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Canonical correlation analysis (CCA) is a popular statistical technique for exploring relationships between datasets. In recent years, the estimation of sparse canonical vectors has emerged as an important but challenging variant of the CCA problem, with widespread applications. Unfortunately, existing rate-optimal estimators for sparse canonical vectors have high computational cost. We propose a quasi-Bayesian estimation procedure that not only achieves the minimax estimation rate, but also is easy to compute by Markov chain Monte Carlo (MCMC). The method builds on (Tan et al.) and uses a rescaled Rayleigh quotient function as the quasi-log-likelihood. However, unlike (Tan et al.), we adopt a Bayesian framework that combines this quasi-log-likelihood with a spike-and-slab prior to regularize the inference and promote sparsity. We investigate the empirical behavior of the proposed method on both continuous and truncated data, and we demonstrate that it outperforms several state-of-the-art methods. As an application, we use the proposed methodology to maximally correlate clinical variables and proteomic data for better understanding the Covid-19 disease. for this article are available online. [ABSTRACT FROM AUTHOR]

Numerous image restoration algorithms have been proposed in the last several decades. These algorithms usually optimize an objective function consisting of an ℓ 2 norm based fidelity and a regularization term, whose optimality could be justified from the view of maximum a posteriori estimation with an assumption that the noise is Gaussian. However, it is known that the ℓ 2 norm based fidelity is very sensitive to gross errors that may appear in the observation. Since real-world image restoration tasks are usually hindered by abnormal pixels, impulsive noise, and other heavy-tailed noise, the utility of these traditional algorithms is limited. Although some robust algorithms have been proposed by replacing the ℓ 2 norm based fidelity with a robust one, they are designed for specific restoration tasks (e.g., multi-frame super-resolution) with a fixed image prior (e.g., the total-variation) and have not provided a principled way to justify the choice of a robust fidelity term. Currently designing a robust algorithm for general image restoration tasks is still an open problem. This paper studies the problem of robust image restoration in both theoretical and algorithmic manners. In the theoretical part, we point out that Huber function based fidelity could be justified from the pespective of minimax estimation, which facilities the choice of the robust fidelity term. In the algorithmic part, we first propose an adaptive approach to set the threshold of the Huber function, and then we derive an efficient and flexible method to solve the proposed robust formulation of the image restoration problem, which enables the proposed algorithm to incorporate various image priors. Experiments have demonstrated the robustness of the proposed algorithm and its utility in real-world image restoration tasks. [ABSTRACT FROM AUTHOR]

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e., the estimation error), whereas the second is more specifically oriented to the moments of the estimation error. The proposed bounds are relatively easy to calculate numerically (in the sense that their optimization is over relatively few auxiliary parameters), yet they turn out to be tighter (sometimes significantly so) than previously reported bounds that are associated with similar calculation efforts, across many application examples. In addition to their relative simplicity, they also have the following advantages: (i) Essentially no regularity conditions are required regarding the parametric family of distributions. (ii) The bounds are local (in a sense to be specified). (iii) The bounds provide the correct order of decay as functions of the number of observations, at least in all the examples examined. (iv) At least the first family of bounds extends straightforwardly to vector parameters. [ABSTRACT FROM AUTHOR]

Bayes minimax estimation is important because it provides a robust approach to statistical estimation that considers the worst-case scenario while incorporating prior knowledge. In this paper, Bayes minimax estimation of the mean matrix of a matrix variate normal distribution is considered under the quadratic loss function. A large class of (proper and generalized) Bayes minimax estimators of the mean matrix is presented. Two examples are given to illustrate the class of estimators, showing, among other things, that the class includes classes of estimators presented by Tsukuma. [ABSTRACT FROM AUTHOR]

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e., the estimation error), whereas the second is more specifically oriented to the moments of the estimation error. The proposed bounds are relatively easy to calculate numerically (in the sense that their optimization is over relatively few auxiliary parameters), yet they turn out to be tighter (sometimes significantly so) than previously reported bounds that are associated with similar calculation efforts, across many application examples. In addition to their relative simplicity, they also have the following advantages: (i) Essentially no regularity conditions are required regarding the parametric family of distributions. (ii) The bounds are local (in a sense to be specified). (iii) The bounds provide the correct order of decay as functions of the number of observations, at least in all the examples examined. (iv) At least the first family of bounds extends straightforwardly to vector parameters.

Bayes minimax estimation is important because it provides a robust approach to statistical estimation that considers the worst-case scenario while incorporating prior knowledge. In this paper, Bayes minimax estimation of the mean matrix of a matrix variate normal distribution is considered under the quadratic loss function. A large class of (proper and generalized) Bayes minimax estimators of the mean matrix is presented. Two examples are given to illustrate the class of estimators, showing, among other things, that the class includes classes of estimators presented by Tsukuma.

Consider a finite sample from an unknown distribution over a countable alphabet. The missing mass refers to the probability of symbols that do not appear in the sample. Estimating the missing mass is a basic problem in statistics and related fields, which dates back to the early work of Laplace, and the more recent seminal contribution of Good and Turing. In this article, we introduce a generalized Good-Turing (GT) framework for missing mass estimation. We derive an upper-bound for the risk (in terms of mean squared error) and minimize it over the parameters of our framework. Our analysis distinguishes between two setups, depending on the (unknown) alphabet size. When the alphabet size is bounded from above, our risk-bound demonstrates a significant improvement compared to currently known results (which are typically oblivious to the alphabet size). Based on this bound, we introduce a numerically obtained estimator that improves upon GT. When the alphabet size holds no restrictions, we apply our suggested risk-bound and introduce a closed-form estimator that again improves GT performance guarantees. Our suggested framework is easy to apply and does not require additional modeling assumptions. This makes it a favorable choice for practical applications. for this article are available online. [ABSTRACT FROM AUTHOR]

For the weighted symmetric loss function, the Bayes estimation method was used to study the Bayes estimation of the shape parameter of two-parameter generalized Pareto distribution and its properties.The general form of Bayes estimation of the shape parameters and the exact form under two different prior distributions were obtained, and a form of the minimax estimation of the shape parameter was obtained.So the connection among different estimates were established.The accuracy of the estimate was studied by simulation, and the result showed that the accuracy of the estimate is higher. [ABSTRACT FROM AUTHOR]

Time-frequency analysis deals with signals for which the underlying spectral characteristics change over time. The essential tool is the short-time Fourier transform, which localizes the Fourier transform in time by means of a window function. In a white noise model, we derive rate-optimal and adaptive estimators of signals in modulation spaces, which measure smoothness in terms of decay properties of the short-time Fourier transform. The estimators are based on series expansions by means of Gabor frames and on thresholding the coefficients. The minimax rates have interesting new features, and the derivation of the lower bounds requires the use of test functions which approximately localize both in time and in frequency. Simulations and applications to audio recordings illustrate the practical relevance of our methods. We also discuss the best $N$ -term approximation and the approximation of variational problems in modulation spaces by Gabor frame expansions. [ABSTRACT FROM AUTHOR]

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. 47(1), 177–204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a C k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C 3 and C 4 cases, with a gap given by a logarithmic factor. [ABSTRACT FROM AUTHOR]

Statistical divergences (SDs), which quantify the dissimilarity between probability distributions, are a basic constituent of statistical inference and machine learning. A modern method for estimating those divergences relies on parametrizing an empirical variational form by a neural network (NN) and optimizing over parameter space. Such neural estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. We establish non-asymptotic absolute error bounds for a neural estimator realized by a shallow NN, focusing on four popular f-divergences| Kullback-Leibler, chi-squared, squared Hellinger, and total variation. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory to bound the two sources of error involved: function approximation and empirical estimation. The bounds characterize the effective error in terms of NN size and the number of samples, and reveal scaling rates that ensure consistency. For compactly supported distributions, we further show that neural estimators of the first three divergences above with appropriate NN growth-rate are minimax rate-optimal, achieving the parametric convergence rate. [ABSTRACT FROM AUTHOR]

Quantum state tomography, which aims to estimate quantum states that are described by density matrices, plays an important role in quantum science and quantum technology. This paper examines the eigenspace estimation and the reconstruction of large low-rank density matrix based on Pauli measurements. Both ordinary principal component analysis (PCA) and iterative thresholding sparse PCA (ITSPCA) estimators of the eigenspace are studied, and their respective convergence rates are established. In particular, we show that the ITSPCA estimator is rate-optimal. We present the reconstruction of the large low-rank density matrix and obtain its optimal convergence rate by using the ITSPCA estimator. A numerical study is carried out to investigate the finite sample performance of the proposed estimators. • This paper examines the eigenspace estimation and matrix reconstruction problems given a high-dimensional low-rank density matrix based on Pauli measurements. • This paper analyzes the asymptotic behaviors of the principal component analysis (PCA) estimators: the ordinary PCA and the iterative thresholding sparse PCA (ITSPCA); and establish their convergence rates under both dense and sparse eigenvector settings. • Under the sparse eigenvector condition, we show that the ITSPCA is rate-optimal and the corresponding reconstructed low-rank density matrix is also rate-optimal. [ABSTRACT FROM AUTHOR]

Recently, many studies have shed light on the high adaptivity of deep neural network methods in nonparametric regression models, and their superior performance has been established for various function classes. Motivated by this development, we study a deep neural network method to estimate the drift coefficient of a multi-dimensional diffusion process from discrete observations. We derive generalization error bounds for least squares estimates based on deep neural networks and show that they achieve the minimax rate of convergence up to a logarithmic factor when the drift function has a compositional structure. [ABSTRACT FROM AUTHOR]

This paper studies the rates of convergence for learning distributions implicitly with the adversarial framework and Generative Adversarial Networks (GANs), which subsumeWasserstein, Sobolev, MMD GAN, and Generalized/Simulated Method of Moments (GMM/SMM) as special cases. We study a wide range of parametric and nonparametric target distributions under a host of objective evaluation metrics. We investigate how to obtain valid statistical guarantees for GANs through the lens of regularization. On the nonparametric end, we derive the optimal minimax rates for distribution estimation under the adversarial framework. On the parametric end, we establish a theory for general neural network classes (including deep leaky ReLU networks) that characterizes the interplay on the choice of generator and discriminator pair. We discover and isolate a new notion of regularization, called the generator-discriminator-pair regularization, that sheds light on the advantage of GANs compared to classical parametric and nonparametric approaches for explicit distribution estimation. We develop novel oracle inequalities as the main technical tools for analyzing GANs, which are of independent interest. [ABSTRACT FROM AUTHOR]

The problem of estimating the mean matrix of a matrix-variate normal distribution with a covariance matrix is considered under two loss functions. We construct a class of empirical Bayes estimators which are better than the maximum likelihood estimator under the first loss function and hence show that the maximum likelihood estimator is inadmissible. We find a general class of minimax estimators. Also we give a class of estimators that improve on the maximum likelihood estimator under the second loss function and hence show that the maximum likelihood estimator is inadmissible.

The authors consider the problem of minimax estimation of a functional of the solution to the wave equation with rapidly oscillating coefficients. The observation (output signal) is nonlinear (has the operator of superposition type). For the small parameter ε > 0, the existence of the solution of the original problem is proved using the traditional minimax approach. Transition to a homogenized parameter problem allows us to remove the nonlinearity in the observation. The main result of the paper is that the minimax estimate of the problem with homogenized parameters is an approximate minimax estimate of the original problem. [ABSTRACT FROM AUTHOR]

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Topic models have become popular tools for dimension reduction and exploratory analysis of text data which consists in observed frequencies of a vocabulary of p words in n documents, stored in a p x n matrix. The main premise is that the mean of this data matrix can be factorized into a product of two non-negative matrices: a p x K word-topic matrix A and a K x n topic-document matrix W. This paper studies the estimation of A that is possibly element-wise sparse, and the number of topics K is unknown. In this under-explored context, we derive a new minimax lower bound for the estimation of such A and propose a new computationally efficient algorithm for its recovery. We derive a finite sample upper bound for our estimator, and show that it matches the minimax lower bound in many scenarios. Our estimate adapts to the unknown sparsity of A and our analysis is valid for any finite n, p, K and document lengths. Empirical results on both synthetic data and semi-synthetic data show that our proposed estimator is a strong competitor of the existing state-of-the-art algorithms for both non-sparse A and sparse A, and has superior performance is many scenarios of interest. [ABSTRACT FROM AUTHOR]

The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed. [ABSTRACT FROM AUTHOR]

We consider a linear regression model with a vector of bounded parameters and a centered noise vector that has an uncertain unimodal distribution but known covariance matrix. We pose the minimax estimation problem for a linear combination of unknown parameters with the use of the probability criterion. The minimax estimate is determined as a result of minimizing a probability bound over the region of possible values of the variance and squared bias for all possible linear estimates. We establish that the resulting robust solution is less conservative in comparison with wider classes of distributions. [ABSTRACT FROM AUTHOR]

In this work we introduce the notion of the excess risk in the setup of estimation of the Gaussian mean when the observations are corrupted by outliers. It is known that the sample mean loses its good properties in the presence of outliers [5-6]. In addition, even the sample median is not minimax-rate-optimal in the multivariate setting. The optimal rate of the minimax risk in this setting was established by [1]. However, even these minimax-rate-optimality results do not quantify how fast the risk in the contaminated model approaches the risk in the uncontaminated model when the rate of contamination goes to zero. The present paper does a first step in filling this gap by showing that the group hard thresholding estimator has an excess risk that goes to zero when the corruption rate approaches zero. [ABSTRACT FROM AUTHOR]

In this article we develop the minimax estimation approach of general linear models to the semiparametric linear models when the parameters are simultaneously constrained by an ellipsoid and linear restrictions. Combining sample information and prior constraints the minimax estimator is obtained and compared with partially least square estimator by theoretical and simulation methods. [ABSTRACT FROM AUTHOR]

Learning and making inference from a finite set of samples are among the fundamental problems in science. In most popular applications, the paradigmatic approach is to seek a model that best explains the data. This approach has many desirable properties when the number of samples is large. However, in many practical setups, data acquisition is costly and only a limited number of samples is available. In this work, we study an alternative approach for this challenging setup. Our framework suggests that the role of the train-set is not to provide a single estimated model, which may be inaccurate due to the limited number of samples. Instead, we define a class of “reasonable” models. Then, the worst-case performance in the class is controlled by a minimax estimator with respect to it. Further, we introduce a robust estimation scheme that provides minimax guarantees, also for the case where the true model is not a member of the model class. Our results draw important connections to universal prediction, the redundancy-capacity theorem, and channel capacity theory. We demonstrate our suggested scheme in different setups, showing a significant improvement in worst-case performance over currently known alternatives.

In this paper, we focus on optimal estimation of solutions of the Cauchy problem for the first order linear hyperbolic equation systems (or, more generally, estimation of values of some functionals on their solutions) under incomplete data. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of clock skew and offset estimation (CSOE) for the IEEE 1588 precision time protocol. Built on the classical two-way message exchange scheme, IEEE 1588 is a prominent synchronization protocol for packet switched networks. Due to the presence of random queuing delays in a packet switched network, the joint recovery of clock skew and offset from the received packet timestamps can be viewed as a statistical estimation problem. Recently, assuming perfect clock skew information, minimax optimum clock offset estimators were developed for IEEE 1588. Building on this work, we first develop joint optimum invariant clock skew and offset estimators for IEEE 1588 for known queuing delay statistics and unlimited computational complexity. We then show that the developed estimators are minimax optimum, i.e., these estimators minimize the maximum skew normalized mean square estimation error over all possible values of the unknown parameters. Minimax optimum estimators that utilize information from past timestamps to improve accuracy are also introduced. The developed optimum estimators provide useful fundamental limits for evaluating the performance of CSOE schemes. These performance limits can aid system designers to develop algorithms with the desired computational complexity that achieve performance close to the performance of the optimum estimators. If a designer finds an approach with a complexity they find acceptable and which provides performance close to the optimum performance, they can use it and know they have near optimum performance. This is precisely the approach used in communications when comparing to capacity. [ABSTRACT FROM AUTHOR]

We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size. [ABSTRACT FROM AUTHOR]

The problem of estimating of the vector β of the linear regression model y = Aβ + ϵ with ϵ ∼ Np(0, σ2Ip) under quadratic loss function is considered when common variance σ2 is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator. [ABSTRACT FROM AUTHOR]

Theoretical results in the functional linear regression literature have so far focused on minimax estimation where smoothness parameters are assumed to be known and the estimators typically depend on these smoothness parameters. In this paper we consider adaptive estimation in functional linear regression. The goal is to construct a single data-driven procedure that achieves optimality results simultaneously over a collection of parameter spaces. Such an adaptive procedure automatically adjusts to the smoothness properties of the underlying slope and covariance functions. The main technical tools for the construction of the adaptive procedure are functional principal component analysis and block thresholding. The estimator of the slope function is shown to adaptively attain the optimal rate of convergence over a large collection of function spaces. [ABSTRACT FROM AUTHOR]

We consider the minimax estimation problem of a discrete distribution with support size $k$ under privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number $\epsilon $ measures the privacy level of a privatization scheme. For a given $\epsilon $ , we consider the problem of constructing optimal privatization schemes with $\epsilon $ -privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes known in the literature provide order optimal performance in the high privacy regime where $\epsilon $ is very close to 0, and in the low privacy regime where $e^{\epsilon }\approx k$ , respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when $1\ll e^{\epsilon } \ll k$. More concretely, we prove that when $3.8 < \epsilon <\ln (k/9) $ , our schemes reduce the expected estimation loss by 50% under $\ell _{2}^{2}$ metric and by 30% under $\ell _{1}$ metric over the existing schemes. We also prove a lower bound for the region $e^{\epsilon } \ll k$ , which implies that our schemes are order optimal in this regime. [ABSTRACT FROM AUTHOR]

This paper considers an extension of the Kelly criterion used in sports wagering. By recognizing that the probability p of placing a correct wager is unknown, modified Kelly criteria are obtained that take the uncertainty into account. Estimators are proposed that are developed from a decision theoretic framework. We observe that the resultant betting fractions can differ markedly based on the choice of loss function. In the cases that we study, the modified Kelly fractions are smaller than original Kelly. [ABSTRACT FROM AUTHOR]

In recent years, tools from information theory have played an increasingly prevalent role in statistical machine learning. In addition to developing efficient, computationally feasible algorithms for analyzing complex datasets, it is of theoretical importance to determine whether such algorithms are "optimal" in the sense that no other algorithm can lead to smaller statistical error. This paper provides a survey of various techniques used to derive information-theoretic lower bounds for estimation and learning. We focus on the settings of parameter and function estimation, community recovery, and online learning for multi-armed bandits. A common theme is that lower bounds are established by relating the statistical learning problem to a channel decoding problem, for which lower bounds may be derived involving information-theoretic quantities such as the mutual information, total variation distance, and Kullback'Leibler divergence. We close by discussing the use of information-theoretic quantities to measure independence in machine learning applications ranging from causality to medical imaging, and mention techniques for estimating these quantities efficiently in a data-driven manner. [ABSTRACT FROM AUTHOR]

This note considers the problem of estimating the mean matrixof a matrix-variate normal distribution with the covariance matrixwhen the loss function is, whereis unknown. We find a large class of (proper and generalized) Bayes minimax estimators for the mean matrix. This class includes classes of estimators obtained by Tsukuma [Proper Bayes minimax estimators of the normal mean matrix with common unknown variances. J Statist Plan Inference. 2010;140:2596–2606]. [ABSTRACT FROM PUBLISHER]

We tackle the problem of the estimation of a vector of underlying means (signal sizes) from a single vector-valued observation y. Often one is interested in estimating only a subvector of signals corresponding to a set of selected, 'interesting' sample elements. These 'interesting' sample elements tend to have the largest absolute size, gleaned by applying some selection procedure like that of Benjamini & Hochberg (2015). Previous work on this estimation task proposes the reduction in size of the largest (absolute) sample elements either via shrinkage (like James-Stein) or by subtracting biases estimated using empirical Bayes methodology. We take a novel approach and adapt recent developments by Lee et al. (2016) in post-selection inference. Adapting and applying their distributional results to our problem post-selection point and interval estimators for underlying signal sizes are proposed. Simulations suggest that our estimator seems to perform quite well against competitors. Furthermore we prove an upper bound to the so-called 'worst case risk' of our estimator-when combined with the Benjamini-Hochberg selection procedure-and show that it is within a constant multiple of the minimax risk over a rich set of parameter spaces meant to evoke sparsity. The Canadian Journal of Statistics 45: 128-148; 2017 © 2017 Statistical Society of Canada [ABSTRACT FROM AUTHOR]

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classesBsp, q. The theory is illustrated with some numerical examples. A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest. [ABSTRACT FROM AUTHOR]

Suppose a subset of populations is selected fromkexponential populations with unknown location parameters θ1, θ2, …, θkand common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case ofk= 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators. [ABSTRACT FROM AUTHOR]

This paper studies the problem of optimal estimation of a quadratic functional of two normal mean vectors, Q(μ, θ) = (1/n) Σi=1n μi2 θi2, with a particular focus on the case where both mean vectors are sparse. We propose optimal estimators of Q(μ, θ) for different regimes and establish the minimax rates of convergence over a family of parameter spaces. The optimal rates exhibit interesting phase transitions in this family. We also include a simulation study to complement the theoretical results in the paper. [ABSTRACT FROM AUTHOR]

The problem of estimating the mean vectorof a multivariate normal distribution with known covariance matrixis considered under the extended reflected normal and extended balanced loss functions. We extend the class of minimax estimators obtained by Towhidi and Behboodian (2002) and also by Asgharazadeh and Sanjari Farsipour (2008). The class of estimators derived under the extended balanced loss function includes a special case of estimators obtained by Chung et al. (1999). We introduce a class of minimax estimators ofextending Berger’s estimators (1976) and dominating the sample mean in terms of risk under the extended reflected normal and extended balanced loss functions. Using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators under the extended balanced loss function and show that the result of Chung and Kim (1998) is a special case of our result. [ABSTRACT FROM PUBLISHER]

For a Poisson point process X, Itô's famous chaos expansion implies that every square integrable regression function r with covariate X can be decomposed as a sum of multiple stochastic integrals called chaos. In this paper, we consider the case where r can be decomposed as a sum of δchaos. In the spirit of Cadre and Truquet [ESAIM: PS 19 (2015) 251-267], we introduce a semiparametric estimate of r based on i.i.d. copies of the data. We investigate the asymptotic minimax properties of our estimator when δ is known. We also propose an adaptive procedure when δ is unknown. [ABSTRACT FROM AUTHOR]

This paper considers the asymptotic behavior in β -Hölder spaces, and under L p losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison–Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever (p , β) ∈ [ 1 , 3) × (0 , 2 ] or (p , β) ∈ A d , where A d is a specific subset of [ 3 , 4) × (0 , 2 ] that depends on the dimension d of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either p ∈ [ 4 , ∞) or β ∈ (2 , ∞). These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators. [ABSTRACT FROM AUTHOR]