Integrated quantile functions: properties and applications
Volume 4, Issue 4 (2017), pp. 285–314
Pub. online: 8 December 2017
Type: Research Article
Open Access
Open Access
Received
9 August 2017
9 August 2017
Revised
6 November 2017
6 November 2017
Accepted
8 November 2017
8 November 2017
Published
8 December 2017
8 December 2017
Abstract
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.
References
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
Blackwell, D.: Equivalent comparisons of experiments. Ann. Math. Statistics 24, 265–272 (1953). MR0056251
Carlier, G., Chernozhukov, V., Galichon, A.: Vector quantile regression: An optimal transport approach. Ann. Statist. 44(3), 1165–1192 (2016). MR3485957. https://doi.org/10.1214/15-AOS1401
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
Chernozhukov, V., Galichon, A., Hallin, M., Henry, M.: Monge–Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45(1), 223–256 (2017). MR3611491. https://doi.org/10.1214/16-AOS1450
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). https://doi.org/10.1007/978-3-540-77913-1_12
Cox, A.M.G., Hobson, D.G.: Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135(3), 395–414 (2006). https://doi.org/10.1007/s00440-005-0467-y
Embrechts, P., Hofert, M.: A note on generalized inverses. Math. Methods Oper. Res. 77(3), 423–432 (2013). https://doi.org/10.1007/s00186-013-0436-7
Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney (1971). MR0270403
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). https://doi.org/10.1137/S0040585X97T987594
Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54(1), 81–116 (1930). MR1555303. https://doi.org/10.1007/BF02547518
Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. Springer (1986). MR0856411. https://doi.org/10.1007/978-1-4612-4946-7
Lehmann, E.L., Romano, J.P.: Testing Statistical Hypotheses, 3rd edn. Springer (2005). MR2135927
Leskelä, L., Vihola, M.: Stochastic order characterization of uniform integrability and tightness. Stat. Probab. Lett. 83(1), 382–389 (2013). https://doi.org/10.1016/j.spl.2012.09.023
McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995). https://doi.org/10.1215/S0012-7094-95-08013-2
Müller, A.: Orderings of risks: A comparative study via stop-loss transforms. Insurance: Mathematics and Economics 17(3), 215–222 (1996). https://doi.org/10.1016/0167-6687(96)90002-5
Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, Ltd., Chichester (2002). MR1889865
Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002). https://doi.org/10.1137/S1052623400375075
Rüschendorf, L.: Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer (2013). https://doi.org/10.1007/978-3-642-33590-7
Shiryaev, A.N., Spokoiny, V.G.: Statistical Experiments and Decisions: Asymptotic Theory. World Scientific Publishing Co., Inc., River Edge, NJ (2000). https://doi.org/10.1142/9789812779243
Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley Publishing Co., Inc., Reading, Mass. (1965). MR0185620
Strasser, H.: Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory. Walter de Gruyter & Co., Berlin (1985). https://doi.org/10.1515/9783110850826
Torgersen, E.: Comparison of Statistical Experiments. Cambridge University Press, Cambridge (1991). https://doi.org/10.1017/CBO9780511666353
