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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 152 (21 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
Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data
 Ann. Statist
, 2006
"... Control of the False Discovery Rate (FDR) is an important development in multiple hypothesis testing, allowing the user to limit the fraction of rejected null hypotheses which correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in multiparameter estimation pro ..."
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Cited by 24 (5 self)
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Control of the False Discovery Rate (FDR) is an important development in multiple hypothesis testing, allowing the user to limit the fraction of rejected null hypotheses which correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in multiparameter estimation problems to set thresholds for separating signal from noise when the signal is sparse. Success has been proven when the noise is Gaussian; see [3]. In this paper, we consider the application of FDR thresholding to a nonGaussian setting, in hopes of learning whether the good asymptotic properties of FDR thresholding as an estimation tool hold more broadly than just at the standard Gaussian model. We consider a vector Xi, i = 1,..., n, whose coordinates are independent exponential with individual means µi. The vector µ is thought to be sparse, with most coordinates 1 and a small fraction significantly larger than 1. This models a situation where most coordinates are simply ‘noise’, but a small fraction of the coordinates contain ‘signal’. We develop an estimation theory working with log(µi) as the estimand, and use the percoordinate meansquared error in recovering log(µi) to measure risk. We consider minimax
Asymptotic Minimaxity of False Discovery Rate Thresholding for Sparse Exponential Data
, 2004
"... Control of the False Discovery Rate (FDR) is a recent innovation in multiple hypothesis testing, allowing the user to limit the fraction of rejected null hypotheses which correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in multiparameter estimation problems ..."
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Control of the False Discovery Rate (FDR) is a recent innovation in multiple hypothesis testing, allowing the user to limit the fraction of rejected null hypotheses which correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in multiparameter estimation problems to set thresholds for separating signal from noise when the signal is sparse. Success has been proven when the noise is Gaussian; see [1]. In this paper, we consider the application of FDR thresholding to a nonGaussian setting, in hopes of learning whether the good asymptotic properties of FDR thresholding as an estimation tool hold more broadly than just at the standard Gaussian model. We consider a vector Xi, i = 1,..., n whose coordinates are independent exponential with individual means µi. The vector µ is thought to be sparse, with most coordinates 1 and a small fraction significantly larger than 1. This models a situation where most coordinates are simply ‘noise’, but a small fraction of the coordinates contain ‘signal’. We develop an estimation theory working with log(µi) as the estimand, and use the percoordinate meansquared error in recovering log(µi) to measure risk. We consider minimax estimation over parameter spaces defined by constraints on the percoordinate ℓ p norm of log(µi): Avei log p (µi) ≤ η p. Members of such spaces are vectors (µi) which are sparsely heterogeneous. We find that, for large n and small η, FDR thresholding is nearly minimax, increasingly so as η decreases. The FDR control parameter 0 < q < 1 plays an important role: when q ≤ 1 2 prevents near minimaxity. These conclusions mirror those found by Abramovich et al in the Gaussian case. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures, and noni.i.d. dependency structures. We will also compare our results with work in the Gaussian setting [1]. This is joint work with David Donoho., the FDR estimator is nearly minimax, while choosing a fixed q>
DISCUSSION ON THE PAPER ”LEAST ANGLE REGRESSION”
"... The issue of model selection has drawn the attention of both applied and theoretical statisticians for a long time. Indeed, there has been an enormous range of contribution in model selection proposal, which includes the work by Akaike [1], Mallows [7], Foster and George [5], Birge ́ and Massart [3] ..."
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The issue of model selection has drawn the attention of both applied and theoretical statisticians for a long time. Indeed, there has been an enormous range of contribution in model selection proposal, which includes the work by Akaike [1], Mallows [7], Foster and George [5], Birge ́ and Massart [3] or Abramovich, Benjamini, Donoho and Johnstone [2]. Over the last decade, modern computerdriven methods have been developed such as All Subsets, Forward Selection, Forward Stagewise or LASSO. Such methods are useful in the setting of standard linear model, where we observe noisy data, and wish to predict the response variable using only a few number of covariables, since they provide automatically linear models that fit the data. The procedure described in this paper is, on the one hand, numerically very efficient and on the other hand, very general, since, with slight modifications, it enables to recover the estimates given by the LASSO and Stagewise. I. Estimation procedure The “LARS ” method is based on a recursive procedure selecting, at each step, the covariables having largest absolute correlation with the response y. In the case of an orthogonal design, the estimates can then be viewed as an l1penalized estimator. Consider the linear regression model where we observe y with some random noise ε, with orthogonal design assumptions: y = Xβ + ε. Using the softthresholding form of the estimator, we can write it, in an equivalent way, as the minimum of an ordinary leastsquares and a l1 penalty over the coefficients of the regression. As a matter of fact, at step k = 1,..., m, the estimators β̂k = X−1µ̂k are given by µ̂k = arg min µ∈IRn