Electronic Journal of Probability

Optimal two-value zero-mean disintegration of zero-mean random variables

Iosif Pinelis

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For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $\{X,r(X)\}$ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 26, 663-727.

Accepted: 10 March 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A50: Integration and disintegration of measures 60E05: Distributions: general theory 60E15: Inequalities; stochastic orderings 62G10: Hypothesis testing 62G15: Tolerance and confidence regions 62F03: Hypothesis testing 62F25: Tolerance and confidence regions
Secondary: 49K30: Optimal solutions belonging to restricted classes 49K45: Problems involving randomness [See also 93E20] 49N15: Duality theory 60G50: Sums of independent random variables; random walks 62G35: Robustness 62G09: Resampling methods 90C08: Special problems of linear programming (transportation, multi-index, etc.) 90C46: Optimality conditions, duality [See also 49N15]

Disintegration of measures Wasserstein metric Kantorovich-Rubinstein theorem transportation of measures optimal matching most symmetric hypothesis testing confidence regions Student's t-test asymmetry exact inequalities conservative properties

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Pinelis, Iosif. Optimal two-value zero-mean disintegration of zero-mean random variables. Electron. J. Probab. 14 (2009), paper no. 26, 663--727. doi:10.1214/EJP.v14-633. https://projecteuclid.org/euclid.ejp/1464819487

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