Electronic Journal of Probability

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

Iosif Pinelis

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819487

Digital Object Identifier
doi:10.1214/EJP.v14-633

Mathematical Reviews number (MathSciNet)
MR2486818

Zentralblatt MATH identifier
1193.60020

Subjects
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]

Keywords
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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

  • Abramovich, Y. A.; Wickstead, A. W. Remarkable classes of unital AM-spaces. J. Math. Anal. Appl. 180 (1993), no. 2, 398–411.
  • Abramovich, Y. A.; Aliprantis, C. D. Problems in operator theory.Graduate Studies in Mathematics, 51. American Mathematical Society, Providence, RI, 2002. xii+386 pp. ISBN: 0-8218-2147-4
  • Aizenman, Michael; Germinet, François; Klein, Abel; Warzel, Simone. On Bernoulli decompositions for random variables, concentration bounds, and spectral localization. Probab. Theory Related Fields 143 (2009), no. 1-2, 219–238.
  • Bartlett, M. S. (1935). The effect of non-normality on the $t$ distribution. Proc. Camb. Phil. Soc. 31, 223–231.
  • Bentkus, Vidmantas. On Hoeffding's inequalities. Ann. Probab. 32 (2004), no. 2, 1650–1673.
  • Bentkus, V.; Juškevičius, T. Bounds for tail probabilities of martingales using skewness and kurtosis. Lith. Math. J. 48 (2008), no. 1, 30–37.
  • Cambanis, Stamatis; Simons, Gordon; Stout, William. Inequalities for $Ek(X,Y)$ when the marginals are fixed. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 4, 285–294.
  • Eaton, Morris L. A note on symmetric Bernoulli random variables. Ann. Math. Statist. 41 1970 1223–1226.
  • Eaton, M.L. (1974). A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2, 609–614.
  • Eaton, M. L.; Efron, Bradley. Hotelling's $Tsp{2}$ test under symmetry conditions. J. Amer. Statist. Assoc. 65 1970 702–711.
  • Efron, Bradley. Student's $t$-test under symmetry conditions. J. Amer. Statist. Assoc. 64 1969 1278–1302.
  • Gangbo, Wilfrid. The Monge mass transfer problem and its applications. Monge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997), 79–104, Contemp. Math., 226, Amer. Math. Soc., Providence, RI, 1999.
  • Hall, Peter. The bootstrap and Edgeworth expansion.Springer Series in Statistics. Springer-Verlag, New York, 1992. xiv+352 pp. ISBN: 0-387-97720-1
  • Hall, Peter; Wang, Qiying. Exact convergence rate and leading term in central limit theorem for Student's $t$ statistic. Ann. Probab. 32 (2004), no. 2, 1419–1437.
  • Hoaglin, D. C. (1985). Summarizing shape numerically: The $g$- and $h$-distributions. In Exploring Data Tables, Trends, and Shapes (D. C. Hoaglin, F. Mosteller and J. W. Tukey, eds.) 461–514. Wiley, New York.
  • Hoeffding, W. (1940). Masstabinvariante korrelationstheorie. Schr. Math. Inst. Univ. Berlin. 5, 179–233.
  • Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 1963 13–30.
  • Hoeffding, Wassily. The collected works of Wassily Hoeffding.Edited and with a preface by N. I. Fisher and P. K. Sen.Springer Series in Statistics: Perspectives in Statistics. Springer-Verlag, New York, 1994. xvi+658 pp. ISBN: 0-387-94310-2
  • Kafadar, Karen. John Tukey and robustness.Tribute to John W. Tukey. Statist. Sci. 18 (2003), no. 3, 319–331.
  • Kallenberg, Olav. Foundations of modern probability.Second edition.Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Logan, B. F.; Mallows, C. L.; Rice, S. O.; Shepp, L. A. Limit distributions of self-normalized sums. Ann. Probability 1 (1973), 788–809.
  • Oates, D. K. (1971). A non-compact Krein-Milman theorem. Pacific J. Math. 36, 781–785.
  • Parthasarathy, K. R.; Ranga Rao, R.; Varadhan, S. R. S. On the category of indecomposable distributions on topological groups. Trans. Amer. Math. Soc. 102 1962 200–217.
  • Pinelis, Iosif. Extremal probabilistic problems and Hotelling's $Tsp 2$ test under a symmetry condition. Ann. Statist. 22 (1994), no. 1, 357–368.
  • Pinelis, Iosif. Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above. High dimensional probability, 33–52, IMS Lecture Notes Monogr. Ser., 51, Inst. Math. Statist., Beachwood, OH, 2006.
  • Pinelis, Iosif. On normal domination of (super)martingales. Electron. J. Probab. 11 (2006), no. 39, 1049–1070 (electronic).
  • Pinelis, I. (2006). Student's $t$-test without symmetry conditions. arXiv:math/0606160v1 [math.ST], http://arxiv.org/abs/math/0606160.
  • Pinelis, Iosif. Exact inequalities for sums of asymmetric random variables, with applications. Probab. Theory Related Fields 139 (2007), no. 3-4, 605–635.
  • Rachev, Svetlozar T.; Rüschendorf, Ludger. Mass transportation problems. Vol. I.Theory.Probability and its Applications (New York). Springer-Verlag, New York, 1998. xxvi+508 pp. ISBN: 0-387-98350-3
  • Ratcliffe, J. F. (1968). The effect on the $t$-distribution of non-normality in the sampled population. Appl. Statist.17, 42–48.
  • Shao, Qi-Man. Self-normalized large deviations. Ann. Probab. 25 (1997), no. 1, 285–328.
  • Skorokhod, A. V. Studies in the theory of random processes.Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965 viii+199 pp. (32 #3082b)
  • Tchen, André H. Inequalities for distributions with given marginals. Ann. Probab. 8 (1980), no. 4, 814–827.
  • Tukey, J. W. (1948). Some elementary problems of importance to small sample practice. Human Biol. 20, 205–214.
  • Tukey, John W. On the comparative anatomy of transformations. Ann. Math. Statist. 28 (1957), 602–632.