Some Selected Topics for the Bootstrap of the Empirical and Quantile processes

Sergio Alvarez-Andrade ; Salim Bouzebda

Teoriâ slučajnyh processov, Tome 24 (2019) no. 1, pp. 19-48.

Résumé

In the present work, we consider the asymptotic distributions of $L_{p}$ functionals of bootstrapped weighted uniform quantile and empirical processes. The asymptotic laws obtained are represented in terms of Gaussian integrals. We investigate the strong approximations for the bootstrapped Vervaat process and the weighted bootstrap for Bahadur-Kiefer process. We obtain new results on the precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence, under some mild conditions, for the weighted bootstrap of empirical and the quantile processes. In addition we consider the strong approximation of the hybrids of empirical and partial sums processes when the sample size is random.

Détail

Keywords: Empirical processes, Bootstrap, Gaussian processes, Kac's representation, Bahadur-Kiefer process, Complete convergence.
Mots-clés : Vervaat process

Bibliographie

[1] Sergio Alvarez-Andrade, Salim Bouzebda, “Strong approximations for weighted bootstrap of empirical and quantile processes with applications”, Stat. Methodol., 11 (2013), 36–52 | DOI | MR | Zbl

[2] Sergio Alvarez-Andrade, Salim Bouzebda, “Asymptotic results for hybrids of empirical and partial sums processes”, Statist. Papers, 55:4 (2014), 1121–1143 | DOI | MR | Zbl

[3] Sergio Alvarez-Andrade, Salim Bouzebda, “On the hybrids of {$k$}-spacing empirical and partial sum processes”, Rev. Mat. Complut., 30:1 (2017), 185–216 | DOI | MR | Zbl

[4] Sergio Alvarez-Andrade, Salim Bouzebda, “Almost sure central limit theorem for the hybrid process”, Statistics, 52:3 (2018), 519–532 | DOI | MR | Zbl

[5] R. R. Bahadur, “A note on quantiles in large samples”, Ann. Math. Statist., 37 (1966), 577–580 | DOI | MR | Zbl

[6] Philippe Barbe, Patrice Bertail, The weighted bootstrap, Lecture Notes in Statistics, 98, Springer-Verlag, New York, 1995 | MR | Zbl

[7] Leonard E. Baum, Melvin Katz, “Convergence rates in the law of large numbers”, Trans. Amer. Math. Soc., 120 (1965), 108–123 | DOI | MR

[8] Luisa Beghin, “Weak convergence of some randomly indexed empirical processes”, Probab. Math. Statist., 22:2 (2002), 333–342 ; Acta Univ. Wratislav, no. 2470 | MR | Zbl

[9] R J. Beran, L. Le Cam, P. W. Millar, “Convergence of stochastic empirical measures”, J. Multivariate Anal., 23:1 (1987), 159–168 | DOI | MR | Zbl

[10] Peter J. Bickel, David A. Freedman, “Some asymptotic theory for the bootstrap”, Ann. Statist., 9:6 (1981), 1196–1217 | DOI | MR | Zbl

[11] S. Bouzebda, “On the strong approximation of bootstrapped empirical copula processes with applications”, Math. Methods Statist., 21:3 (2012), 153–188 | DOI | MR | Zbl

[12] Salim Bouzebda, “Kac's representation for empirical copula process from an asymptotic viewpoint”, Statist. Probab. Lett., 123 (2017), 107–113 | DOI | MR | Zbl

[13] Salim Bouzebda, Mohamed Cherfi, “General bootstrap for dual {$\phi$}-divergence estimates”, J. Probab. Stat. (Art. ID 834107, 33), 2012 | MR

[14] J. Bretagnolle, P. Massart, “Hungarian constructions from the nonasymptotic viewpoint”, Ann. Probab., 17:1 (1989), 239–256 | DOI | MR | Zbl

[15] Youyou Chen, Tailong Li, “Moment convergence rates for the uniform empirical process and the uniform sample quantile process”, Comm. Statist. Theory Methods, 46:18 (2017), 9086–9091 | DOI | MR | Zbl

[16] Michael R. Chernick, Bootstrap methods, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley Sons, Inc., New York, 1999, A practitioner's guide, A Wiley-Interscience Publication | MR

[17] Michael R. Chernick, Bootstrap methods: a guide for practitioners and researchers, Wiley Series in Probability and Statistics, second ed., Wiley-Interscience [John Wiley Sons], Hoboken, NJ, 2008 | MR | Zbl

[18] Kai-Lai Chung, “An estimate concerning the {K}olmogoroff limit distribution”, Trans. Amer. Math. Soc., 67 (1949), 36–50 | MR | Zbl

[19] Endre Csáki, Miklós Csörgő, Antónia Földes, Zhan Shi, Ričardas Zitikis, “Pointwise and uniform asymptotics of the {V}ervaat error process”, J. Theoret. Probab., 15:4 (2002), 845–875 | DOI | MR | Zbl

[20] M. Csörgő, P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981 | MR | Zbl

[21] Miklós Csörgő, Sándor Csörgő, Lajos Horváth, An asymptotic theory for empirical reliability and concentration processes, Lecture Notes in Statistics, 33, Springer-Verlag, Berlin, 1986 | MR | Zbl

[22] Miklós Csörgő, Sándor Csörgő, Lajos Horváth, David M. Mason, “Weighted empirical and quantile processes”, Ann. Probab., 14:1 (1986), 31–85 | DOI | MR | Zbl

[23] Miklós Csörgő, Sándor Csörgő, Lajos Horváth, David M. Mason, Brian S. Yandell, Asymptotic theory of some bootstrapped empirical processes, Technical report, Laboratory for Research in Statistical Probability, Carleton University, Ottawa, Canada, 1986

[24] Miklós Csörgő, Lajos Horváth, “On the distributions of {$L_p$} norms of weighted uniform empirical and quantile processes”, Ann. Probab., 16:1 (1988), 142–161 | DOI | MR | Zbl

[25] Miklós Csörgő, Lajos Horváth, Weighted approximations in probability and statistics, With a foreword by David Kendall, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley Sons, Ltd., Chichester, 1993 | MR

[26] Miklós Csörgő, Lajos Horváth, Piotr Kokoszka, “Approximation for bootstrapped empirical processes”, Proc. Amer. Math. Soc., 128:8 (2000), 2457–2464 | DOI | MR | Zbl

[27] Miklós Csörgő, Pál Révész, “Strong approximations of the quantile process”, Ann. Statist., 6:4 (1978), 882–894 | DOI | MR | Zbl

[28] Miklós Csörgő, Ričardas Zitikis, “Strassen's {LIL} for the {L}orenz curve”, J. Multivariate Anal., 59:1 (1996), 1–12 | DOI | MR | Zbl

[29] Miklós Csörgő, Ričardas Zitikis, “On the rate of strong consistency of {L}orenz curves”, Statist. Probab. Lett., 34:2 (1997), 113–121 | DOI | MR | Zbl

[30] Sándor Csörgő, David M. Mason, “Bootstrapping empirical functions”, Ann. Statist., 17:4 (1989), 1447–1471 | DOI | MR

[31] A. C. Davison, D. V. Hinkley, Bootstrap methods and their application, With 1 IBM-PC floppy disk (3.5 inch; HD), Cambridge Series in Statistical and Probabilistic Mathematics, 1, Cambridge University Press, Cambridg, 1997 | MR | Zbl

[32] Monroe D. Donsker, “Justification and extension of {D}oob's heuristic approach to the {K}omogorov-{S}mirnov theorems”, Ann. Math. Statistics, 23 (1952), 277–281 | MR | Zbl

[33] A. Dvoretzky, J. Kiefer, J. Wolfowitz, “Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator”, Ann. Math. Statist., 27 (1956), 642–669 | DOI | MR | Zbl

[34] B. Efron, “Bootstrap methods: another look at the jackknife”, Ann. Statist., 7:1 (1979), 1–26 | DOI | MR | Zbl

[35] Bradley Efron, Robert J. Tibshirani, An introduction to the bootstrap, Monographs on Statistics and Applied Probability, 57, Chapman and Hall, New York, 1993 | MR | Zbl

[36] Helen Finkelstein, “The law of the iterated logarithm for empirical distributions”, Ann. Math. Statist., 42 (1971), 607–615 | DOI | MR | Zbl

[37] P. Gaenssler, W. Stute, Seminar on empirical processes, DMV Seminar, v. 9, Birkhäuser Verlag, Basel, 1987 | MR

[38] Peter Gänssler, Empirical processes, v. 3, Institute of Mathematical Statistics Lecture Notes—Monograph Series, Institute of Mathematical Statistics, Hayward, CA, 1983 | MR | Zbl

[39] Evarist Giné, Joel Zinn, “Bootstrapping general empirical measures”, Ann. Probab., 18:2 (1990), 851–869 | DOI | MR | Zbl

[40] Phillip Good, Permutation, parametric and bootstrap tests of hypotheses, Springer Series in Statistics, third ed., Springer-Verlag, New York, 2005 | MR | Zbl

[41] A. Gut, J. Steinebach, “Precise asymptotics– a general approach”, Acta Math. Hungar., 138:4 (2013), 365–385 | DOI | MR | Zbl

[42] Allan Gut, Aurel Spătaru, “Precise asymptotics in the {B}aum-{K}atz and {D}avis laws of large numbers”, J. Math. Anal. Appl., 248:1 (2000), 233–246 | DOI | MR | Zbl

[43] Allan Gut, Aurel Spătaru, “Precise asymptotics in the law of the iterated logarithm”, Ann. Probab., 28:4 (2000), 1870–1883 | DOI | MR | Zbl

[44] Allan Gut, Ulrich Stadtmüller, “On the {H}su-{R}obbins-{E}rdős-{S}pitzer-{B}aum-{K}atz theorem for random fields”, J. Math. Anal. Appl., 387:1 (2012), 447–463 | DOI | MR | Zbl

[45] Allan Gut, Josef Steinebach, “Convergence rates in precise asymptotics”, J. Math. Anal. Appl., 390:1 (2012), 1–14 | DOI | MR | Zbl

[46] Allan Gut, Josef Steinebach, “Convergence rates in precise asymptotics {II}”, Ann. Univ. Sci. Budapest. Sect. Comput., 39 (2013), 95–110 | MR | Zbl

[47] Peter Hall, The bootstrap and {E}dgeworth expansion, Springer Series in Statistics, Springer-Verlag, New York, 1992 | MR

[48] Lajos Horváth, “A note on the rate of {P}oisson approximation of empirical processes”, Ann. Probab., 18:2 (1990), 724–726 | MR | Zbl

[49] Lajos Horváth, “Kac's representation from an asymptotic viewpoint”, J. Statist. Plann. Inference, 46:1 (1995), 27–45 | DOI | MR | Zbl

[50] Lajos Horváth, “Approximations for hybrids of empirical and partial sums processes”, J. Statist. Plann. Inference, 88:1 (2000), 1–18 | DOI | MR | Zbl

[51] Lajos Horváth, Piotr Kokoszka, Josef Steinebach, “Approximations for weighted bootstrap processes with an application”, Statist. Probab. Lett., 48:1 (2000), 59–70 | DOI | MR | Zbl

[52] P. L. Hsu, Herbert Robbins, “Complete convergence and the law of large numbers”, Proc. Nat. Acad. Sci. U.S.A., 33 (1947), 25–31 | DOI | MR | Zbl

[53] Barry R. James, “A functional law of the iterated logarithm for weighted empirical distributions”, Ann. Probability, 3:5 (1975), 762–772 | DOI | MR | Zbl

[54] M. Kac, “On deviations between theoretical and empirical distributions”, Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 252–257 | DOI | MR | Zbl

[55] J. Kiefer, “On {B}ahadur's representation of sample quantiles”, Ann. Math. Statist., 38 (1967), 1323–1342 | DOI | MR | Zbl

[56] J. Kiefer, “Deviations between the sample quantile process and the sample ${\rm df}$, Nonparametric {T}echniques in {S}tatistical {I}nference”, {P}roc. {S}ympos., {I}ndiana {U}niv., {B}loomington, {I}nd, 1969, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, Cambridge Univ. Pres, London, 1970,, 299–319 | MR

[57] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent {${\rm RV}$}'s and the sample {${\rm DF}$}. {I}”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32 (1975), 111–131 | DOI | MR | Zbl

[58] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent {RV}'s, and the sample {DF}. {II}”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34:1 (1976), 33–58 | DOI | MR | Zbl

[59] Michael R. Kosorok, Introduction to empirical processes and semiparametric inference, Springer Series in Statistics, Springer, New York, 2008 | MR | Zbl

[60] De Li Li, Fu Xing Zhang, Andrew Rosalsky, “A supplement to the {B}aum-{K}atz-{S}pitzer complete convergence theorem”, Acta Math. Sin. (Engl. Ser.), 23:3 (2007), 557–562 | DOI | MR | Zbl

[61] Péter Major, “A note on the approximation of the uniform empirical process”, Ann. Probab., 18:1 (1990), 129–139 | DOI | MR

[62] Péter Major, 2000 Notes available from (2000) http://www.renyi.hu/~major/probability/empir.html

[63] Bryan F. J. Manly, Randomization, bootstrap and {M}onte {C}arlo methods in biology, third ed., Chapman Hall/CRC Texts in Statistical Science Series, Chapman Hall/CRC, Boca Raton, FL, 2007 | MR

[64] David M. Mason, “Some characterizations of almost sure bounds for weighted multidimensional empirical distributions and a {G}livenko-{C}antelli theorem for sample quantiles”, Z. Wahrsch. Verw. Gebiete, 59:4 (1982), 505–513 | DOI | MR | Zbl

[65] David M. Mason, Michael A. Newton, “A rank statistics approach to the consistency of a general bootstrap”, Ann. Statist., 20:3 (1992), 1611–1624 | DOI | MR | Zbl

[66] David M. Mason, Willem R. van Zwet, “A refinement of the {KMT} inequality for the uniform empirical process”, Ann. Probab., 15:3 (1987), 871–884 | DOI | MR | Zbl

[67] Yan-Jiao Meng, “General laws of precise asymptotics for sums of random variables”, J. Korean Math. Soc., 49:4 (2012), 795–804 | MR | Zbl

[68] Ja. Ju. Nikitin, “Limit distributions and comparative asymptotic efficiency of the {K}olmogorov-{S}mirnov statistics with a random index”, 202, Problems in the theory of probability distributions, III, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 55 (1976), 185–194 | MR | Zbl

[69] Valentin V. Petrov, “Limit theorems of probability theory”, Sequences of independent random variables, Oxford Science Publications, Oxford Studies in Probability, 4 (1995), The Clarendon Press, Oxford University Press, New York | MR

[70] Galen R. Shorack, “Bootstrapping robust regression”, Comm. Statist. A—Theory Methods, 11:9 (1982), 961–972 | DOI | MR | Zbl

[71] Galen R. Shorack, Jon A. Wellner, Empirical processes with applications to statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley Sons, Inc., New York, 1986 | MR

[72] Winfried Stute, “The oscillation behavior of empirical processes”, Ann. Probab., 10:1 (1982), 86–107 | DOI | MR | Zbl

[73] Aad W. van der Vaart, Jon A. Wellner, Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996 | MR | Zbl

[74] Jon A. Wellner, “Limit theorems for the ratio of the empirical distribution function to the true distribution function”, Z. Wahrsch. Verw. Gebiete, 45:1 (1978), 73–88 | DOI | MR | Zbl

[75] Qing-pei Zang, “Precise asymptotics for complete moment convergence of {$U$}-statistics of i.i.d. random variables”, Appl. Math. Lett., 25:2 (2012), 120–127 | MR | Zbl

[76] Qing Pei Zang, Wei Huang, “A general law of moment convergence rates for uniform empirical process”, Acta Math. Sin. (Engl. Ser.), 27:10 (2011), 1941–1948 | MR | Zbl

[77] Yong Zhang, Xiao-Yun Yang, “Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process”, Statist. Probab. Lett., 78:9 (2008), 1051–1055 | DOI | MR | Zbl

[78] Ričardas Zitikis, “The {V}ervaat process”, Asymptotic methods in probability and statistics ({O}ttawa, {ON}, 1997), North-Holland, Amsterdam, 1998, 667–694 | MR