ$ be a joint density of a random sample $X$. A frequently used criterion asserts that a statistic $T(X)$ is minimal sufficient if, for any sample points $x$ and $y$, $T(x)=T(y)$ exactly when there exists a finite constant $h_{xy}>0$, independent of $θ$, such that $f_θ(y)=f_θ(x)h_{xy}$ for every $θ$. Although valid under regularity assumptions identified by Lehmann (1950), this statement is false as stated in general: we give a counterexample based on the choice of versions of Radon--Nikodym derivatives. We then propose a version-robust criterion that can be applied once sufficiency is established, and we use it to extend the method of Sato (1996) beyond Euclidean settings to analytic Borel sample spaces and standard Borel statistic spaces. While Lehmann--Scheffé--type regularity conditions and Sato's approach can be difficult to verify in practice, our method is straightforward to check when the statistic is known to be sufficient. Finally, we analyze a distinct criterion for identifying minimal sufficient statistics developed by (Pfanzagl,1994, Pfanzagl, 2017) and provide a discrete counterexample showing that it also requires additional assumptions.

Methods for Identifying Minimal Sufficient Statistics

Abstract

Let be a joint density of a random sample . A frequently used criterion asserts that a statistic is minimal sufficient if, for any sample points and , exactly when there exists a finite constant , independent of , such that for every . Although valid under regularity assumptions identified by Lehmann (1950), this statement is false as stated in general: we give a counterexample based on the choice of versions of Radon--Nikodym derivatives. We then propose a version-robust criterion that can be applied once sufficiency is established, and we use it to extend the method of Sato (1996) beyond Euclidean settings to analytic Borel sample spaces and standard Borel statistic spaces. While Lehmann--Scheffé--type regularity conditions and Sato's approach can be difficult to verify in practice, our method is straightforward to check when the statistic is known to be sufficient. Finally, we analyze a distinct criterion for identifying minimal sufficient statistics developed by (Pfanzagl,1994, Pfanzagl, 2017) and provide a discrete counterexample showing that it also requires additional assumptions.
Paper Structure (6 sections, 13 theorems, 29 equations)

This paper contains 6 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Counterexamples
  3. Corrected and Generalized Methods
  4. Proofs
  5. Acknowledgments
  6. Basic definitions

Key Result

Proposition 4.1

Let $\mathcal{E}:=(\mathcal{X},\Sigma,\{P_\theta\}_{\theta\in\Theta})$ be a statistical model, $\mathfrak{A}$ a sub-$\sigma$-algebra of $\Sigma$, and $(\mathcal{Y},\Sigma_\mathcal{Y})$ a standard Borel space. If $f:\mathcal{X}\to \mathcal{Y}$ is $\overline{\mathfrak{A}}^\mathcal{E}$-measurable, then

Theorems & Definitions (41)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 31 more