Results 1  10
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18
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 20 (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 17 (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.
Rates of convergence and adaption over Besov spaces under pointwise risk
 STATISTICA SINICA
, 2003
"... 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 a ..."
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Cited by 9 (1 self)
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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.
Estimation of the density of regression errors
 Annals of Statistics
, 2005
"... ..."
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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 7 (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.
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 6 (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.
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
Sharp adaptive estimation by a blockwise method
 J. Nonparametr. Stat
, 2009
"... We consider a blockwise JamesStein estimator for nonparametric function estimation in suitable wavelet or Fourier bases. The estimator can be readily explained and implemented. We show that the estimator is asymptotically sharpadaptive in minimax risk over any Sobolev ball containing the true fun ..."
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We consider a blockwise JamesStein estimator for nonparametric function estimation in suitable wavelet or Fourier bases. The estimator can be readily explained and implemented. We show that the estimator is asymptotically sharpadaptive in minimax risk over any Sobolev ball containing the true function. Further, for a moderately broad range of bounded sets in Besov space our estimator is asymptotically nearly sharp adaptive in the sense that it comes within the DonohoLiu constant, 1.24, of being exactly sharp adaptive. Other parameter spaces are also considered. The paper concludes with a MonteCarlo study comparing the performance of our estimator to that of three other popular wavelet estimators. Our procedure generally (but not always) outperforms two of these and is overall comparable, or perhaps slightly superior, to the third.
On Adaptability And Information Pooling in Nonparametric Function Estimation
"... It is well known that it is possible to achieve adaptation for “free” in function estimation under a global loss. It is unclear, however, why and how the adaptability is achieved. In this article we show that adaptability is achieved through information pooling. It is first shown that separable rule ..."
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It is well known that it is possible to achieve adaptation for “free” in function estimation under a global loss. It is unclear, however, why and how the adaptability is achieved. In this article we show that adaptability is achieved through information pooling. It is first shown that separable rules, which figure prominently in wavelet and other orthogonal series methods, lack adaptability; they are necessarily not rateadaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We then derive a tight lower bound on the amount of information pooling required for achieving global adaptability. Moreover, 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 demonstate that information pooling is the key to increase estimation precision and achieve adaptability and even superefficiency.