Given two random variables X and Y , the condition of unbiasedness states that E(X|Y = y) = y and E(Y |X = x) = x both almost surely (a.s.). If the prior on Y is proper and has finite expectation or nonnegative support, unbiasedness implies X = Y a.s. This paper examines the implications of unbiasedness when the prior on Y is improper. Since the improper case can be meaningfully analysed in a finitely additive framework, we revisit the whole issue of unbiasedness from this perspective. First we argue that a notion weaker than equality a.s., named coincidence, is more appropriate in a finitely additive setting. Next we discuss the meaning of unbiasedness from a Bayesian and fiducial perspective.We then show that unbiasedness and finite expectation of Y imply coincidence between X and Y , while a weaker conclusion follows if the improper prior on Y is only assumed to have positive support. We illustrate our approach throughout the paper by revisiting some examples discussed in the recent literature.

Unbiased Bayes estimates and improper prior

VERONESE, PIERO
1992

Abstract

Given two random variables X and Y , the condition of unbiasedness states that E(X|Y = y) = y and E(Y |X = x) = x both almost surely (a.s.). If the prior on Y is proper and has finite expectation or nonnegative support, unbiasedness implies X = Y a.s. This paper examines the implications of unbiasedness when the prior on Y is improper. Since the improper case can be meaningfully analysed in a finitely additive framework, we revisit the whole issue of unbiasedness from this perspective. First we argue that a notion weaker than equality a.s., named coincidence, is more appropriate in a finitely additive setting. Next we discuss the meaning of unbiasedness from a Bayesian and fiducial perspective.We then show that unbiasedness and finite expectation of Y imply coincidence between X and Y , while a weaker conclusion follows if the improper prior on Y is only assumed to have positive support. We illustrate our approach throughout the paper by revisiting some examples discussed in the recent literature.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11565/269391
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