A Sufficiency Paradox: An Insufficient Statistic Preserving the Fisher Information
نویسندگان
چکیده
An example of a regular statistical experiment is constructed where an insufficient statistic preserves the Fisher information contained in the data. The data are a pair (∆, X) where ∆ is a binary random variable and, given∆,X has a density f(x−θ|∆) depending on a location parameter θ. The phenomenon is based on the fact that f(x|∆) smoothly vanishes at one point; it can be eliminated by adding to the regularity of a statistical experiment positivity of the density function.
منابع مشابه
Probabilistic Sufficiency and Algorithmic Sufficiency from the point of view of Information Theory
Given the importance of Markov chains in information theory, the definition of conditional probability for these random processes can also be defined in terms of mutual information. In this paper, the relationship between the concept of sufficiency and Markov chains from the perspective of information theory and the relationship between probabilistic sufficiency and algorithmic sufficien...
متن کاملMarginal information for expectation parameters
The authors consider a special case of inference in the presence of nuisance parameters. They show that when the orthogonalized score function is a function of a statistic S, no Fisher information for the interest parameter is lost by using the marginal distribution of S rather than the full distribution of the observations. Therefore, no information for the interest parameter is recovered by c...
متن کاملA survey with examples
This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated paralelly to the the measure theoretic case. It turns out that several classical examples and results have a non-commutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The ...
متن کاملMaximally Informative Statistics
In this paper we propose a Bayesian, information theoretic approach to dimensionality reduction. The approach is formulated as a variational principle on mutual information, and seamlessly addresses the notions of sufficiency, relevance, and representation. Maximally informative statistics are shown to minimize a Kullback-Leibler distance between posterior distributions. Illustrating the approa...
متن کاملInformation in equity markets with ambiguity averse investors
In this paper, I show that persistent pricing anomalies are consistent with a market that includes ambiguity averse investors. In particular, I show that ambiguity averse investors may prefer to trade based on aggregate signals that reduce ambiguity at the cost of a loss in information. Because the aggregate signals preferred by ambiguity averse investors are not sufficient statistics for the c...
متن کامل