Properties of Conditional Expectation Operators and Sufficient Subfields
Andrew Tausz
TL;DR
This paper explores properties of conditional expectation operators and constructs a counterexample showing that two sufficient variables can jointly not be sufficient, highlighting subtle aspects of sufficiency in statistics.
Contribution
It provides a detailed analysis of conditional expectation properties and presents a novel counterexample demonstrating limitations of joint sufficiency.
Findings
Existence of sufficient variables that are not jointly sufficient
Counterexample clarifies subtlety in sufficiency concepts
Complete proofs and missing steps supplied for prior work
Abstract
We discuss some properties of conditional expectation operators, and use these facts to prove an interesting counterexample regarding sufficient statistics. In particular, we show that there exists sufficient random variables X and Y, such that (X, Y) are jointly not sufficient. We follow the work of Burkholder and Chow, presenting complete proofs and supplying missing steps.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical Analysis and Transform Methods · Stochastic processes and financial applications