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In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function f in a class of β -Hölder smooth functions. Let 'worst-case' refer to the supremum over all functions f in the Hölder class. It is shown that any estimator with worst-case bias ≲ n − β / (2 β + 1) ≕ ψ n must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ ψ n. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation. [ABSTRACT FROM AUTHOR]

This note considers the problem of simultaneous estimation of matrix-variate normal mean matrix using a balanced loss function when common variance σ 2 is unknown. We first find a class of minimax estimators for this problem and show that the maximum likelihood estimator is inadmissible. We obtain a large class of (proper and generalized) Bayes minimax estimators for the mean matrix. [ABSTRACT FROM AUTHOR]

We study the relation between Stein’s theorem and Stokes’ theorem (or the divergence theorem) and show, using completeness of certain exponential families, that they are equivalent, in a certain sense, by using each to prove a version of the other. [ABSTRACT FROM AUTHOR]

We consider non-linear wavelet-based estimators of density functions with stationary random fields, which are indexed by the integer lattice points in the N -dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B p , q s . Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability. [ABSTRACT FROM AUTHOR]

We investigate the problem of simultaneous estimation of multivariate normal mean vector using Zellner (1994)'s balance loss function when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Chung et al. (1999). This result is then used to obtain a large class of Bayes minimax estimators for mean vector. [ABSTRACT FROM AUTHOR]

Abstract: The problem of estimating the mean vector of a multivariate normal distribution with the covariance matrix is considered under the loss function, , where is unknown and is a known positive definite diagonal matrix. A large class of Bayes minimax estimators of is found. This class includes classes of estimators obtained by Lin and Mousa (1982) and Zinodiny et al. (2011). [Copyright &y& Elsevier]

Abstract: Autin [2007. Maxisets for -thresholding rules. Test, to appear, see http://www.seio.es/test/Archivos/issues/Ab_TESTAutin.html ] has established the following estimation result: by considering the Gaussian white noise model and the Besov risk , the BlockShrink estimator is better in the maxiset sense than the hard thresholding estimator. In the present paper, we show that this maxiset superiority is strict and can be extended to the risk for numerous sophisticated models (regression with random uniform design, convolution model in Gaussian white noise,…). [Copyright &y& Elsevier]

Abstract: For estimating an unknown parameter , we introduce and motivate the use of the balanced-type loss function: , where , is a positive weight function, and is a general “target” estimator. Developments and various examples are given with regards to the issues of admissibility, dominance, Bayesianity, and minimaxity. In many cases, as in Dey et al. [1999. On estimation with balanced loss functions. Statist. Probab. Lett. 45, 97–101], we show that results for loss may be inferred directly from corresponding results for weighted squared error loss (i.e., ). Specific issues related to constrained parameter spaces, which include the choice of the target estimator, are addressed. Finally, we derive minimax estimators of a bounded normal mean under loss with being the maximum-likelihood estimator of . [Copyright &y& Elsevier]

Abstract: For a vast class of discrete model families with cdf''s , and for estimating under squared error loss under a constraint of the type , we present a general and unified development concerning the minimaxity of a boundary supported prior Bayes estimator. While the sufficient conditions obtained are of the expected form , the approach presented leads, in many instances, to both necessary and sufficient conditions, and/or explicit values for . Finally, the scope of the results is illustrated with various examples that, not only include several common distributions (e.g., Poisson, Binomial, Negative Binomial), but many others as well. [Copyright &y& Elsevier]

A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a lower-bounded parameter in the family of transformed chi-square distributions is given. These estimators are the pointwise limits of a sequence of Bayes estimators. Some examples are given. [Copyright &y& Elsevier]