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Mathematical Foundations of Infinite-Dimensional Statistical Models
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  • Cited by 4
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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Belomestny, Denis Hübner, Tobias Krätschmer, Volker and Nolte, Sascha 2019. Minimax theorems for American options without time-consistency. Finance and Stochastics, Vol. 23, Issue. 1, p. 209.

    Trabs, Mathias 2018. Bayesian inverse problems with unknown operators. Inverse Problems, Vol. 34, Issue. 8, p. 085001.

    Coca, Alberto J. 2018. Efficient nonparametric inference for discretely observed compound Poisson processes. Probability Theory and Related Fields, Vol. 170, Issue. 1-2, p. 475.

    Kerkyacharian, Gerard Ogawa, Shigeyoshi Petrushev, Pencho and Picard, Dominique 2018. Regularity of Gaussian Processes on Dirichlet Spaces. Constructive Approximation, Vol. 47, Issue. 2, p. 277.

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Book description

In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood estimators and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory, and on the basic theory of function spaces. The theory of statistical inference in such models - hypothesis testing, estimation and confidence sets - is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, but also Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding, and adaptive inference for self-similar functions.

Reviews

‘Finally - a book that goes all the way in the mathematics of nonparametric statistics. It is reasonably self-contained, despite its depth and breadth, including accessible overviews of the necessary analysis and approximation theory.’

Aad van der Vaart - Universiteit Leiden

‘This remarkable book provides a detailed account of a great wealth of mathematical ideas and tools that are crucial in modern statistical inference, including Gaussian and empirical processes (where the first author, Evarist Giné, was one of the key contributors), concentration inequalities and methods of approximation theory. Building upon these ideas, the authors develop and discuss a broad spectrum of statistical applications such as minimax lower bounds and adaptive inference, nonparametric likelihood methods and Bayesian nonparametrics. The book will be exceptionally useful for a great number of researchers interested in nonparametric problems in statistics and machine learning, including graduate students.’

Vladimir Koltchinskii - Georgia Institute of Technology

‘This is a very welcome contribution. The wealth of material on the empirical processes and nonparametric statistics is quite exceptional. It is a masterly written treatise offering an unprecedented coverage of the classical theory of nonparametric inference, with glimpses into advanced research topics. For the first time in the monographic literature, estimation, testing and confidence sets are treated in a unified way from the nonparametric perspective with a comprehensive insight into adaptation issues. A delightful major reading that I warmly recommend to anyone wanting to explore the mathematical foundations of these fields.’

Alexandre Tsybakov - ENSAE ParisTech

'This is a remarkably comprehensive, detailed and rigorous treatment of mathematical theory for non-parametric and high-dimensional statistics. Special emphasis is on density and regression estimation and corresponding confidence sets and hypothesis testing. The minimax paradigm and adaptivity play a key role.'

Natalie Neumeyer Source: MathSciNet

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References
Adamczak, R.Moment inequalities for U-statistics. Ann. Probab. 34 (2006), 2288–314.
Adamczak, R.A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electronic J. Probab. 34 (2008), 1000–34.
Akaike, H.Approximation to the density function. Ann. Stat. Math. Tokyo 6 (1954), 127–32.
Albin, J. M. P., and Choi, H.A new proof of an old result by Pickands. Electronic Comm. Probab. 15 (2010), 339–45.
Alexander, K.Probability inequalities for empirical processes and a law of the iterated logarithm. Ann.Probab. 12 (1984), 1041–67
Alexander, K. S.The central limit theorem for empirical processes on Vapnik-Cervonenkis classes. Ann. Probab. 15 (1987), 178–203.
Andersen, N. T.The Calculus of Non-measurable Functions and Sets. (Various Publications Series No. 36). Aarhus Universitet, Matematisk Institut, Aarhus, Denmark, 1985.
Andersen, N. T., and Dobrić, V.The central limit theorem for stochastic processes. Ann. Probab. 15 (1987), 164–7.
Andersen, N. T., Giné, E., and Zinn, J.The central limit theorem and the law of iterated logarithm for empirical processes under local conditions. Probab. Theory Related Fields 77 (1988), 271–305.
Anderson, T. W.The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc. 6 (1955), 170–6.
Arcones, M. A., and Giné, E.Limit theorems for U-processes. Ann. Probab. 21 (1993), 1494–542.
Aronszajn, N.Theory of reproducing kernels. Trans. Am. Math. Soc. 68 (1950), 337–404.
Baernstein, II. A., and Taylor, B. A.Spherical rearrangements, subharmonic functions, and *-functions in n-space. Duke Math. J. 43 (1976), 245–68.
Balabdaoui, F., Rufiback, K., and Wellner, J.Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Stat. 37 (2009), 1299–331.
Balabdaoui, F., and Wellner, J.A Kiefer-Wolfowitz theorem for convex densities. IMS Lectures Notes 55 (2007), 1–31.
Baldi, P., and Roynette, B.Some exact equivalents for Brownian motion in Hölder norms. Probab. Theory Rel. Fields 93 (1992), 457–84.
Ball, K. Isometric problems in lp and sections of convex sets. Ph.D. thesis, University of Cambridge, 1986.
Ball, K., and Pajor, A.The entropy of convex bodies with ‘few’ extreme points. London Mathematical Society Lecture Note Series 158 (1990), 25–32 [Geometry of Banach spaces, Proceedings of a Conference, Strobl, Austria, 1999. P. F. X. Müller and W. Schachermayer, eds.].
Baraud, Y.Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 (2002), 577–606.
Baraud, Y.Confidence balls in Gaussian regression. Ann. Stat. 32 (2004), 528–51.
Baraud, Y.Estimator selection with respect to Hellinger-type risks. Probab. Theory Relat. Fields 151 (2011), 353–401.
Baraud, Y., Huet, S., and Laurent, B.Adaptive tests of linear hypotheses by model selection. Ann. Stat. 31 (2003), 225–51.
Baraud, Y., Huet, S., and Laurent, B.Testing convex hypotheses on the mean of a Gaussian vector: application to testing qualitative hypotheses on a regression function. Ann. Stat. 33 (2005), 214–57.
Barron, A., Birgé, L., and Massart, P.Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999), 301–413.
Barron, A., Schervish, M. J., and Wasserman, L.The consistency of posterior distributions in nonparametric problems. Ann. Stat. 27 (1999), 536–61.
Bartlett, P., Boucheron, S., and Lugosi, G.Model selection and error estimation. Mach. Learn. 48 (2002), 85–113.
Bass, R. F.Law of the iterated logarithm for set-indexed partial sum processes with finite variance. Z. Wahrsch. Verw. Gebiete 70 (1985), 591–608.
Beckner, W.Inequalities in Fourier analysis. Ann. Math. 102 (1975), 159–82.
Bednorz, W., and Latała, R.On the boundedness of Bernoulli processes. Ann. Math. 180 (2014), 1167–203.
Beers, G.Topologies on Closed and Closed Convex Sets. Klüwer, Dordrecht, 1993.
Beirlant, J., and Mason, D. M.On the asymptotic normality of Lp-norms of empirical functionals. Math. Methods Stat. 4 (1995), 1–19.
Bennett, G.Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57 (1962), 33–45.
Benyamini, Y.Two point symmetrization, the isoperimetric inequality on the sphere and some applications. Longhorn Notes, Texas Functional Analysis Seminar 1983–1984. University of Texas, Austin, 1984, pp. 53–76.
Beran, R., and Dümbgen, L.Modulation of estimators and confidence sets. Ann. Stat. 26 (1998), 1826–56.
Bergh, J., and Löfström, J.Interpolation Spaces. Springer, Berlin, 1976.
Berman, S. M.Limit theorems for the maximum term in stationary sequences. Ann. Math. Stat. 35 (1964), 502–16.
Berman, S. M.Excursions above high levels for stationary Gaussian processes. Pacific J. Math. 36 (1971), 63–79.
Berman, S. M.Asymptotic independence of the number of high and low level crossings of stationary Gaussian processes. Ann. Math. Stat. 42 (1971a), 927–45.
Besov, O. V.On a family of function spaces: Embedding theorems and applications. Dokl. Akad. Nauk SSSR 126 (1959), 1163–65 (in Russian).
Besov, O. V.On a family of function spaces in connection with embeddings and extensions. Trudy Mat. Inst. Steklov 60 (1961), 42–81 (in Russian).
Bickel, P. and Ritov, Y.Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya 50 (1988), 381–93.
Bickel, P. J., and Ritov, Y.Nonparametric estimators that can be ‘plugged-in’. Ann. Stat. 31 (2003), 1033–53.
Bickel, P. J., and Rosenblatt, M.On some global measures of the deviations of density function estimates. Ann. Stat. 1 (1973), 1071–95; correction, ibid. 3, p. 1370.
Billingsley, P., and Topsoe, F.Uniformity in weak convergence. Z. Wahrsch. Verw. Gebiete 7 (1967), 1–16.
Birgé, L.Approximation dans les espace métriques et théorie de l'estimation. Z. für Wahrscheinlichkeitstheorie und Verw. Geb. 65 (1983), 181–238.
Birgé, L.Sur un théoréme de minimax et son application aux tests. Probab. Math. Stat. 3 (1984), 259–82.
Birgé, L.Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Ann. Inst. H. Poincaré B 20 (2006), 201–23.
Birgé, L.Robust tests for model selection. IMS Collections 9 (2012), 47–64.
Birgé, L., and Massart, P.Rates of convergence for minimum contrast estimators. Probab. Theory Rel. Fields 97 (1993), 113–50.
Birgé, L., and Massart, P.Estimation of integral functionals of a density. Ann. Stat. 23 (1995), 11–29.
Birgé, L., and Massart, P.Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 (1998), 329–75.
Birgé, L., and Massart, P.Gaussian model selection. J. Eur. Math. Soc. 3 (2001), 203–68.
Birgé, L., and Massart, P.Minimal penalties for Gaussian model selection. Probab. Theory Related Fields 138 (2007), 33–73.
Birman, M. S., and Solomjak, M. Z.Piecewise polynomial approximations of functions of the classes Wαp Math. Sb. 73 (1967), 331–55 (in Russian).
Blum, J. R.On the convergence of empirical distribution functions. Ann. Math. Stat. 26 (1955), 527–9.
Bobkov, S. G., and Ledoux, M.From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. and Funct. Analysis 10 (2000), 1028–52
Bogachev, V. L.Gaussian Measures. American Mathematical Society, Providence, RI, 1998.
Bonami, A.Étude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier 20 (1970), 335–402.
Borell, C.The Brunn-Minkowski inequalty in Gauss space. Invent. Math. 30 (1975), 207–16.
Borell, C.Gaussian Radon measures on locally convex spaces. Math. Scand. 38 (1976), 265–84.
Borell, C.On the integrability of Banach space valued Walsh Polynomials. Seminaire de Probabilités XIII, Lect. Notes in Math. 721 (1979), 1–3.
Borisov, I. S.Some limit theorems for empirical distributions. Abstracts of Reports, Third Vilnius Conf. Probab. Theory Math. Stat. 1 (1981), 71–2.
Borovkov, A., and Mogulskii, A.On probabilities of small deviations for stochastic processes. Siberian Adv. Math. 1 (1991), 39–63.
Boucheron, S., Bousquet, O., Lugosi, G., and Massart, P.Moment inequalities for functions of independent random variables. Ann. Prob. 33 (2005), 514–60.
Boucheron, S., Lugosi, G., and Massart, P.A sharp concentration inequality with applications. Random Structures and Algorithms 16 (2000), 277–92.
Boucheron, S., Lugosi, G., and Massart, P.Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford, UK, 2013.
Bourdaud, G., de Cristoforis, M. L., and Sickel, W.Superposition operators and functions of bounded p-variation. Rev. Mat. Iberoam. 22 (2006), 455–87.
Bousquet, O.Concentration inequalities for sub-additive functions using the entropy method. In Stochastic Inequalities and Applications, E., Giné, C., Houdré, and D., Nualart, eds. Birkhäuser, Boston, 2003, pp. 180–212.
Bretagnolle, J., and Huber, C.Estimation des densités: risque minimax. Z. für Wahrscheinlichkeitstheorie und verw. Geb. 47 (1979), 199–37.
Brown, L. D., Cai, T. T., Low, M., and Zhang, C. H.Asymptotic equivalence theory for nonparametric regression with random design. Ann. of Stat. 30 (2002), 688–707.
Brown, L. D., and Low, M.Asymptotic equivalence of nonparametric regression and white noise. Ann. Stat. 24 (1996), 2384–98.
Brown, L. D., and Zhang, C. H.Asymptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2. Ann. Stat. 26 (1998), 279–87.
Bull, A. D.Honest adaptive confidence bands and self-similar functions. Elect. J. Stat. 6 (2012), 1490–516.
Bull, A. D.A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets. Construct. Approx. 37 (2013), 295–309.
Bull, A. D., and Nickl, R.Adaptive confidence sets in L2. Probability Theory and Related Fields 156 (2013), 889–919.
Bunea, F., Tsybakov, A. B., and Wegkamp, M.Aggregation for Gaussian regression. Ann. Stat. 35 (2007), 1674–97.
Butucea, C., Matias, C., and Pouet, C.Adaptive goodness-of-fit testing from indirect observations. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), 352–72.
Butucea, C., and Tsybakov, A. B.Sharp optimality in density deconvolution with dominating bias, I (Russian summary). Teor. Veroyatn. Primen. 52 (2007), 111–28; trans. in Theory Probab. Appl. 52 (2008a), 24–39.
Butucea, C., and Tsybakov, A. B.Sharp optimality in density deconvolution with dominating bias, II (Russian summary). Teor. Veroyatn. Primen. 52 (2007), 336–49; trans. in Theory Probab. Appl. 52 (2008b), 237–49.
Cai, T. T., and Low, M. G.An adaptation theory for nonparametric confidence intervals. Ann. Stat. 32 (2004), 1805–40.
Cai, T. T., and Low, M. G.On adaptive estimation of linear functionals, Ann. Stat. 33 (2005), 2311–43.
Cai, T. T., and Low, M.Nonquadratic estimators of a quadratic functional. Ann. Stat. 33 (2005), 2930–56.
Cai, T. T., and Low, M. G.Adaptive confidence balls. Ann. Stat. 34 (2006), 202–28.
Cai, T. T., Low, M. G., and Ma, Z.Adaptive confidence bands for nonparametric regression functions. J. Amer. Stat. Assoc. 109 (2014), 1054–70.
Cai, T. T., Low, M. G., and Xia, Y.Adaptive confidence intervals for regression functions under shape constraints. Ann. Stat. 41 (2013), 722–50.
Cameron, R. H., and Martin, W. T.Transformations of Wiener integrals under translations. Ann. Math. 45 (1944), 386–96.
Cantelli, F. P.Sulla determinazione empirica delle leggi di probabilit;á. Giorn. Ist. Ital. Attuari 4 (1933), 421–4.
Carl, B.Metric entropy of convex hulls in Hilbert spaces. Bull. London Math. Soc. 29 (1997), 452–8.
Carpentier, A.Honest and adaptive confidence sets in Lp. Elect. J. Stat. 7 (2013), 2875–2923.
Carpentier, A.Testing the regularity of a smooth signal (2014). Bernoulli, 21 (2015), 465–88.
Castillo, I.On Bayesian supremum norm contraction rates, Ann. Stat. 42 (2014), 2058–91.
Castillo, I., and Nickl, R.Nonparametric Bernstein-von Mises theorems in Gaussian white noise. Ann. Stat. 41 (2013), 1999–2028.
Castillo, I., and Nickl, R.On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures. Ann. Stat. 42 (2014), 1941–69.
Cavalier, L.Nonparametric statistical inverse problems. Inverse Problems 24 (2008).
Cavalier, L., and Tsybakov, A. B.Penalized blockwise Stein's method, monotone oracles and sharp adaptive estimation. Math. Meth. Stat. 10 (2001), 247–82.
Čencov, N. N.A bound for an unknown distribution density in terms of the observations. Dokl. Akad. Nauk. SSSR 147 (1962), 45–8.
Cencov, N. N. (1972) Statistical Decision Rules and Optimal Inference. Nauka, Moscow. English translation in Translations of Mathematical Monographs, 53, AMS, Providence, RI, 1982.
Chernozhukov, V., Chetverikov, D., and Kato, K.Gaussian approximation of suprema of empirical processes. Ann. Stat. 42 (2014), 1564–97.
Chernozhukov, V., Chetverikov, D., and Kato, K.Anti-concentration and honest adaptive confidence bands. Ann. Stat. 42 (2014a), 1787–818.
Chevet, S.Mesures de Radon sur Rn et mesures cylindriques. Ann. Fac. Sci. Univ. Clermont 43 (1970), 91–158.
Ciesielski, Z., Kerkyacharian, G., and Roynette, B.Quelques espaces fonctionnels associeś à des processus Gaussiens. Studia Math. 107 (1993), 171–204.
Claeskens, G., and van Keilegom, I.Bootstrap confidence bands for regression curves and their derivatives. Ann. Stat. 31 (2003), 1852–84.
Cohen, A., Daubechies, I., and Vial, P.Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 (1993), 54–84.
Coulhon, T., Kerkyacharian, G., and Petrushev, P.Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18 (2012), 995–1066.
Cox, D. D.An analysis of Bayesian inference for nonparametric regression. Ann. Stat. 21 (1993), 903–23.
Csörgö, M., and Horv;áth, L.Central limit theorems for Lp-norms of density estimators. Z. Wahrscheinlichkeitstheorie Verw. Geb. 80 (1988), 269–91.
Dalalyan, A., and Reiß, M.Asymptotic statistical equivalence for scalar ergodic diffusions. Probability Theory and Related Fields, 134 (2006), 248–82.
Daubechies, I.Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), 909–96.
Daubechies, I.Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, 1992.
Davies, E. B., and Simon, B.Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 (1984), 335–95.
Davies, P. L., and Kovac, A.Local extremes, runs, strings and multiresolution. Ann. Stat. 29 (2001), 1–65.
Davies, P. L., Kovac, A., and Meise, M.Nonparametric regression, confidence regions and regularization. Ann. Stat. 37 (2009), 2597–625.
de Acosta, A.Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Prob. 11 (1983), 78–101.
DeHardt, J.Generalizations of the Glivenko-Cantelli theorem. Ann. Math. Stat. 42 (1971), 2050–5.
Deheuvels, P.Uniform limit laws for kernel density estimators on possibly unbounded intervals. In Recent Advances in Reliability Theory: Methodology, Practice and Inference, N., Limnios and M., Nikulin, eds., Birkhäuser, Boston, 2000, pp. 477–92.
de la Peña, V., and Giné, E.Decoupling, from Dependence to Independence. Springer, Berlin, 1999.
DeVore, R., and Lorentz, G. G.Constructive Approximation. Springer, Berlin, 1993.
Devroye, L. ACourse in Density Estimation. Birkhäuser, Boston, 1987.
Devroye, L.Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics, G., Roussas, ed. NATO ASI Series. Kluwer, Dordrecht, 1991, pp. 31–44.
Devroye, L.A note on the usefulness of super kernels in density estimation. Ann. Stat. 20 (1992), 2037–56.
Devroye, L., and Lugosi, G.Combinatorial Methods in Density Estimation. Springer, New York, 2001.
Donoho, D. L., and Johnstone, I. M.Adapting to unknown smoothness via wavelet shrinkage, J. Am. Stat. Assoc. 90 (1995), 1200–24.
Donoho, D. L., and Johnstone, I. M.Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998), 879–921.
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D.Wavelet shrinkage: asymptopia? J. R.Stat. Soc. B 57 (1995), 301–69.
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D.Density estimation by wavelet thresholding, Ann. Stat. 24 (1996), 508–39.
Donoho, D., and Nussbaum, M.Minimax quadratic estimation of a quadratic functional. J. Complexity 6 (1990), 290–323.
Donsker, M. D.Justification and extension of Doob's heuristic approach to the Komogorov-Smirnov theorems. Ann. Math. Stat. 23 (1952), 277–81.
Doob, J. L.Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Stat. 20 (1949), 393–403.
Doob, J. L.Application of the theory of martingales. In Le Calcul des Probabilités et ses Applications (1949a), pp. 23–7. Colloques Internationaux du Centre National de la Recherche Scientifique, no. 13, Centre National de la Recherche Scientifique, Paris.
Doss, C., and Wellner, J. Global rates of convergence of the MLEs of logconcave and s-concave densities, arxiv preprint (2015).
Dudley, R. M.Weak convergences of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 (1966), 109–26.
Dudley, R.M.The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 (1967), 290–330.
Dudley, R. M.Measures on non-separable metric spaces. Illinois J. Math. 11 (1967a), 449–53.
Dudley, R. M.Sample functions of the Gaussian process. Ann. Probab. 1 (1973), 66–103.
Dudley, R. M.Central limit theorems for empirical processes. Ann. Probab. 6 (1978), 899–929.
Dudley, R. M.A course on empirical processes. Lecture Notes in Math. 1097 (1984) 1–142.
Dudley, R. M.An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions, (Probability in Banach spaces, V). Lecture Notes in Math. 1153 (1985), 141–78.
Dudley, R. M.Universal Donsker classes and metric entropy. Ann. Probab. 15 (1987), 1306–26.
Dudley, R. M.Nonlinear functionals of empirical measures and the bootstrap (Probability in Banach spaces, 7). Progr. Probab. 21 (1990), 63–82.
Dudley, R. M.Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20 (1992), 1968–82.
Dudley, R. M.Uniform Central Limit Theorems (1st ed. 1999), 2nd ed. Cambridge University Press, 2014.
Dudley, R. M.Real Analysis and Probability, 2nd ed. Cambridge University Press, 2002.
Dudley, R. M., Giné, E., and Zinn, J.Uniform and universal Glivenko-Cantelli classes. J. Theoret. Probab. 4 (1991), 485–510.
Dudley, R.M., and Kanter, M.Zero-one laws for stable measures. Proc. Am. Math. Soc. 45 (1974), 245–52.
Dudley, R. M., and Norvaisa, R.Concrete Functional Calculus. Springer, New York, 2011.
Dudley, R. M., and Philipp, W. Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62 (1983), 509–52.
Dümbgen, L.Optimal confidence bands for shape-restricted curves. Bernoulli 9 (2003), 423–49.
Dümbgen, L., and Rufibach, K.Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency. Bernoulli 15 (2009), 40–68.
Dümbgen, L., and Spokoiny, V.Multiscale testing of qualitative hypotheses. Ann. Stat. 29 (2001), 124–52.
Durot, C., Kulikov, V. N., and Lopuhaä, H. P.The limit distribution of the L8-error of Grenander-type estimators. Ann. Stat. 40 (2012), 1578–608.
Durrett, R.Probability: Theory and Examples. Duxbury, Pacific Grove, CA, 1996.
Durst, M., and Dudley, R. M.Empirical processes, Vapnik-C? ervonenkis classes and Poisson processes. Probab. Math. Stat. 1 (1981), 109–15.
Dvoretzky, A., Kiefer, J., and Wolfowitz, J.Asymptotic minimax character of a sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 33 (1956), 642–69.
Edmunds, D. E., and Triebel, H.Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, 1996.
Efroimovich, S. Y.Nonparametric estimation of a density of unknown smoothness. Theory Probability and Applications 30 (1985), 524–34.
Efromovich, S. Y.Adaptive estimation of and oracle inequalities for probability densities and characteristic functions. Ann. Stat. 36 (2008), 1127–55.
Efroimovich, S. Y., and Pinsker, M. S.Learning algorithm for nonparametric filtering, Automation and Remote Control 11 (1984), 1434–40.
Eggermont, P. P. B., and LaRiccia, V. N.Maximum Penalized Likelihood Estimation, Vol. 1; Density Estimation. Springer, New York, 2000.
Einmahl, U., and Mason, D. M.Some universal results on the behavior of increments of partial sums. Ann. Probab. 24 (1996), 1388–407.
Einmahl, U., and Mason, D. M.An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 (2000), 1–37.
Einmahl, U., and Mason, D.Uniform in bandwidth consistency of kernel type function estimators. Ann. Stat. 33 (2005), 1380–403.
Ermakov, M. S.Minimax detection of a signal in a white Gaussian noise, Theory Probab. Appl. 35 (1990), 667–79.
Fan, J.Global behavior of deconvolution kernel estimates. Stat. Sinica 1 (1991), 541–51.
Fan, J.Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Stat. 21 (1993), 600–10.
Fernique, X.Une demonstration simple du théorème de R.M. Dudley et M. Kanter sur les lois zéro-un pour les mésures stables. Lecture Notes in Math. 381 (1974), 78–9.
Fernique, X.Regularité des trajectoires des fonctions aléatoires gaussiennes. Lecture Notes in Math. 480 (1975), 1–96.
Fernique, X.Fonctions Aléatoires Gaussiennes, Vecteurs Aléatoires Gaussiens. Les Publications CRM, Montreal, 1997.
Figiel, T., Lindenstrauss, J., and Milman, V. D.The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 53–94.
Fisher, R. A.On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. A. 222 (1922), 309–68.
Fisher, R. A.Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22 (1925a), 700–25.
Fisher, R.A.Statistical Methods for Research Workers. Oliver and Body, Edinburgh, 1925b.
Folland, G. B.Real Analysis: Modern Techniques and Their Applications. Wiley, New York, 1999.
Frazier, M., Jawerth, B., and Weiss, G.Littlewood-Paley Theory and the Study of Function Spaces. AMS monographs, Providence, RI, 1991.
Freedman, D. A.On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Stat. 34 (1963), 1386–403.
Freedman, D. A.On the Bernstein-von Mises theorem with infinite-dimensional parameters. Ann. Stat. 27 (1999), 1119–40.
Fromont, M., and Laurent, B.Adaptive goodness-of-fit tests in a density model. Ann. Stat. 34 (2006), 680–720.
Gach, F., and Pötscher, B. M.Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators. Math. Methods Stat. 20 (2011), 288–326.
Gänssler, P., and Stute, W.Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Probab. 7 (1979), 193–243.
Gayraud, G., and Pouet, C.Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133 (2005), 531–58.
Gardner, R. J.The Brunn-Minkowski inequality. Bull AMS 39 (2002), 355–405.
Gauß, C. F.Theoria Motus Corporum Coelestium. Perthes, Hamburg, 1809; reprint of the original at Cambridge University Press, 2011.
Gelfand, I. M., and Vilenkin, N. Y.Les distributions, Tome 4: Applications de l'analyse harmonique. Dunod, Paris, 1967 (translated from Russian).
Geller, D., and Pesenson, I. Z.Band-limited localized parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21 (2011), 334–71.
Genovese, C., and Wasserman, L.Adaptive confidence bands. Ann. Stat. 36 (2008), 875–905.
Ghosal, S.The Dirichlet process, related priors and posterior asymptotics. In Bayesian Nonparametrics, Cambridge University Press, 2010, pp. 35–79.
Ghosal, S., Ghosh, J. K., and van der Vaart, A. W.Convergence rates for posterior distributions. Ann. Stat. 28 (2000), 500–31.
Ghosal, S., and van der Vaart, A.W.Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Stat. 35 (2007), 192–223.
Giné, E.Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Stat. 3 (1975), 1243–66.
Giné, E.Empirical processes and applications: an overview. Bernoulli 2 (1996), 1–28.
Giné, E., and Guillou, A.On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervalsAnn. Inst. H. Poincaré Probab. Stat. 37 (2001), 503–22.
Giné, E., and Guillou, A.Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Prob. Stat. 38 (2002), 907–21.
Giné, E., Güntürk, C. S., and Madych, W. R.On the periodized square of L2 cardinal splines. Exp. Math. 20 (2011), 177–88.
Giné, E., and Koltchinskii, V.Concentration inequalities and asymptotic results for ratio type empirical processes. Ann. Probab. 34 (2006), 1143–216.
Giné, E., Koltchinskii, V., and Sakhanenko, L.Kernel density estimators: convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 (2004), 167–98.
Giné, E., Koltchinskii, V., and Wellner, J.Ratio limit theorems for empirical processes. In Stochastic Inequalities and Applications, Vol. 56. Birkhäuser, Basel, 2003, pp. 249–78.
Giné, E., Koltchinskii, V., and Zinn, J.Weighted uniform consistency of kernel density estimators. Ann. Probab. 32 (2004), 2570–605.
Giné, E., Latala, R., and Zinn, J.Exponential and moment inequalities for U-statistics. In High Dimensional Probability, Vol. 2. Birkhäuser, Boston, 2000, pp. 13–38.
Giné, E., and Madych, W.On wavelet projection kernels and the integrated squared error in density estimation. Probab. Lett. 91 (2014), 32–40.
Giné, E., and Mason, D. M.On the LIL for self-normalized sums of i.i.d. random variables. J. Theoret. Probab. 11 (1998), 351–70.
Giné, E., and Mason, D. M.On local U-statistic processes and the estimation of densities of functions of several sample variables. Ann. Stat. 35 (2007), 1105–45.
Giné, E., Mason, D., and Zaitsev, A.The L1-norm density estimator process. Ann. Probab. 31 (2003), 719–68.
Giné, E., and Nickl, R.Uniform central limit theorems for kernel density estimators. Probab. Theory Relat. Fields 141 (2008), 333–87.
Giné, E., and Nickl, R. Asimple adaptive estimator of the integrated square of a density. Bernoulli 14 (2008a), 47–61.
Giné, E., and Nickl, R.Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 (2009), 1605–46.
Giné, E., and Nickl, R.An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields 143 (2009a), 569–96.
Giné, E., and Nickl, R.Confidence bands in density estimation. Ann. Stat. 38 (2010), 1122–70.
Giné, E., and Nickl, R.Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli 16 (2010a), 1137–63.
Giné, E., and Nickl, R.Rates of contraction for posterior distributions in Lr-metrics, 1 = r=8. Ann. Stat. 39 (2011), 2883–911.
Giné, E., and Zinn, J.Central limit theorems and weak laws of large numbers in certain Banach spaces. Z. Warscheinlichkeitstheorie Verw. Geb. 62 (1983), 323–54.
Giné, E., and Zinn, J.Some limit theorems for empirical processes. Ann. Probab. 12 (1984), 929–89.
Giné, E., and Zinn, J.Lectures on the central limit theorem for empirical processes (probability and Banach spaces). Lecture Notes in Math. 1221 (1986), 50–113.
Giné, E., and Zinn, J.Empirical processes indexed by Lipschitz functions. Ann. Probab. 14 (1986a), 1329–38.
Giné, E., and Zinn, J.Bootstrapping general empirical measures. Ann. Probab. 18 (1990), 851–69.
Giné, E., and Zinn, J.Gaussian characterization of uniform Donsker classes of functions. Ann. Probab. 19 (1991), 758–82.
Glivenko, V. I.Sulla determiniazione empirica delle leggi di probabilit;á. Giorn. Ist. Ital. Attuari 4 (1933), 92–9.
Goldenshluger, A., and Lepski, O. V.Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality. Ann. Stat. 39 (2011), 1608–32.
Goldenshluger, A., and Lepski, O. V.On adaptive minimax density estimation on Rd. Probab. Theory Relat. Fields 159 (2014), 479–543.
Golubev, G. K.Adaptive asymptotically minimax estimates of smooth signals, Prob. Inf. Transm. 23 (1987), 57–67.
Golubev, G. K.Nonparametric estimation of smooth densities of a distribution in L2. Problems of Information Transmission, 28 (1992), 44–54.
Golubev, G. K., and Nussbaum, M.Adaptive spline estimates in a nonparametric regression model. Theory Prob. Appl. 37 (1992), 521–9.
Golubev, G. K., Nussbaum, M., and Zhou, H. H.Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Stat. 38 (2010), 181–214.
Grama, I., and Nussbaum, M.Asymptotic equivalence for nonparametric regression. Math. Methods Stat. 11 (2002), 1–36.
Grenander, U.On the theory of mortality measurement. Scand. Actuarial J. 2 (1956), 125–53.
Groeneboom, P.Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II. Berkeley, CA, 1985, pp. 539–55.
Groeneboom, P., Jongbloed, G., and Wellner, J. A.Estimation of a convex function: characterizations and asymptotic theory. Ann. Stat. 29 (2001), 1653–98.
Gross, L.Abstract Wiener spaces. In Proceedings of the Fifth Berkeley Symposium on Mathematics Statistics, and Probability, Vol. II: Contributions to Probability Theory, Part 1. University of California Press, Berkeley, 1967, pp. 31–42.
Gross, L.Logarithmic Sobolev inequalities. Am. J. Math. 97 (1975), 1061–83.
Haagerup, U.The best constants in the Khintchine inequality. Studia Math. 70 (1982), 231–83.
Haar, A.Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69 (1910), 331–71.
Hall, P.The rate of convergence of normal extremes. J. Appl. Probab. 16 (1979), 433–9.
Hall, P.Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Analysis 14 (1984), 1–16.
Hall, P.Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann. Stat. 20 (1992), 675–94.
Hanson, D. L., and Wright, F. T.A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 (1971), 1079–83.
Härdle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A.Wavelets, Approximation and Statistical Applications (Springer Lecture Notes in Statistics 129), Springer, New York, 1998.
Hardy, G. H., Littlewood, J. E., and Pölya, G.Inequalities. Cambridge University Press, 1967.
Haussler, D.Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-C? ervonenkis dimension. J. Comb. Theory 69 (1995), 217–32.
Hoeffding, V.Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58 (1963), 13–30.
Hoffmann, M., and Lepski, O. V.Random rates in anisotropic regression. Ann. Stat. 30 (2002), 325–96 (with discussion).
Hoffmann, M., and Nickl, R.On adaptive inference and confidence bands, Ann. Stat. 39 (2011), 2383–409.
Hoffmann, M., Rousseau, J., and Schmidt-Hieber, J.On adaptive posterior concentration rates. Ann. Stat. (2015) (in press).
Hoffmann-Jørgensen, J.Sums of independent Banach space valued random variables. Studia Math. 52 (1974), 159–86.
Hoffmann-Jørgensen, J.The law of large numbers for non-measurable and non-separable random elements. Asterisque 131 (1985), 299–356.
Hoffmann-Jørgensen, J.Stochastic Processes on Polish Spaces (Various Publications Series 39). Aarhus Universitet, Mathematisk Institute, 1991, Aarhus, Denmark.
Hoffmann-Jørgensen, J., Shepp, L. A., and Dudley, R. M.On the lower tail of Gaussian seminorms. Ann. Probab. 7 (1979), 319–42.
Houdré, C., and Reynaud-Bouret, P.Exponential inequalities with constants for U-statistics of order two. In Stochastic Inequalities and Applications (Progress in Probability 56). Birkhäuser, Boston, 2003, pp. 55–69.
Ibragimov, I. A., and Khasminskii, R. Z.On the estimation of an infinite-dimensional parameter in Gaussian white noise. Sov. Math. Dokl. 18 (1977), 1307–8.
Ibragimov, I. A., and Khasminskii, R. Z.On the nonparametric estimation of functionals. In Symposium in Asymptotic Statistics, J., Kozesnik, ed. Reidel, Dordrecht, 1978, pp. 42–52.
Ibragimov, I. A., and Khasminskii, R. Z.Statistical Estimation. Asymptotic Theory. Springer, New York, 1981.
Ibragimov, I. A., and Khasminskii, R. Z.Bounds for the risks of nonparametric regression estimates. Theory of Probability and its Applications, 27 (1982), 84–99.
Ibragimov, I. A., Nemirovskii, A. S., and Khasminskii, R. Z.Some problems of nonparametric estimation in Gaussian white noise. Theory of Probability and its Applications, 31 (1987), 391–406.
Ingster, Y. I.On the minimax nonparametric detection of a signal with Gaussian white noise. Prob. Inform. Transm. 28 (1982), 61–73 (in Russia).
Ingster, Y. I.Minimax testing of nonparametric hypotheses on a distribution density in the Lp-metrics, Theory Probab. Appl. 31 (1986), 333–7.
Ingster, Y. I.Asymptotically minimax hypothesis testing for nonparametric alternatives, Part I, II, and III. Math. Methods Stat. 2 (1993), 85–114, 171–89, 249–68.
Ingster, Y. I., and Suslina, I. A.Nonparametric Goodness-of-Fit Testing under Gaussian Models (Lecture Notes in Statistics). Springer, New York, 2003.
Jaffard, S.On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79 (2000), 525–52.
Jain, N. C., and Marcus, M. B.Central limit theorems for C(S)-valued random variables. J. Funct. Anal. 19 (1975), 216–31.
Johnson, W. B., Schechtman, G., and Zinn, J.Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 (1985), 234–53.
Juditsky, A., and Lambert-Lacroix, S.Nonparametric confidence set estimation. Math. Methods Stat. 12 (2003), 410–28.
Juditsky, A., Rigollet, P., and Tsybakov, A. B. Learning by mirror averaging, Ann. Stat. 36 (2008), 2183–206.
Kahane, J. P.Sur les sommes vectorielles Σ±un. Comptes Rendus Acad. Sci. Paris 259 (1964), 2577–80.
Kahane, J. P.Some Random Series of Functions. D. C. Heath, Lexington, MA, 1968.
Kakutani, S.On equivalence of infinite product measures. Ann. Math. 49 (1948), 214–24.
Kallianpur, G.AbstractWiener processes and their reproducing kernel Hilbert spaces. Zeits. Wahrsch. Verb. Geb. 17 (1971), 113–23.
Karhunen, K.Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 37 (1947), 1–79.
Katz, M. F., and Rootzén, H.On the rate of convergence for extremes of mean square differentiable stationary normal processes. J. Appl. Probab. 34 (1997), 908–23.
Kerkyacharian, G., Nickl, R., and Picard, D.Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Related Fields 153 (2012), 363–404.
Kerkyacharian, G., and Picard, D.Density estimation in Besov spaces. Stat. Probab. Letters 13 (1992), 15–24.
Khatri, C. G.On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Stat. 38 (1967), 1853–67.
Khinchin, A.Uber dyadische Brüche. Math. Zeits. 18 (1923), 109–16.
Kiefer, J., and Wolfowitz, J.Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), 73–85.
Klein, T.Une inégalité de concentration à gauche pour les processus empiriques. C. R. Math. Acad. Sci. Paris 334 (2002), 501–4.
Klein, T., and Rio, E.Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 (2005), 1060–77.
Klemelä, J.Sharp adaptive estimation of quadratic functionals. Probab. Theory Relat. Fields 134 (2006), 539–64.
Kolmogorov, A. N.Grundbegriffe der Wahrscheinlichkeitstheorie. Springer, Berlin, 1933.
Kolmogorov, A. N.Sulla determiniazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4 (1933a), 83–91.
Kolmogorov, A. N., and Tikhomirov, V. M.The ε-entropy and ε-capacity of sets in functional spaces. Am. Math. Soc. Trans 17 (1961), 277–364.
Koltchinskii, V. I.On the central limit theorem for empirical measures. Theor. Probab. Math. Stat. 24 (1981), 63–75.
Koltchinskii, V.Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 (2001), 1902–14.
Koltchinskii, V.Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Stat. 34 (2006), 2593–656.
Konstant, D. G., and Piterbarg, V. I.Extreme values of the cyclostationary Gaussian processes. J. Appl. Probab. 30 (1993), 82–97.
Korostelev, A. P., and Tsybakov, A. B.Minimax Theory of Image Reconstruction (Lecture Notes in Statistics 82). Springer, New York, 1993.
Kuelbs, J., and Li, W.Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 (1993), 133–57.
Kuelbs, J., Li, W., and Linde, W.The Gaussian measure of shifted balls. Probab. Theory Related Fields 98 (1994), 143–62.
Kullback, S.A lower bound for discrimination information in terms of variation. IEEE Trans. Inform. Theory 13 (1967), 126–7.
Kwapień, S., and Woyczynski, W.Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, 1992.
Latała, R.Estimation of moments of sums of independent random variables. Ann. Probab. 25 (1997), 1502–13.
Latała, R.Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006), 2315–31.
Latała, R., and Oleskiewicz, K.On the best constant in the Khinchin-Kahane inequality. Studia Math. 109 (1994), 101–4.
Laurent, B.Efficient estimation of integral functionals of a density. Ann. Stat. 24 (1996), 659–81.
Leadbetter, M. R., Lindgren, G., and Rootzén, H.Extremes and Related Properties of Random Sequences and Processes. Springer, New York, 1983.
Leahu, H.On the Bernstein–von Mises phenomenon in the Gaussian white noise model. Elect. J. Stat. 5 (2011), 474–4.
Le Cam, L. M.On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates. Univ. California Publ. Stat. 1 (1953), 277–329.
Le Cam, L. M.Convergence of estimates under dimensionality restrictions. Ann. Stat. 1 (1973), 38–53.
Le Cam, L.Asymptotic Methods in Statistical Decision Theory. Springer, New York, 1986.
Le Cam, L., and Yang, G. L.Asymptotics in Statistics: Some Basic Concepts. Springer, New York, 1990.
Ledoux, M.Isoperimetry and Gaussian analysis. Lecture Notes in Math. Lecture Notes in Statistics (1996), 165–294.
Ledoux, M.On Talagrand's deviation inequalities for product measures. ESAIM: Probab. Stat. 1 (1997), 63–87.
Ledoux, M.The Concentration of Measure Phenomenon. American Math. Soc., Providence, RI, 2001.
Ledoux, M., and Talagrand, M.Conditions d'integrabilité pour les multiplicateurs dans le TLC banachique. Ann. Probab. 14 (1986), 916–21.
Ledoux, M., and Talagrand, M.Comparison theorems, random geometry and some limit theorems for empirical processes. Ann. Probab. 17 (1989), 596–631.
Ledoux, M., and Talagrand, M.Probability in Banach Spaces. Springer-Verlag, Berlin, 1991.
Leindler, L.On a certain converse of Hölder's inequality. Acta Sci. Math. 34 (1973), 335–43.
Lepski, O. V.On a problem of adaptive estimation in Gaussian white noise, Theory Prob. Appl. 35 (1990), 454–66.
Lepski, O. V.How to improve the accuracy of estimation. Math. Meth. Stat. 8 (1999), 441–86.
Lepski, O. V.Multivariate density estimation under sup-norm loss: oracle approach, adaptation and independence structure. Ann. Stat. 41 (2013), 1005–34.
Lepski, O. V., Mammen, E., and Spokoiny, V.Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors. Ann. Stat. 25 (1997), 929–47.
Lepski, O. V., and Tsybakov, A. B.Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Related Fields 117 (2000), 17–48.
Levit, B. Ya.Asymptotically efficient estimation of nonlinear functionals. Prob. Inform. Transm. 14 (1978), 204–9.
Lévy, P.Problémes concrets d'analyse fonctionelle. Gauthier-Villars, Paris, 1951.
Li, K. C.Honest confidence regions for nonparametric regression, Ann. Stat. 17 (1989), 1001–8.
Li, W., and Linde, W.Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 (1999), 1556–78.
Li, W. V., and Shao, Q|-M.Gaussian processes: inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods (Handbook of Statistics), Vol. 19, C. R. Rao and D. Shanbhag, eds. Elsevier, New York, 2001, pp. 533–98.
Lijoi, A., and Prünster, I.Models beyond the Dirichlet process. In Bayesian Nonparametrics. 35–79, (Camb. Ser. Stat. Probab. Math.). Cambridge University Press, 2010, pp. 35–79.
Littlewood, J. E.On bounded bilinear forms in an infinite number of variables. Q. J. Math., Oxford Ser. 1 (1930), 164–74.
Littlewood, J. E., and Paley, R. E. A. C.Theorems on Fourier series and power series I, II.J. Lond. Math. Soc. 6 (1931), 230–3 (I); Proc. Lond. Math. Soc. 42 (1936), 52–89 (II).
Loéve, M.Probability Theory (Graduate Texts in Mathematics 46), Vol. II, 4th ed. Springer-Verlag, Berlin, 1978.
Lorentz, G. G., Golitscheck, M. V., and Makovoz, Y.Constructive Approximation: Advanced Problems. Springer, Berlin, 1996.
Lounici, K., and Nickl, R.Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 39 (2011) 201–31.
Love, E. R., and Young, L. C.Sur une classe de fonctionelles linéaires. Fund. Math. 28 (1937), 243–57.
Low, M. G.On nonparametric confidence intervals, Ann. Stat. 25 (1997), 2547–54.
Mallat, S.Multiresolution approximation and wavelet orthonormal bases of L2(R). Trans. Am. Math. Soc. 315 (1989), 69–87.
Marcus, D. J.Relationships between Donsker classes and Sobolev spaces, Z. Wahrscheinlichkeitstheorie verw. Gebiete 69 (1985), 323–30.
Marcus, M. B., and Shepp, L. A.Sample behavior of Gaussian processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970–1971), Vol. II: Probability Theory. University California Press, Berkeley, 1972, pp. 423–41.
Massart, P.The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990), 1269–83.
Massart, P.About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 (2000), 863–84.
Massart, P.Some applications of concentration inequalities in statistics. Ann. Fac. Sci. Toulouse Math. 9 (2000), 245–303.
Massart, P.Concentration Inequalities and Model Selection (Lectures from the 33rd Summer School on Probability Theory Held in Saint-Flour, July 6–23, 2003; Lecture Notes in Mathematics 1896). Springer, Berlin, 2007.
Meister, A.Deconvolution Problems in Nonparametric Statistics (Lecture Notes in Statistics 193). Springer, Berlin, 2009.
Maurer, A.Thermodynamics and concentration. Bernoulli 18 (2012), 434–54.
Mattila, P.Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995.
McDiarmid, C.On the method of bounded differences. In Surveys in Combinatorics. Cambridge University Press, 1989, pp. 148–88.
McDiarmid, C.Concentration. In Probabilistic Methods for Algorithmic DiscreteMathematics (Algorithms Combin, 16). Springer, Berlin, 1998, pp. 195–248.
Meyer, Y.Wavelets and Operators. Cambridge University Press, 1992.
Meyer, Y., Sellan, F., and Taqqu, M. S.Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion.J. Fourier Anal. Appl. 5 (1999), 465–94.
Montgomery-Smith, S.Comparison of sums of independent identically distributed random variables. Prob. Math. Stat. 14 (1994), 281–5.
Mourier, E.Lois de grandes nombres et théorie ergodique. C. R. Acad. Sci. Paris 232 (1951), 923–5.
Nadaraya, E. A.On estimating regression. Theory Probab. Appl., 9 (1964), 141–2.
Nemirovskii, A. S.Nonparametric estimation of smooth regression functions. Soviet J. of Computer and Systems Sciences, 23 (1985), 1–11.
Nemirovskii, A. S.On necessary conditions for the efficient estimation of functionals of a nonparametric signal which is observed in white noise. Theory of Probability and its Applications, 35 (1990), 94–103.
Nemirovski, A.Topics in Non-parametric Statistics. Ecole d'Ete de Probabilités de Saint-Flour XXVIII – 1998. Lecture Notes in Mathematics, v. 1738. Springer, New York (2000).
Nemirovskii A, S., Polyak, B. T., and Tsybakov, A. B.Rate of convergence of nonparametric estimators of maximum-likelihood type. Problems of Information Transmission, 21 (1985), 258–72.
Nickl, R.Empirical and Gaussian processes on Besov classes. In High Dimensional Probability IV (IMS Lecture Notes 51), E. Giné, V. Koltchinskii, W. Li, and J. Zinn, eds. Springer, Berlin, 2006, pp. 185–95.
Nickl, R.Donsker-type theorem for nonparametric maximum likelihood estimators. Probab. Theory Related Fields 138 (2007), 411–49; erratum (2008) ibid.
Nickl, R.Uniform central limit theorems for sieved maximum likelihood and trigonometric series estimators on the unit circle. In High Dimensional Probability V. Birkhaeuser, Boston, 2009, pp. 338–56.
Nickl, R., and Pötscher, B. M.Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. J. Theoret. Probab. 20 (2007), 177–99.
Nickl, R., and Reiß, M.A Donsker theorem for Lévy measures. J. Funct. Anal. 263 (2012), 3306–32.
Nickl, R., Reiß, M., Söhl, J., and Trabs, M.High-frequency Donsker theorems for Lévy measures. Probab. Theory Related Fields (2015).
Nickl, R., and Szabó|B. A sharp adaptive confidence ball for self-similar functions. Stoch. Proc. Appl., in press (2015).
Nickl, R., and van de Geer, S.Confidence sets in sparse regression, Ann. Stat. 41 (2013), 2852–76.
Nolan, ,D., and Pollard, D.U-processes: rates of convergence. Ann. Stat. 15 (1987), 780–99.
Nussbaum, M.Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat. 24 (1996), 2399–430.
Ossiander, M.A central limit theorem under metric entropy with L2 bracketing. Ann. Probab. 15 (1987), 897–919.
Ottaviani, G.Sulla teoria astratta del calcolo delle probabilitá proposita dal Cantelli. Giorn. Ist. Ital. Attuari 10 (1939), 10–40.
Oxtoby, J. C., and Ulam, S.On the existence of a measure invariant under a transformation. Ann. Math. 40 (1939), 560–6.
Paley, R. E. A. C., and Zygmund, A.On some series of functions, part 1. Proc. Cambridge Philos. Soc. 28 (1930), 266–72.
Paley, R. E. A. C., and Zygmund, A.A note on analytic functions on the unit circle. Proc. Cambridge Philos. Soc. 28 (1932), 266–72.
Parzen, E.On estimation of a probability density function and mode. Ann. Math. Stat. 33 (1962), 1065–76.
Pearson, K.On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. Series 5 50 (1900), 157–75.
Peetre, J.New Thoughts on Besov Spaces (Duke University Math. Series). Duke University Press, Durham, NC, 1976.
Pensky, M., and Vidakovic, B.Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 27 (1999), 2033–53.
Picard, D., and Tribouley, K.Adaptive confidence interval for pointwise curve estimation. Ann. Stat. 28 (2000), 298–335.
Pickands, J., III. Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Am. Math. Soc. 145 (1969), 75–86.
Pinelis, I.Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 (1994), 1679–706.
Pinsker, M. S.Information and Information Stability of Random Variables and Processes. Holden-Day, San Francisco, 1964.
Pisier, G.Remarques sur un résultat non publié de B. Maurey. In Séminaire d'Analyse Fonctionelle 1980–1981, Vols. 1–12. École Polytechnique, Palaiseau, France.
Pisier, G.Some applications of the metric entropy condition to harmonic analysis. Lecture Notes in Math. Vol. 995. Springer, Berlin, pp. 123–54.
Pisier, G.Probabilistic methods in the geometry of Banach spaces. Lecture Notes in Math., Vol. 1206. Springer, Berlin, 1986, 105–136.
Pisier, G.The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, 1989.
Piterbarg, V. I.Asymptotic Methods in the Theory of Gaussian Processes and Fields (Translations of Math. Monographs 148). AMS, Providence, RI, 1996.
Piterbarg, V. I., and Seleznjev, O.Linear interpolation of random processes and extremes of a sequence of Gaussian non-stationary processes. Technical report, Center for Stochastic Processes, North Carolina University; Chapel Hill, NC, 1994, p. 446.
Plackett, R. L.The discovery of the method of least squares. Biometrika 59 (1972), 239–51.
Pollard, D.Limit theorems for empirical processes. Z. Wahrsch. Verw. Gebiete 57 (1981), 181–95.
Pollard, D.A central limit theorem for empirical processes. J. Austral. Math. Soc., Ser. A 33 (1982), 235–48.
Pouet, C.Test asymptotiquement minimax pour une hypothése nulle composite dans le modéle de densité. C.R. Math. Acad. Sci. Paris 334 (2002), 913–16.
Prakasa Rao, B. L. S.Estimation of a unimodal density. Sankhya Ser. A 31 (1969), 23–36.
Prékopa, A.On logarithmically concave measures and functions. Acta Sci. Math. 33 (1972), 217–23.
Radulović, D., and Wegkamp, M.Uniform central limit theorems for pregaussian classes of functions. In High Dimensional Probability V: The Luminy Volume (Inst. Math. Stat. Collect. 5). IMS, Beachwood, OH, 2009, pp. 84–102.
Ray, K.Random Fourier series with applications to statistics. Part III Essay in Mathematics, University of Cambridge, 2010.
Ray, K.Bayesian inverse problems with non-conjugate priors, Elect. J. Stat. 7 (2013), 2516–49.
Ray, K. Bernstein–von Mises theorems for adaptive Bayesian nonparametric procedures, Preprint (2014), arxiv 1407.3397.
Reiß, M.Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Stat. 36 (2008), 1957–82.
Rhee, WanSoo, T.Central limit theorem and increment conditions. Stat. Probab. Lett. 4 (1986), 191–5.
Rio, E.Local invariance principles and their application to density estimation. Probab. Theory Related Fields 98 (1994), 21–45.
Rio, E.Inégalités de concentration pour les processus empiriques de classes de parties. Probab. Theory Related. Fields 119 (2001), 163–75.
Rio, E.Une inégalité de Bennett pour les maxima de processus empiriques. Ann. I. H. Poincaré Prob. 38 (2002), 1053–7.
Rio, E.Inegalités exponentielles et inegalités de concentration. Lectures at University Bordeaux Sud-Ouest, 2009.
Rio, E.Sur la function de taux dans les inegalités de Talagrand pour les processus empiriques. C. R. Acad. Sci. Paris Ser. I 350 (2012), 303–5.
Robins, J., and van der Vaart, A. W.Adaptive nonparametric confidence sets. Ann. Stat. 34 (2006), 229–53.
Rootzén, H.The rate of convergence of extremes of stationary normal sequences. Adv. Appl. Probab. 15 (1983), 54–80.
Rosenblatt, M.Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27 (1956), 832–7.
Salem, R., and Zygmund, A.Some properties of trigonometric series whose terms have random signs. Acta Math. 91 (1954), 245–301.
Samson, P. -M.Concentration of measure inequalities for Markov chains and φ-mixing processes. Ann. Probab. 28 (2000), 416–61.
Sauer, N.On the density of families of sets. J. Comb. Theory 13 (1972), 145–7.
Schmidt, E.Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. Math. Nachr. 1 (1948), 81–157.
Schwartz, L.On Bayes procedures. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 10–26.
Shannon, C. E.Communication in the presence of noise, Proc. Institute of Radio Engineers 37 (1949), 10–21.
Sheehy, A., and Wellner, J.Uniform Donsker classes of functions. Ann. Probab. 20 (1992), 1983–2030.
Shelah, S.A combinatorial problem: stability and order for models and theories in infinitary languages. Pacific J. Math. 41 (1992), 247–61.
Shen, X., and Wasserman, L.Rates of convergence of posterior distributions. Ann. Stat. 29 (2001), 687–714.
Sidak, Z.Rectangular confidence regions for the means of multivariate normal distributions. J. Am. Stat. Assoc. 62 (1967), 626–33.
Sidak, Z.On multivariate normal probabilities of rectangles: their dependence on correlations. Ann. Math. Stat. 39 (1968), 1425–34.
Slepian, D.The one sided barrier problem for Gaussian noise. Bell Systems Tech. J. 41 (1962), 463–501.
Smirnov, N. V.Estimation of the deviation between empirical distribution curves of two independent samples. Bull. Univ. Moscow 2 (1939), 3–14 (in Russian).
Smirnov, N. V.On the construction of confidence regions for the density of distribution of random variables. Dokl. Akad. Nauk SSSR 74, (1950), 184–91 (in Russian).
Sobolev, S. L.The Cauchy problem in a functional space. Dokl. Akad. Nauk SSSR 3 (1935), 291–4 (in Russian).
Sobolev, S. L.On a theorem of functional analysis. Mat. Sb. 4 (1938), 471–97 (in Russian).
Söhl, J.Uniform central limit theorems for the Grenander estimator. Elect. J. Stat. 9 (2015) 1404–23.
Spokoiny, V.Adaptive hypothesis testing using wavelets. Ann. Stat. 24 (y), 2477–98.
Stefanski, L. A., and Carroll, R. J.Deconvoluting kernel density estimators. Stat. 21 (1990), 169–84.
Stigler, S. M.Gauss and the invention of least squares. Ann. Stat. 9 (1981), 465–74.
Stolz, W.Une méthode elementaire pour l'evaluation de petities boules browniennes. C. R. Acad. Sci. Paris 316 (1994), 1217–20.
Stone, C. J.Optimal rates of convergence for nonparametric estimators, Ann. Stat. 8 (1980), 1348–60.
Stone, C. J.Optimal global rates of convergence for nonparametric regression, Ann. Stat. 10 (1982), 1040–53.
Strassen, V., and Dudley, R|M.The central limit theorem and epsilon-entropy. In Probability and Information Theory (Lecture Notes in Math. 1247). Springer, Berlin, 1969, pp. 224–31.
Strobl, F.On the reversed submartingale property of empirical discrepancies in arbitrary sample spaces. J. Theoret. Probab. 8 (1995), 825–31.
Stute, W.The oscillation behavior of empirical processes. Ann. Probab. 10 (1982), 86–107.
Stute, W.The oscillation behavior of empirical processes: the multivariate case. Ann. Probab. 12 (1984), 361–79.
Sudakov, V. N.Gaussian measures, Cauchy measures and ε-entropy. Soviet Math. Dokl. 10 (1969), 310–13.
Sudakov, V. N.A remark on the criterion of continuity of Gaussian sample functions. Lecture Notes in Math. 330 (1973), 444–54.
Sudakov, V. N., and Tsirelson, B. S.Extremal properties of half-spaces for spherically invariant measures. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24.
Szabó, B., A. W., van der Vaart, and J. H., van ZantenFrequentist coverage of adaptive nonparametric Bayesian credible sets (with discussion). Ann. Stat. 43 (2015), 1391–428.
Szarek, S. J.On the best constants in the Khinchin inequality. Studia Math. 58 (1976), 197–208.
Talagrand, M.Donsker classes and random geometry. Ann. Probab. 15 (1987), 1327–38.
Talagrand, M.Regularity of Gaussian processes. Acta Math. 159 (1987a), 99–149.
Talagrand, M.Donsker classes of sets. Probab. Theory Related Fields 78 (1988), 169–91.
Talagrand, M.An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities. Proc. Am. Math. Soc. 104 (1988a), 905–9.
Talagrand, M.Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probab. 17 (1989), 1546–70.
Talagrand, M.Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 (1994), 28–76.
Talagrand, M.The supremum of some canonical processes. Am. J. Math. 116 (1994a), 283–325.
Talagrand, M.Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73–205.
Talagrand, M.New concentration inequalities in product spaces. Invent. Math. 126 (1996), 505–63.
Talagrand, M.The Generic Chaining. Springer, Berlin, 2005.
Teh, Y. W., and Jordan, M. I.Hierarchical Bayesian nonparametric models with applications. In Bayesian Nonparametrics, (Camb. Ser. Stat. Probab. Math.)Cambridge University Press, 2010, pp. 35–79.
Triebel, H.Theory of Function Spaces. Birkhäuser, Basel, 1983.
Tsybakov, A. B.Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Annals of Statistics, 26 (1998), 2420–69.
Tsybakov, A. B.Optimal rates of aggregation. Computational Learning Theory and Kernel Machines. Proc. 16th Annual Conference on Learning Theory (COLT), B. Scholkopf and M. Warmuth, eds. Lecture Notes in Artificial Intelligence, v.2777. Springer, Heidelberg, 303–13 (2003).
Tsybakov, A. B.Introduction to Nonparametric Estimation, Springer Series in Statistics, Springer, New York (2009).
van de Geer, S.The entropy bound for monotone functions. Report TW91–10, University of Leiden, 1991.
S., van de GeerHellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Stat. 21 (1993), 14–44.
van de Geer, S.The method of sieves and minimum contrast estimators, Math. Methods Stat. 4 (1995), 20–28.
van de Geer, S.Empirical Processes in M-Estimation. Cambridge University Press, 2000.
van der Vaart, A. W.Weak convergence of smoothed empirical processes. Scand. J. Stat. 21 (1994), 501–4.
van der Vaart, A. W.Asymptotic Statistics. Cambridge University Press, 1998.
van der Vaart, A. W., and van Zanten, J. H.Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Stat. 36 (2008), 1435–63.
van der Vaart, A. W., and van Zanten, J. H.Reproducing kernel Hilbert spaces of Gaussian priors. IMS Collections: Pushing the Limits of Contemporary Statistics: Contributions in honor of Jayantha K. Ghosh 3 (2008a), 200–22.
van der Vaart, A. W., and Wellner, J.Weak convergence and empirical processes. With Applications to Statistics. Springer, New York, 1996.
van der Vaart, A. W., and Wellner, J. Alocal maximal inequality under uniform entropy. Electronic J. Stat. 5 (2011), 192–203.
Vapnik, V. N., and Cervonenkis, A. Ya.On the uniform convergence of frequencies of occurrence of events to their probabilities. Probab. Theory Appl. 26 (1968), 264–80.
Vapnik, V. N., and Cervonenkis, A. Ya.Necessary and sufficient conditions for the uniform convergence of means to their expectations. Theory Probab. Appl. 16 (1971), 264–80.
Vapnik, V. N., and Cervonenkis, A. Ya.Theory of Pattern Recognition: Statistical Problems on Learning. Nauka, Moscow, 1974 (in Russian).
Watson, G. S.Smooth regression analysis. Sankhya, Ser. A 26 (1964), 359–72.
Wong, W. H., and Severini, T. A.On maximum likelihood estimation in infinite dimensional parameter spaces. Ann. Stat. 19 (1991), 603–32.
Wong, W. H., and Shen, X.Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Stat. 23 (1995), 33–362.
Yukich, J.Weak convergence of smoothed empirical processes. Scand. J. Stat. 19 (1992), 271–9.
Zhao, L. H.Bayesian aspects of some nonparametric problems. Ann. Stat. 28 532–52.
Zygmund, A.Smooth functions. Duke Math. J. 12 (1945), 47–76.
Zygmund, A.Trigonometric Series Vols. I and II, 3rd edn. Cambridge University Press, 2002.