Integrated quantile functions: properties and applications

Alexander A. Gushchin, Dmitriy A. Borzykh


In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and  tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon–Walsh construction in the Skorokhod embedding problem.


Quantile functions; integrated quantile functions; integrated distribution functions; convex stochastic order; binary experiments; Chacon–Walsh construction

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[1] Blackwell, D.: Comparison of experiments. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pp. 93–102. University of California Press, Berkeley and Los Angeles (1951). MR0046002

[2] Blackwell, D.: Equivalent comparisons of experiments. Ann. Math. Statistics 24, 265–272 (1953).


[3] Carlier, G., Chernozhukov, V., Galichon, A.: Vector quantile regression: An optimal transport approach. Ann. Statist. 44(3), 1165–1192 (2016). doi:10.1214/15-AOS1401 MR3485957

[4] Chacon, R.V., Walsh, J.B.: One-dimensional potential embedding. In: Séminaire de Probabilités, X. Lecture Notes in Math., vol. 511, pp. 19–23. Springer (1976). MR0445598

[5] Chernozhukov, V., Galichon, A., Hallin, M., Henry, M.: Monge–Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45(1), 223–256 (2017). doi:10.1214/16-AOS1450 MR3611491

[6] Cox, A.M.G.: Extending Chacon-Walsh: minimality and generalised starting distributions. In: Séminaire de Probabilités XLI. Lecture Notes in Math., vol. 1934, pp. 233–264. Springer (2008). doi:10.1007/978-3-540-77913-1_12

[7] Cox, A.M.G., Hobson, D.G.: Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135(3), 395–414 (2006). doi:10.1007/s00440-005-0467-y

[8] Embrechts, P., Hofert, M.: A note on generalized inverses. Math. Methods Oper. Res. 77(3), 423–432 (2013). doi:10.1007/s00186-013-0436-7

[9] Faugeras, O., Rüschendorf, L.: Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles. Mathematica Applicanda 45(1), 21–63 (2017)

[10] Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney (1971). MR0270403

[11] Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 4th edn. Walter de Gruyter & Co., Berlin (2016)

[12] Gushchin, A.A., Urusov, M.A.: Processes that can be embedded in a geometric Brownian motion. Theory of Probability & Its Applications 60(2), 246–262 (2016). doi:10.1137/S0040585X97T987594

[13] Hallin, M.: On distribution and quantile functions, ranks and signs in Rd. ECARES Working Paper 2017-34 (2017)

[14] Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54(1), 81–116 (1930). doi:10.1007/BF02547518 MR1555303

[15] Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. Springer (1986). doi:10.1007/978-1-4612-4946-7 MR0856411

[16] Lehmann, E.L., Romano, J.P.: Testing Statistical Hypotheses, 3rd edn. Springer (2005). MR2135927

[17] Leskelä, L., Vihola, M.: Stochastic order characterization of uniform integrability and tightness.Stat. Probab. Lett. 83(1), 382–389 (2013). doi:10.1016/j.spl.2012.09.023

[18] McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995). doi:10.1215/S0012-7094-95-08013-2

[19] Müller, A.: Orderings of risks: A comparative study via stop-loss transforms. Insurance: Mathematics and Economics 17(3), 215–222 (1996). doi:10.1016/0167-6687(96)90002-5

[20] Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, Ltd., Chichester (2002). MR1889865

[21] Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002). doi:10.1137/S1052623400375075

[22] Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. Journal of Risk 2, 21–42 (2000)

[23] Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. Journal of Banking & Finance 26(7), 1443–1471 (2002)

[24] Rüschendorf, L.: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer (2013). doi:10.1007/978-3-642-33590-7

[25] Shiryaev, A.N., Spokoiny, V.G.: Statistical Experiments and Decisions: Asymptotic Theory. World Scientific Publishing Co., Inc., River Edge, NJ (2000). doi:10.1142/9789812779243

[26] Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley Publishing Co., Inc., Reading, Mass. (1965). MR0185620 (32 #3082b) MR0185620

[27] Strasser, H.: Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory. Walter de Gruyter & Co., Berlin (1985). doi:10.1515/9783110850826

[28] Torgersen, E.: Comparison of Statistical Experiments. Cambridge University Press, Cambridge (1991). doi:10.1017/CBO9780511666353



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