Results 1  10
of
17
Higher criticism for detecting sparse heterogeneous mixtures
 Ann. Statist
, 2004
"... Higher Criticism, or secondlevel significance testing, is a multiple comparisons concept mentioned in passing by Tukey (1976). It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested to compare the ..."
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Cited by 154 (22 self)
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Higher Criticism, or secondlevel significance testing, is a multiple comparisons concept mentioned in passing by Tukey (1976). It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested to compare the fraction of observed significances at a given αlevel to the expected fraction under the joint null, in fact he suggested to standardize the difference of the two quantities and form a zscore; the resulting zscore tests the significance of the body of significance tests. We consider a generalization, where we maximize this zscore over a range of significance levels 0 < α ≤ α0. We are able to show that the resulting Higher Criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ‘sparse normal means ’ testing problem can be seen from work of Ingster (1999) and Jin (2002), who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so
Lectures on the central limit theorem for empirical processes
 Probability and Banach Spaces
, 1986
"... Abstract. Concentration inequalities are used to derive some new inequalities for ratiotype suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratiotype suprema and to recover anumber of the results from [1] and [2]. As a statistical applica ..."
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Cited by 135 (9 self)
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Abstract. Concentration inequalities are used to derive some new inequalities for ratiotype suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratiotype suprema and to recover anumber of the results from [1] and [2]. As a statistical application, an oracle inequality for nonparametric regression is obtained via ratio bounds. 1.
Concentration inequalities and asymptotic results for ratio type empirical processes
 ANN. PROBAB
, 2006
"... Let F be a class of measurable functions on a measurable space (S, S) with values in [0, 1] and let Pn = n −1 n ∑ δXi i=1 be the empirical measure based on an i.i.d. sample (X1,...,Xn) from a probability distribution P on (S, S). We study the behavior of suprema of the following type: sup rn<σP f ..."
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Cited by 40 (5 self)
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Let F be a class of measurable functions on a measurable space (S, S) with values in [0, 1] and let Pn = n −1 n ∑ δXi i=1 be the empirical measure based on an i.i.d. sample (X1,...,Xn) from a probability distribution P on (S, S). We study the behavior of suprema of the following type: sup rn<σP f ≤δn Pnf − Pf  φ(σPf) where σP f ≥ Var 1/2 P f and φ is a continuous, strictly increasing function with φ(0) = 0. Using Talagrand’s concentration inequality for empirical processes, we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior, expressing the conditions in terms of expectations of localized suprema of empirical processes. We also prove new bounds for expected values of supnorms of empirical processes in terms of the largest σP f and the L2(P) norm of the envelope of the function class, which are especially suited for estimating localized suprema. With this technique, we extend to function classes most of the known results on ratio type suprema of empirical processes, including some of Alexander’s results for VC classes of sets. We also consider applications of these results to several important problems in nonparametric statistics and in learning theory (including general excess risk bounds in empirical risk minimization and their versions for L2regression and classification and ratio type bounds for margin distributions in classification).
Goodnessoffit tests via phidivergences
, 2006
"... A unified family of goodnessoffit tests based on φdivergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cas ..."
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Cited by 26 (2 self)
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A unified family of goodnessoffit tests based on φdivergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cases (s = 2 and s = 1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk
A note on the asymptotic distribution of BerkJones type statistics under the null hypothesis
 High Dimensional Probability III, pp321332. Birkhäuser
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The quantiletransform–empiricalprocess approach to limit theorems for sums of order statistics
 Sums, trimmed sums and extremes, 215–267, Progr. Probab
, 1991
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KernelType Estimators for the Extreme Value Index
"... A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best known estimator for that case is the Hill estimator. This estimator can be considered to be either a moment estimator or a (quasi) maximum likelihood estimator and was generalized t ..."
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Cited by 4 (0 self)
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A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best known estimator for that case is the Hill estimator. This estimator can be considered to be either a moment estimator or a (quasi) maximum likelihood estimator and was generalized to a kerneltype estimator, still only valid for positive extreme value indices. The Hill estimator has been extended to a momenttype estimator valid for all extreme value indices. Also the quasi maximum likelihood estimators based on the generalized Pareto distribution, have been given for a restricted region of negative extreme value indices We derive kerneltype estimators valid for all real extreme value indices and compare their performance with the (generalized) moment estimator and (quasi) maximum likelihood estimator. # TU Delft, Faculty ITS, Dep. CROSS & VU Amsterdam, p.groeneboom@its.tudelft.nl + TU Delft, Faculty ITS, Dep. CROSS, h.p.Lopuhaa@its.tudelft.nl # Statistics Netherlands, PWOF@cbs.nl 1 1
On the “Poisson boundaries” of the family of weighted Kolmogorov statistics
 IMS Monograph
, 2004
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Non parametric maximum Likelihood Approach to Multiple change point problems”, The Annals of Statistics,42
, 2014
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Improvement of two Hungarian bivariate theorems.
, 2009
"... Nous introduisons une nouvelle technique pour établir des théorèmes hongrois multivariés. Appliquée dans cet article aux théorèmes bivariés d’approximation forte du processus empirique uniforme, cette technique améliore le résultat de Komlós, Major et Tusnády (1975) ainsi que les nôtres (1998). Plus ..."
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Nous introduisons une nouvelle technique pour établir des théorèmes hongrois multivariés. Appliquée dans cet article aux théorèmes bivariés d’approximation forte du processus empirique uniforme, cette technique améliore le résultat de Komlós, Major et Tusnády (1975) ainsi que les nôtres (1998). Plus précisément, nous montrons que l’erreur dans l’approximation du nprocessus empirique uniforme bivarié par un pont brownien bivarié est d’ordre n −1/2 (log(nab)) 3/2 sur le pavé [0,a]×[0,b], 0 ≤ a,b ≤ 1, et que l’erreur dans l’approximation du nprocessus empirique uniforme univarié par un processus de Kiefer est d’ordre n −1/2 (log(na)) 3/2 sur l’intervalle [0,a], 0 ≤ a ≤ 1. Dans les deux cas la borne d’erreur globale est donc d’ordre n −1/2 (log(n)) 3/2. Les résultats précédents donnaient depuis l’article de 1975 de Komlós, Major et Tusnády une borne d’erreur globale d’ordre n −1/2 (log(n)) 2, et depuis notre article de 1998 des bornes d’erreur locales d’ordre n −1/2 (log(nab)) 2 ou n −1/2 (log(na)) 2. Le nouvel argument de cet article consiste à reconnaître des martingales dans les termes d’erreur, puis à leur appliquer une inégalité exponentielle de Van de Geer (1995) ou de de la Peña (1999). L’idée est de borner le compensateur du terme d’erreur, au lieu de borner le terme d’erreur luimême. We introduce a new technique to establish Hungarian multivariate theorems. In this article, we apply