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13
Sharp optimality for density deconvolution with dominating bias
 Theor. Probab. Appl
, 2005
"... bias ..."
General empirical Bayes wavelet methods and exactly adaptive minimax estimation

, 2005
"... In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risk ..."
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Cited by 17 (1 self)
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In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals. In particular, our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in Besov balls of all smoothness and shape indices, and they are uniformly superefficient in convergence rates in all compact sets in Besov spaces with a finite secondary shape parameter. Furthermore, in classes nested between Besov balls of the same smoothness index, our estimators dominate threshold and James–Stein estimators within an infinitesimal fraction of the minimax risks. More general block empirical Bayes estimators are developed. Both white noise with drift and nonparametric regression are considered.
Can the Strengths of AIC and BIC Be Shared?
 BIOMETRICA
, 2003
"... It is well known that AIC and BIC have different properties in model selection. BIC is consistent in the sense that if the true model is among the candidates, the probability of selecting the true model approaches 1. On the other hand, AIC is minimaxrate optimal for both parametric and nonparame ..."
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Cited by 15 (1 self)
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It is well known that AIC and BIC have different properties in model selection. BIC is consistent in the sense that if the true model is among the candidates, the probability of selecting the true model approaches 1. On the other hand, AIC is minimaxrate optimal for both parametric and nonparametric cases for estimating the regression function. There are several successful results on constructing new model selection criteria to share some strengths of AIC and BIC. However, we show that in a rigorous sense, even in the setting that the true model is included in the candidates, the above mentioned main strengths of AIC and BIC cannot be shared. That is, for any model selection criterion to be consistent, it must behave supoptimally compared to AIC in terms of mean average squared error.
STEIN SHRINKAGE AND SECONDORDER EFFICIENCY FOR SEMIPARAMETRIC ESTIMATION OF THE SHIFT
, 2007
"... Abstract. The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function f observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile likelihood with a datadriven penalization is introduced so t ..."
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Cited by 6 (0 self)
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Abstract. The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function f observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile likelihood with a datadriven penalization is introduced so that the estimator of the center of symmetry is defined as the maximizer of the penalized profile likelihood. This estimator has the advantage of being independent of the functional class to which the signal f si assumed to belong and, furthermore, is shown to be semiparametrically adaptive and efficient. Moreover, the secondorder term of the risk expansion of the proposed estimator is proved to behave at least as well as the secondorder term of the risk of the best possible estimator using monotone smoothing filter. Under mild assumptions, this estimator is shown to be secondorder minimax sharp adaptive over the whole scale of Sobolev balls with smoothness β> 1. Thus, these results extend those of [10], where secondorder asymptotic minimaxity is proved for an estimator depending on the functional class containing f and β ≥ 2 is required. 1.
Rates of convergence and adaption over Besov spaces under pointwise risk, Statistica Sinica 13
, 2003
"... Abstract: Function estimation over the Besov spaces under pointwise ℓ r (1 ≤ r< ∞) risks is considered. Minimax rates of convergence are derived using a constrained risk inequality and wavelets. Adaptation under pointwise risks is also considered. Sharp lower bounds on the cost of adaptation are obt ..."
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Cited by 5 (1 self)
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Abstract: Function estimation over the Besov spaces under pointwise ℓ r (1 ≤ r< ∞) risks is considered. Minimax rates of convergence are derived using a constrained risk inequality and wavelets. Adaptation under pointwise risks is also considered. Sharp lower bounds on the cost of adaptation are obtained and are shown to be attainable by a wavelet estimator. The results demonstrate important differences between the minimax properties under pointwise and global risk measures. The minimax rates and adaptation for estimating derivatives under pointwise risks are also presented. A general ℓ rrisk oracle inequality is developed for the proofs of the main results. Key words and phrases: Adaptability, adaptive estimation, Besov spaces, constrained risk inequality, minimax estimation, nonparametric functional estimation,
Prediction/estimation With Simple Linear Models: Is It Really That Simple?
, 2004
"... Consider the simple normal linear regression model for estimation/prediction at a new design point. When the slope parameter is not obviously nonzero, hypothesis testing and model selection methods can be used for identifying the right model. We compare performance of such methods both theoretically ..."
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Cited by 5 (0 self)
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Consider the simple normal linear regression model for estimation/prediction at a new design point. When the slope parameter is not obviously nonzero, hypothesis testing and model selection methods can be used for identifying the right model. We compare performance of such methods both theoretically and empirically from different perspectives for more insight. The testing approach, in spite of being the "standard approch", performs poorly. We also found that the frequently told story "BIC is good when the true model is finitedimensional and AIC is good when the true model is infinitedimensional" is far from being accurate. In addition, despite some successes in the effort to go beyond the debate between AIC and BIC by adaptive model selection, it turns out that it is not possible to share the most essential properties of them by any model selection method. When model selection methods have difficulty in selection, model combining is seen to be a better alternative.
ESTIMATION OF THE DENSITY OF REGRESSION ERRORS
, 2004
"... Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a longstanding problem, formulated two d ..."
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Cited by 5 (2 self)
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Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a longstanding problem, formulated two decades ago by Mark Pinsker, about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying regression errors. The solution implies that, under a mild assumption on the differentiability of the design density and regression function, the MISE of a datadriven error density estimator attains minimax rates and sharp constants known for the case of directly observed regression errors. The result holds for error densities with finite and infinite supports. Some extensions of this result for more general heteroscedastic models with possibly dependent errors and predictors are also obtained; in the latter case the marginal error density is estimated. In all considered cases a blockwiseshrinking Efromovich– Pinsker density estimate, based on pluggedin residuals, is used. The obtained results imply a theoretical justification of a customary practice in applied regression analysis to consider residuals as proxies for underlying regression errors. Numerical and real examples are presented and discussed, and the SPLUS software is available. 1. Introduction. A
On Information Pooling, Adaptability And Superefficiency in Nonparametric Function Estimation
"... The connections between information pooling and adaptability as well as superefficiency are considered. Separable rules, which figure prominently in wavelet and other orthogonal series methods, are shown to lack adaptability; they are necessarily not rateadaptive. A sharp lower bound on the cost of ..."
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Cited by 1 (0 self)
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The connections between information pooling and adaptability as well as superefficiency are considered. Separable rules, which figure prominently in wavelet and other orthogonal series methods, are shown to lack adaptability; they are necessarily not rateadaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We show that adaptability is achieved through information pooling. A tight lower bound on the amount of information pooling required for achieving rateoptimal adaptation is given. Furthermore, in a sharp contrast to the separable rules, it is shown that adaptive nonseparable estimators can be superefficient at every point in the parameter spaces. The results demonstrate that information pooling is the key to increasing estimation precision as well as achieving adaptability and even superefficiency.
ADAPTIVE BAYESIAN INFERENCE ON THE MEAN OF
"... We consider the problem of estimating the mean of an infinitedimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a “smoothness condition, ” we first derive the convergence rate of the posterior distribution for a prior that is the ..."
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Cited by 1 (1 self)
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We consider the problem of estimating the mean of an infinitedimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a “smoothness condition, ” we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the optimal rate of convergence, the required prior depends on a “smoothness parameter ” q. When this parameter q is unknown, besides the estimation of the mean, we encounter the problem of selecting a model. In a Bayesian approach, this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model. We show that if q takes values only in a discrete set, the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model. A slightly weaker result is presented when q is unrestricted. An adaptive point estimator based on the posterior distribution is also constructed. 1. Introduction. Suppose
Universites de Paris 6 Paris 7  CNRS (UMR 7599) PR
"... We give necessary and su#cient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process. ..."
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We give necessary and su#cient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying Levy process # in the Lamperti's representation converges weakly at infinity and E T1 where T 1 = inf{t : # t 1}. Under these conditions, we give a pathwise construction of the limit law. Key words: Selfsimilar process, Levy process, Lamperti's representation, overshoot, weak convergence, first passage time.