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45
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 20 (5 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Adaptive multilevel splitting for rare event analysis
, 2005
"... The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10^−9, na ..."
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Cited by 17 (4 self)
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The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10^−9, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand. This is unfortunately not always possible. In this paper, we propose an adaptive algorithm to cope with this problem: the estimation is asymptotically consistent, costs just a little bit more than classical multilevel splitting and has the same efficiency in terms of asymptotic variance. In the one dimensional case, we prove rigorously the a.s. convergence and the asymptotic normality of our estimator, with the same variance as with other algorithms that use fixed crossing levels. In our proofs we mainly use tools from the theory of empirical processes, which seems to be quite new in the field of rare events.
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 14 (4 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
Estimation of a function under shape restrictions. Applications to reliability
 Ann. Statist
, 2005
"... This paper deals with a nonparametric shape respecting estimation method for Ushaped or unimodal functions. A general upper bound for the nonasymptotic L1risk of the estimator is given. The method is applied to the shape respecting estimation of several classical functions, among them typical inte ..."
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Cited by 6 (0 self)
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This paper deals with a nonparametric shape respecting estimation method for Ushaped or unimodal functions. A general upper bound for the nonasymptotic L1risk of the estimator is given. The method is applied to the shape respecting estimation of several classical functions, among them typical intensity functions encountered in the reliability field. In each case, we derive from our upper bound the spatially adaptive property of our estimator with respect to the L1metric: it approximately behaves as the best variable binwidth histogram of the function under estimation. 1. Introduction. In
Interference alignmentbased sum capacity bounds for random dense Gaussian interference networks
 IEEE Transactions on Information Theory
"... We consider a dense K user Gaussian interference network formed by paired transmitters and receivers placed independently at random in a fixed spatial region. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average perus ..."
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Cited by 5 (2 self)
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We consider a dense K user Gaussian interference network formed by paired transmitters and receivers placed independently at random in a fixed spatial region. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average peruser capacity CΣ/K to 1 2 E log(1+2SNR). The achievability result follows directly from results based on an interference alignment scheme presented in recent work of Nazer et al. Our main contribution comes through an upper bound, motivated by ideas of ‘bottleneck capacity ’ developed in recent work of Jafar. By controlling the physical location of transmitter–receiver pairs, we can match a large proportion of these pairs to form socalled ǫbottleneck links, with consequent control of the sum capacity. 1 Introduction and main result 1.1 Interference networks and bottleneck states
Convergence Rates for Differentially Private Statistical Estimation
"... Differential privacy is a cryptographicallymotivated definition of privacy which has gained significant attention over the past few years. Differentially private solutions enforce privacy by adding random noise to a function computed over the data, and the challenge in designing such algorithms is t ..."
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Cited by 5 (1 self)
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Differential privacy is a cryptographicallymotivated definition of privacy which has gained significant attention over the past few years. Differentially private solutions enforce privacy by adding random noise to a function computed over the data, and the challenge in designing such algorithms is to control the added noise in order to optimize the privacyaccuracysample size tradeoff. This work studies differentiallyprivate statistical estimation, and shows upper and lower bounds on the convergence rates of differentially private approximations to statistical estimators. Our results reveal a formal connection between differential privacy and the notion of Gross Error Sensitivity (GES) in robust statistics, by showing that the convergence rate of any differentially private approximation to an estimator that is accurate over a large class of distributions has to grow with the GES of the estimator. We then provide an upper bound on the convergence rate of a differentially private approximation to an estimator with bounded range and bounded GES. We show that the bounded range condition is necessary if we wish to ensure a strict form of differential privacy. 1.