The $L\_2$-minimax risk in Sobolev classes of densities with non-integer smoothness index is shown to have an analog form to that in integer Sobolev classes. To this end, the notion of Sobolev classes is generalized to fractional derivatives of order $β\in\mathbb R^+$. A minimax kernel density estimator for such a classes is found. Although there exists no corresponding proof in the literature so far, the result of this article was used implicitly in numerous papers. A certain necessity that this gap had to be filled, can thus not be denied.
We study minimax density estimation on the product space $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$. We consider $L^p$-risk for probability density functions defined over regularity spaces that allow for different level of smoothness in each of the variables. Precisely, we study probabilities on Sobolev spaces with dominating mixed-smoothness. We provide the rate of convergence that is optimal even for the classical Sobolev spaces.
We construct an adaptive wavelet estimator that attains minimax near-optimal rates in a wide range of Besov balls. The convergence rates are affected only by the weakest dependence amongst the channels, and take into account both noise sources.
In this paper, we are interested in the study of beta kernel estimators from an asymptotic minimax point of view. It is well known that beta kernel estimators are, on the contrary of classical kernel estimators, "free of boundary effect" and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.
We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman (2005) to include non-monotone shrinkage functions.
In this paper, we observe a sparse mean vector through Gaussian noise and we aim at estimating some additive functional of the mean in the minimax sense. More precisely, we generalize the results of (Collier et al., 2017, 2019) to a very large class of functionals. The optimal minimax rate is shown to depend on the polynomial approximation rate of the marginal functional, and optimal estimators achieving this rate are built.
In this paper, we introduce a new estimator for the emission densities of a nonparametric hidden Markov model. It is adaptive and minimax with respect to each state's regularity--as opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. Our method is based on Goldenshluger and Lepski's methodology. It is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral…
In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.
In this paper we study the problem of testing if an $L_2-$function $f$ belonging to a certain $l_2$-Sobolev-ball $B_t(R)$ of radius $R>0$ with smoothness level $t>0$ indeed exhibits a higher smoothness level $s>t$, that is, belongs to $B_s(R)$. We assume that only a perturbed version of $f$ is available, where the noise is governed by a standard Brownian motion scaled by $\frac{1}{\sqrt{n}}$. More precisely, considering a testing problem of the form $$H_0:~f\in B_s(R)~~\mathrm{vs.}~~H_1:~f\in B_t(R),~\inf_{h\in B_s}\Vert f-h\Vert_{L_2}>ρ$$ for some $ρ>0$, we approach the task of identifying th…
For the problem of nonparametric estimation of signal in Gaussian noise we point out the strong asymptotically minimax estimators on maxisets for linear estimators (see \cite{ker93,rio}). It turns out that the order of rates of convergence of Pinsker estimator on this maxisets is worse than the order of rates of convergence for the class of linear estimators considered on this maxisets. We show that balls in Sobolev spaces are maxisets for Pinsker estimators.
In this article, we consider two forms of shrinkage estimators of the mean $θ$ of a multivariate normal distribution $X\sim N_{p}\left(θ, σ^{2}I_{p}\right)$ where $σ^{2}$ is unknown. We take the prior law $θ\sim N_{p}\left(\upsilon, τ^{2}I_{p}\right)$ and we constuct a Modified Bayes estimator $δ_{B}^{\ast}$ and an Empirical Modified Bayes estimator $δ_{EB}^{\ast}$. We are interested in studying the minimaxity and the limits of risks ratios of these estimators, to the maximum likelihood estimator $X$, when $n$ and $p$ tend to infinity.
This paper is concerned with the problem of goodness-of-fit for curve registration, and more precisely for the shifted curve model, whose application field reaches from computer vision and road traffic prediction to medicine. We give bounds for the asymptotic minimax separation rate, when the functions in the alternative lie in Sobolev balls and the separation from the null hypothesis is measured by the l2-norm. We use the generalized likelihood ratio to build a nonadaptive procedure depending on a tuning parameter, which we choose in an optimal way according to the smoothness of the ambient s…
In this paper, we consider the estimation of a mean vector of a multivariate normal population where the mean vector is suspected to be nearly equal to mean vectors of $k-1$ other populations. As an alternative to the preliminary test estimator based on the test statistic for testing hypothesis of equal means, we derive empirical and hierarchical Bayes estimators which shrink the sample mean vector toward a pooled mean estimator given under the hypothesis. The minimaxity of those Bayesian estimators are shown, and their performances are investigated by simulation.
In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator $\widehat{f}$, and a goodness-of-fit test also based on $\widehat{f}$. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X}\subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Ad…
Many statistical practices involve choosing between a full model and reduced models where some coefficients are reduced to zero. Data were used to select a model with estimated coefficients. Is it possible to do so and still come up with an estimator always better than the traditional estimator based on the full model? The James-Stein estimator is such an estimator, having a property called minimaxity. However, the estimator considers only one reduced model, namely the origin. Hence it reduces no coefficient estimator to zero or every coefficient estimator to zero. In many applications includi…
This article discusses estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators shrinking more on the coordinates with larger variances, seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that such James-Stein type estimators can be ensemble minimax, minimax with respect to the ensemble risk, related to empirical Bayes perspective of Efron and Morris.
Consider the problem of estimating a multivariate normal mean with a known variance matrix, which is not necessarily proportional to the identity matrix. The coordinates are shrunk directly in proportion to their variances in Efron and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes approach, whereas inversely in proportion to their variances in Berger's (Ann. Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax estimator, by approximately minimizing the Bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is o…
The paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic Hölder class estimator. Next applying Lepski's method, we propose the estimator which is optimally adaptive over the collection of isotropic Hölder classes. To prove the optimality of the proposed procedure we establish, in particular, the exponential inequality for the deviation of locally bayesian estimator from the parameter to be estimated. These theoretical results are illustrated by simulati…
We consider the problem of estimating a continuous distribution function $F$, as well as meaningful functions $τ(F)$ under a large class of loss functions. We obtain best invariant estimators and establish their minimaxity for Hölder continuous $τ$'s and strict bowl-shaped losses with a bounded derivative. We also introduce and motivate the use of integrated balanced loss functions which combine the criteria of an integrated distance between a decision $d$ and $F$, with the proximity of $d$ with a target estimator $d_0$. Moreover, we show how the risk analysis of procedures under such an integ…
Let y=Aβ+ε, where y is an N\times1 vector of observations, βis a p\times1 vector of unknown regression coefficients, A is an N\times p design matrix and εis a spherically symmetric error term with unknown scale parameter σ. We consider estimation of βunder general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980) 1036-1056, J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which hav…
Two major research tasks lie at the heart of high dimensional data analysis: accurate parameter estimation and correct support recovery. The existing literature mostly aims for either the best parameter estimation or the best model selection result, however little has been done to understand the potential interaction between the estimation precision and the selection behavior. In this work, our minimax result shows that an estimator's performance of type I error control directly links with its $L_2$ estimation error rate, and reveals a trade-off phenomenon between the rate of convergence and t…
We address the problem of adaptive minimax density estimation on $\bR^d$ with $\bL_p$--loss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-drive…
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 …
A new class of minimax Stein-type shrinkage estimators of a multivariate normal mean is studied where the shrinkage factor is based on an l_p norm. The proposed estimators allow some but not all coordinates to be estimated by 0 thereby allow sparsity as well as minimaxity.
In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where th…
We present estimators for a well studied statistical estimation problem: the estimation for the linear regression model with soft sparsity constraints ($\ell_q$ constraint with $0<q\leq1$) in the high-dimensional setting. We first present a family of estimators, called the projected nearest neighbor estimator and show, by using results from Convex Geometry, that such estimator is within a logarithmic factor of the optimal for any design matrix. Then by utilizing a semi-definite programming relaxation technique developed in [SIAM J. Comput. 36 (2007) 1764-1776], we obtain an approximation algor…
In this article, we consider the problem of inverting the exponential Radon transform of a function in the presence of noise. We propose a kernel estimator to estimate the true function, analogous to the one proposed by Korostelëv and Tsybakov in their article `Optimal rates of convergence of estimators in a probabilistic setup of tomography problem', Problems of Information Transmission, 27:73-81,1991. For the estimator proposed in this article, we then show that it converges to the true function at a minimax optimal rate.
Standard approaches to stochastic gradient estimation, with only noisy black-box function evaluations, use the finite-difference method or its variants. While natural, it is open to our knowledge whether their statistical accuracy is the best possible. This paper argues so by showing that central finite-difference is a nearly minimax optimal zeroth-order gradient estimator for a suitable class of objective functions and mean squared risk, among both the class of linear estimators and the much larger class of all (nonlinear) estimators.
We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to quantization in terms of the number of bits, the signal size, and the noise level. This establishes the Pareto optimal tradeoff between storage and risk under quantization constraints for Sobolev spaces. Our results and proof techniques combine elements of rate distortion theory an…
Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate.
This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of semimartingales. Given $ n $ observations of increments over intervals of length $1/n$, the rates of convergence are $1 / \sqrt{n} $ if $ r \leq 1 $ and $ (n\log n)^{(r-2)/2} $ if $ r>1 $, which are optimal in a minimax sense. We bound the co-jump index activity from below with the harmonic mean. Finally, we as…
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 $(p,β) \in [1, 3) \times (0, 2]$ or $(p, β) \in \mathcal{A}_d$, where $\mathcal{A}_d$ is a sp…
We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R.~C. Liu, and B. MacGibbon, {\it Minimax risk over hyperrectangles, and implications}, Ann.~ Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation i…
We address the problem of adaptive minimax estimation in white gaussian noise model under $L_p$--loss, $1\leq p\leq\infty,$ on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called Lp-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm inde…
We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval $[0,T]$, when the risk is given by the energy functional associated to some fractional Sobolev space $H^1_0\subset W^{α,2}\subset L^2$. In both situations, Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).
This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in a multivariate setting. We first demonstrate that conditional density estimation is impossible if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$. This motivates us to consider a sub-class of absolutely continuous distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only be Hölder smooth in $x$, but also be total variation smooth in $z$. We propose a corresponding kernel-based estimator and prove that it achieves the mi…
In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the syste…
Inference about dependencies in a multiway data array can be made using the array normal model, which corresponds to the class of multivariate normal distributions with separable covariance matrices. Maximum likelihood and Bayesian methods for inference in the array normal model have appeared in the literature, but there have not been any results concerning the optimality properties of such estimators. In this article, we obtain results for the array normal model that are analogous to some classical results concerning covariance estimation for the multivariate normal model. We show that under …
We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, one does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares estimator, i.e., they achieve lower mean-squared er…
We investigate minimax results for the anisotropic functional deconvolution model when observations are affected by the presence of long-memory. Under specific conditions about the covariance matrices of the errors, we follow a standard procedure to construct an adaptive wavelet-based estimator that attains asymptotically near-optimal convergence rates. These rates depend on the parameter associated with the weakest long-range dependence, and deteriorate as the intensity of long-memory increases. This behavior suggests that the estimator adjusts to the best case scenario and that the weakest L…
We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback--Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.
A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of non-adaptive tomography scales as $O(1/\sqrt{N})$, in contrast to that of classical probability estimation which is $O(1/N)$. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with…
In the present paper, we consider the estimation of a periodic two-dimensional function $f(\cdot,\cdot)$ based on observations from its noisy convolution, and convolution kernel $g(\cdot,\cdot)$ unknown. We derive the minimax lower bounds for the mean squared error assuming that $f$ belongs to certain Besov space and the kernel function $g$ satisfies some smoothness properties. We construct an adaptive hard-thresholding wavelet estimator that is asymptotically near-optimal within a logarithmic factor in a wide range of Besov balls. The proposed estimation algorithm implements a truncation to e…
In a sparse high-dimensional elliptical model we consider a hard threshold estimator for the correlation matrix based on Kendall's tau with threshold level $α(\frac{\log p}{n})^{1/2}$. Parameters $α$ are identified such that the threshold estimator achieves the minimax rate under the squared Frobenius norm and the squared spectral norm. This allows a reasonable calibration of the estimator without any quantitative information about the tails of the underlying distribution. For Gaussian observations we even establish a critical threshold constant $α^\ast$ under the squared Frobenius norm, i.e. …
We study the problem of estimating a mean pattern from a set of similar curves in the setting where the variability in the data is due to random geometric deformations and additive noise. We propose an estimator based on the notion of Frechet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. We derive a minimax rate for this estimation problem, and we show that our estimator achieves this optimal rate under the asymptotics where both the number of curves and the number of sampling points go to infinity.
We consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ composed of points close to a $d$-dimensional submanifold $M$, with or without outliers drawn in the ambient space, is observed. Based on the Tangential Delaunay Complex, we construct an estimator $\hat{M}$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. The estimator $\hat{M}$ is built from existing algorithms. In a model with additive noise of small amplitude, we show that this estimator is asymptotically minimax optim…
In this paper, we consider estimators for an additive functional of $ϕ$, which is defined as $θ(P;ϕ)=\sum_{i=1}^kϕ(p_i)$, from $n$ i.i.d. random samples drawn from a discrete distribution $P=(p_1,...,p_k)$ with alphabet size $k$. We propose a minimax optimal estimator for the estimation problem of the additive functional. We reveal that the minimax optimal rate is characterized by the divergence speed of the fourth derivative of $ϕ$ if the divergence speed is high. As a result, we show there is no consistent estimator if the divergence speed of the fourth derivative of $ϕ$ is larger than $p^{-…
We consider the regression model with (known) random design. We investigate the minimax performances of an adaptive wavelet block thresholding estimator under the $\mathbb{L}^p$ risk with $p\ge 2$ over Besov balls. We prove that it is near optimal and that it achieves better rates of convergence than the conventional term-by-term estimators (hard, soft,...).
