## Directional differentiability for supremum-type functionals: statistical applications.(English)Zbl 1442.62110

This paper proposes a very useful method for calculating the asymptotic distributions of various statistics in an objective way. This methodology is based on the Hadamard directional derivative and the Delta method. In particular, an application is presented for the asymptotic distribution of Kolmogorov-Smirnov type statistics under the alternative hypothesis. This article presents several inedited results and is relevant for researchers in the area of asymptotic distributions because the methodology developed has few restrictions and consequently can be applied to more contexts.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62E20 Asymptotic distribution theory in statistics 62A01 Foundations and philosophical topics in statistics 62G20 Asymptotic properties of nonparametric inference

EMD
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 [1] Álvarez-Esteban, P.C., del Barrio, E., Cuesta-Albertos, J.A. and Matrán, C. (2012). Similarity of samples and trimming. Bernoulli 18 606-634. · Zbl 1239.62005 [2] Álvarez-Esteban, P.C., del Barrio, E., Cuesta-Albertos, J.A. and Matrán, C. (2016). A contamination model for the stochastic order. TEST 25 751-774. · Zbl 1373.60040 [3] Baíllo, A., Cárcamo, J. and Getman, K. (2019). New distance measures for classifying X-ray astronomy data into stellar classes. Adv. Data Anal. Classif. 13 531-557. · Zbl 07073917 [4] Banach, S. (1936). Théorie des Opérations Linéaires. Monografie Mat. 1. Warsaw. [5] Beare, B.K. and Fang, Z. (2017). Weak convergence of the least concave majorant of estimators for a concave distribution function. Electron. J. Stat. 11 3841-3870. · Zbl 1390.62078 [6] Beare, B.K. and Moon, J.-M. (2015). Nonparametric tests of density ratio ordering. Econometric Theory 31 471-492. · Zbl 1441.62104 [7] Beare, B.K. and Shi, X. (2019). An improved bootstrap test of density ratio ordering. Écon. Stat. 10 9-26. [8] Berk, R.H. and Jones, D.H. (1979). Goodness-of-fit test statistics that dominate the Kolmogorov statistics. Z. Wahrsch. Verw. Gebiete 47 47-59. · Zbl 0379.62026 [9] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 1656-1670. · Zbl 0265.60011 [10] Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. New York: Springer. · Zbl 1220.46002 [11] Cárcamo, J. (2017). Integrated empirical processes in $$L^p$$ with applications to estimate probability metrics. Bernoulli 23 3412-3436. · Zbl 1407.60045 [12] Conway, J.B. (1990). A Course in Functional Analysis, 2nd ed. Graduate Texts in Mathematics 96. New York: Springer. · Zbl 0706.46003 [13] DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. New York: Springer. · Zbl 1154.62001 [14] Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. New York: Wiley. [15] Dette, H., Kokot, K. and Aue, A. (2018). Functional data analysis in the Banach space of continuous functions. Preprint. Available at arXiv:1710.07781v2 [math.ST]. [16] Dette, H., Möllenhoff, K., Volgushev, S. and Bretz, F. (2018). Equivalence of regression curves. J. Amer. Statist. Assoc. 113 711-729. · Zbl 1398.62045 [17] Dudley, R.M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge: Cambridge Univ. Press. · Zbl 0951.60033 [18] Dudley, R.M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge: Cambridge Univ. Press. [19] Dümbgen, L. (1993). On nondifferentiable functions and the bootstrap. Probab. Theory Related Fields 95 125-140. · Zbl 0811.62050 [20] Fang, Z. and Santos, A. (2019). Inference on directionally differentiable functions. Rev. Econ. Stud. 86 377-412. · Zbl 1414.62498 [21] Fermanian, J.-D. (2013). An overview of the goodness-of-fit test problem for copulas. In Copulae in Mathematical and Quantitative Finance. Lect. Notes Stat. 213 61-89. Heidelberg: Springer. · Zbl 1273.62101 [22] Fortet, R. and Mourier, E. (1953). Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Éc. Norm. Supér. (3) 70 267-285. · Zbl 0053.09601 [23] Freitag, G., Lange, S. and Munk, A. (2006). Non-parametric assessment of non-inferiority with censored data. Stat. Med. 25 1201-1217. [24] Genest, C. and Nešlehová, J.G. (2014). On tests of radial symmetry for bivariate copulas. Statist. Papers 55 1107-1119. · Zbl 1310.62069 [25] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics. New York: Cambridge Univ. Press. · Zbl 1358.62014 [26] Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B. and Smola, A. (2012). A kernel two-sample test. J. Mach. Learn. Res. 13 723-773. · Zbl 1283.62095 [27] Huber, P.J. (1981). Robust Statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. [28] Jager, L. and Wellner, J.A. (2004). On the “Poisson boundaries” of the family of weighted Kolmogorov statistics. In A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes - Monograph Series 45 319-331. Beachwood, OH: IMS. · Zbl 1268.62043 [29] Jager, L. and Wellner, J.A. (2007). Goodness-of-fit tests via phi-divergences. Ann. Statist. 35 2018-2053. · Zbl 1126.62030 [30] Kaido, H. (2016). A dual approach to inference for partially identified econometric models. J. Econometrics 192 269-290. · Zbl 1419.62511 [31] Leonard, I.E. and Taylor, K.F. (1983). Supremum norm differentiability. Int. J. Math. Math. Sci. 6 705-713. · Zbl 0545.46028 [32] Leonard, I.E. and Taylor, K.F. (1985). Essential supremum norm differentiability. Int. J. Math. Math. Sci. 8 433-439. · Zbl 0593.46016 [33] Müller, A. (1997). Integral probability metrics and their generating classes of functions. Adv. in Appl. Probab. 29 429-443. · Zbl 0890.60011 [34] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378-418. · Zbl 1041.60024 [35] Neininger, R. and Rüschendorf, L. (2004). On the contraction method with degenerate limit equation. Ann. Probab. 32 2838-2856. · Zbl 1060.60005 [36] Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat. 42 1285-1295. · Zbl 0222.60013 [37] Nickl, R. and Pötscher, B.M. (2007). Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. J. Theoret. Probab. 20 177-199. · Zbl 1130.46020 [38] Rachev, S.T., Klebanov, L.B., Stoyanov, S.V. and Fabozzi, F.J. (2013). The Methods of Distances in the Theory of Probability and Statistics. New York: Springer. · Zbl 1280.60005 [39] Raghavachari, M. (1973). Limiting distributions of Kolmogorov-Smirnov type statistics under the alternative. Ann. Statist. 1 67-73. · Zbl 0276.62028 [40] Rao, M.M., ed. (1997). Real and Stochastic Analysis: Recent Advances. Probability and Stochastics Series. Boca Raton, FL: CRC Press. [41] Römisch, W. (2004). Delta method, infinite dimensional. In Encyclopedia of Statistical Sciences. New York: Wiley. [42] Rubner, Y., Tomasi, C. and Guibas, L.J. (2000). The Earth Mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40 99-121. · Zbl 1012.68705 [43] Schmoyer, R.L. (1988). Linear interpolation with a nonparametric accelerated failure-time model. J. Amer. Statist. Assoc. 83 441-449. · Zbl 0648.62040 [44] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764-782. · Zbl 1243.62066 [45] Seijo, E. and Sen, B. (2011). A continuous mapping theorem for the smallest argmax functional. Electron. J. Stat. 5 421-439. · Zbl 1329.60090 [46] Seo, J. (2018). Tests of stochastic monotonicity with improved power. J. Econometrics 207 53-70. · Zbl 1452.62321 [47] Shapiro, A. (1990). On concepts of directional differentiability. J. Optim. Theory Appl. 66 477-487. · Zbl 0682.49015 [48] Shapiro, A. (1991). Asymptotic analysis of stochastic programs 30 169-186. · Zbl 0745.90057 [49] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley. · Zbl 1170.62365 [50] Sommerfeld, M. and Munk, A. (2018). Inference for empirical Wasserstein distances on finite spaces. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 219-238. · Zbl 1380.62121 [51] Sriperumbudur, B. (2016). On the optimal estimation of probability measures in weak and strong topologies. Bernoulli 22 1839-1893. · Zbl 1360.62163 [52] Sriperumbudur, B.K., Fukumizu, K., Gretton, A., Schölkopf, B. and Lanckriet, G.R.G. (2012). On the empirical estimation of integral probability metrics. Electron. J. Stat. 6 1550-1599. · Zbl 1295.62035 [53] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press. [54] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer. · Zbl 0862.60002 [55] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Berlin: Springer. · Zbl 1156.53003 [56] Wellner, J.A. and Koltchinskii, V. (2003). A note on the asymptotic distribution of Berk-Jones type statistics under the null hypothesis. In High Dimensional Probability, III (Sandjberg, 2002). Progress in Probability 55 321-332. Basel: Birkhäuser. · Zbl 1042.62009 [57] Zolotarev, V.M. (1983). Probability metrics. Theory Probab. Appl. 28 278-302. · Zbl 0533.60025
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