Debasis Bhattacharya, A. K. Basu
Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal family of distributions have been investigated under asymmetric loss functions and the asymptotic distribution of the optimal estimator that attains the bound has been obtained.
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For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mea…
We propose the kl-UCB ++ algorithm for regret minimization in stochastic bandit models with exponential families of distributions. We prove that it is simultaneously asymptotically optimal (in the sense of Lai and Robbins' lower bound) and minimax optimal. This is the first algorithm proved to enjoy these two properties at the same time. This work thus merges two different lines of research with simple and clear proofs.
Minimax rates of convergence for nonparametric location-scale models
本文在Hellinger可微性等温和条件下，证明非参数位置-尺度模型中回归函数估计的极小极大收敛速率由函数类的度量熵决定，并通过多种误差分布实例及低阶交互模型和多指标模型加以验证。
Asymptotically Minimax Robust Likelihood Ratio Test
本文提出了一种统一框架，用于在分布不确定性下进行渐近最小最大稳健假设检验，通过非线性方程组计算，并验证了理论结果。
A weighted Shiryaev-Roberts change detection procedure is shown to approximately minimize the expected delay to detection as well as higher moments of the detection delay among all change-point detection procedures with the given low maximal local probability of a false alarm within a window of a fixed length in pointwise and minimax settings for general non-i.i.d. data models and for the composite post-change hypothesis when the post-change parameter is unknown. We establish very general conditions for the models under which the weighted Shiryaev-Roberts procedure is asymptotically optimal. T…
We consider the quickest change-point detection problem in pointwise and minimax settings for general dependent data models. Two new classes of sequential detection procedures associated with the maximal "local" probability of a false alarm within a period of some fixed length are introduced. For these classes of detection procedures, we consider two popular risks: the expected positive part of the delay to detection and the conditional delay to detection. Under very general conditions for the observations, we show that the popular Shiryaev--Roberts procedure is asymptotically optimal, as the …
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.
This paper concerns the estimation of the regression function at a given point in nonparametric heteroscedastic models with Gaussian noise or with noise having unknown distribution. In the two cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.
Sharp Asymptotic Minimaxity for Multiple Testing Using One-Group Shrinkage Priors
研究稀疏高斯序列模型中基于全局-局部尺度混合正态先验的贝叶斯多重检验规则的渐近极小化极大性质，证明在已知或未知稀疏度下，适当选择全局收缩参数可使检验规则达到精确极小化极大风险。
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.
For the classical continuous-time quickest change-point detection problem it is shown that the randomized Shiryaev-Roberts-Pollak procedure is asymptotically nearly minimax-optimal (in the sense of Pollak 1985) in the class of randomized procedures with vanishingly small false alarm risk. The proof is explicit in that all of the relevant performance characteristics are found analytically and in a closed form. The rate of convergence to the (unknown) optimum is elucidated as well. The obtained optimality result is a one-order improvement of that previously obtained by Burnaev et al. (2009) for …
We study predictive density estimation under Kullback-Leibler loss in $\ell_0$-sparse Gaussian sequence models. We propose proper Bayes predictive density estimates and establish asymptotic minimaxity in sparse models. A surprise is the existence of a phase transition in the future-to-past variance ratio $r$. For $r < r_0 = (\surd 5 - 1)/4$, the natural discrete prior ceases to be asymptotically optimal. Instead, for subcritical $r$, a `bi-grid' prior with a central region of reduced grid spacing recovers asymptotic minimaxity. This phenomenon seems to have no analog in the otherwise parallel …
We study the robust quickest change detection under unknown pre- and post-change distributions. To deal with uncertainties in the data-generating distributions, we formulate two data-driven ambiguity sets based on the Wasserstein distance, without any parametric assumptions. The minimax robust test is constructed as the CUSUM test under least favorable distributions, a representative pair of distributions in the ambiguity sets. We show that the minimax robust test can be obtained in a tractable way and is asymptotically optimal. We investigate the effectiveness of the proposed robust test over…
We consider model selection in generalized linear models (GLM) for high-dimensional data and propose a wide class of model selection criteria based on penalized maximum likelihood with a complexity penalty on the model size. We derive a general nonasymptotic upper bound for the expected Kullback-Leibler divergence between the true distribution of the data and that generated by a selected model, and establish the corresponding minimax lower bounds for sparse GLM. For the properly chosen (nonlinear) penalty, the resulting penalized maximum likelihood estimator is shown to be asymptotically minim…
We consider the problems of confidence estimation and hypothesis testing on a parameter of signal observed in Gaussian white noise. For these problems we point out lower bounds of asymptotic efficiency in the zone of moderate deviation probabilities. These lower bounds are versions of local asymptotic minimax Hajek-Le Cam lower bound in estimation and the lower bound for Pitman efficiency in hypothesis testing. The lower bounds were obtained for both logarithmic and sharp asymptotic of moderate deviation probabilities.
We consider estimating the predictive density under Kullback-Leibler loss in an $\ell_0$ sparse Gaussian sequence model. Explicit expressions of the first order minimax risk along with its exact constant, asymptotically least favorable priors and optimal predictive density estimates are derived. Compared to the sparse recovery results involving point estimation of the normal mean, new decision theoretic phenomena are seen. Suboptimal performance of the class of plug-in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk…
We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U-statistic for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U-statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic re…
Minimax $L_2$ risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on $d=O(\log n)$ important predictors among a list of $p$ predictors, with $\log p=o(n)$; (2) the true regression surface depends on $O(n)$ predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness level different from other components. For either modeling assumption, a practicable extension of the widely used Bayesian …
From Asymptotic to Finite-Sample Minimax Robust Hypothesis Testing
本文提出了一种基于渐近理论的有限样本最小最大稳健假设检验方法，通过总变差距离和带模型分析，推导了参数形式的最不利分布和稳健似然比函数，并通过模拟验证了理论结果。
Minimax Optimal Simple Regret in Two-Armed Best-Arm Identification
本研究证明在双臂固定预算最优臂识别中，Neyman分配在简单遗憾上达到极小极大最优，且无需局部分布假设，并在伯努利分布下退化为均匀分配。
Estimating Kullback-Leibler divergence from identical and independently distributed samples is an important problem in various domains. One simple and effective estimator is based on the k nearest neighbor distances between these samples. In this paper, we analyze the convergence rates of the bias and variance of this estimator. Furthermore, we derive a lower bound of the minimax mean square error and show that kNN method is asymptotically rate optimal.
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 consider the problem of comparing probability densities between two groups. A new probabilistic tensor product smoothing spline framework is developed to model the joint density of two variables. Under such a framework, the probability density comparison is equivalent to testing the presence/absence of interactions. We propose a penalized likelihood ratio test for such interaction testing and show that the test statistic is asymptotically chi-square distributed under the null hypothesis. Furthermore, we derive a sharp minimax testing rate based on the Bernstein width for nonparametric two-s…
We consider the problem of predictive density estimation under Kullback-Leibler loss in a high-dimensional Gaussian model with exact sparsity constraints on the location parameters. We study the first order asymptotic minimax risk of Bayes predictive density estimates based on product discrete priors where the proportion of non-zero coordinates converges to zero as dimension increases. Discrete priors that are product of clustered univariate priors provide a tractable configuration for diversification of the future risk and are used for constructing efficient predictive density estimates. We e…
Theory and algorithms are developed for detecting changes in the distribution of statistically periodic random processes. The statistical periodicity is modeled using independent and periodically identically distributed processes, a new class of stochastic processes proposed by us. An algorithm is developed that is minimax asymptotically optimal as the false alarm rate goes to zero. Algorithms are also developed for the cases when the post-change distribution is not known or when there are multiple streams of observations. The modeling is inspired by real datasets encountered in cyber-physical…
Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression
本文研究二维平面上可逆双Lipschitz函数逆估计的最小极大风险，提出水平集表示法并构造渐近可逆估计器，证明其速率与非可逆情形一致，表明可逆约束不增加估计复杂度。
Minimax and adaptive estimation of general linear functionals under sparsity
研究高维稀疏均值向量下线性函数L(θ)=η^Tθ的非渐近最小最大估计，提出结合插值估计和阈值处理的估计器，通过加载依赖稀疏先验得到匹配下界，扩展至非对称噪声和未知噪声方差情况，展示异质性对最小最大和自适应速率的影响。
Sharp Asymptotic Minimaxity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure in Sparse Gaussian Sequence Models
研究稀疏高斯序列模型中GBS逐步多重检验程序的尖锐渐近极小极大性，通过移位顺序统计不等式等方法，证明其在特定损失下是渐近极小极大的，还建立了相关风险上界，扩展了渐近极小极大性理论。
Efficiency of estimators for locally asymptotically normal quantum statistical models
本文为局部渐近正态量子统计模型建立渐近表示定理，借此研究量子估计量的渐近效率并得出超越独立同分布假设的普适紧致下界，为弱量子局部渐近正态性理论奠定基础。
Minimax estimation for Varying Coefficient Model via Laguerre Series
本文通过拉格朗日多项式方法，提出了一种在最小最大意义下最优收敛速率的变系数模型参数估计方法，并建立了参数的渐近正态性，构造了置信区间和假设检验。
We study asymptotically minimax predictive distributions in an infinite sequence model. First, we discuss the connection between the prediction in the infinite sequence model and the prediction in the function model. Second, we construct an asymptotically minimax predictive distribution when the parameter space is a known ellipsoid. We show that the Bayesian predictive distribution based on the Gaussian prior distribution is asymptotically minimax in the ellipsoid. Third, we construct an asymptotically minimax predictive distribution for any Sobolev ellipsoid. We show that the Bayesian predict…
For the problem of nonparametric detection of signal in Gaussian white noise we point out strong asymptotically minimax tests. The sets of alternatives are a ball in Besov space $B^r_{2\infty}$ with "small" balls in $L_2$ removed.
Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the r…
One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.
This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon i…
Minimax Optimal rates of convergence in the shuffled regression, unlinked regression, and deconvolution under vanishing noise
本文研究消失噪声下的单调函数估计问题，推导了混洗回归、无链接回归的极小极大风险阶数关系及反卷积误差界，且未对链接函数加光滑性假设，填补了相关领域空白。
We consider the nonparametric estimation problem of time-dependent multivariate functions observed in a presence of additive cylindrical Gaussian white noise of a small intensity. We derive minimax lower bounds for the $L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero, when the underlying unknown response function is assumed to belong to a ball of appropriately constructed inhomogeneous time-dependent multivariate functions, motivated by practical applications. Furthermore, we propose both non-adaptive linear and adaptive non-linear wavelet estimators that are asym…
We investigate the adversarial bandit problem with multiple plays under semi-bandit feedback. We introduce a highly efficient algorithm that asymptotically achieves the performance of the best switching $m$-arm strategy with minimax optimal regret bounds. To construct our algorithm, we introduce a new expert advice algorithm for the multiple-play setting. By using our expert advice algorithm, we additionally improve the best-known high-probability bound for the multi-play setting by $O(\sqrt{m})$. Our results are guaranteed to hold in an individual sequence manner since we have no statistical …
The paper considers model selection in regression under the additional structural constraints on admissible models where the number of potential predictors might be even larger than the available sample size. We develop a Bayesian formalism as a natural tool for generating a wide class of model selection criteria based on penalized least squares estimation with various complexity penalties associated with a prior on a model size. The resulting criteria are adaptive to structural constraints. We establish the upper bound for the quadratic risk of the resulting MAP estimator and the correspondin…
Minimax spectral estimation of weighted Laplace operators
本文研究了加权拉普拉斯算子谱的最小最大估计问题，提出通过插值方法在不同正则性条件下估计特征函数和特征值的最优速率，并展示了渐近有效估计器的存在性。
This work considers the sample and computational complexity of obtaining an $ε$-optimal policy in a discounted Markov Decision Process (MDP), given only access to a generative model. In this work, we study the effectiveness of the most natural plug-in approach to model-based planning: we build the maximum likelihood estimate of the transition model in the MDP from observations and then find an optimal policy in this empirical MDP. We ask arguably the most basic and unresolved question in model based planning: is the naive "plug-in" approach, non-asymptotically, minimax optimal in the quality o…
We consider the convolution model where i.i.d. random variables $X_i$ having unknown density $f$ are observed with additive i.i.d. noise, independent of the $X$'s. We assume that the density $f$ belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional $\int f^2$ based o…
Minimax bounds for watermarked and masked recursive discrete distribution estimation
本研究探究带水印的递归离散分布估计的极小极大损失，提出掩码缩小损失差距，推测更严格下界可消除剩余差距。
Given a random sample from a distribution with density function that depends on an unknown parameter $θ$, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback--Leibler loss function $D(f_θ||{\hat{f}})=\int{f_θ \log{(f_θ/ hat{f})}}$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk…
Optimal Recovery Meets Minimax Estimation
本文解决了现有极小化极大速率未仔细考虑噪声水平依赖的问题，确定了Besov类在$L_q$-范数误差下的噪声水平感知（NLA）极小化极大速率，实现了与最优恢复速率的调和。
Adapting to general quadratic loss via singular value shrinkage
本研究提出多变量高斯序列模型下的自适应估计方法，通过奇异值收缩估计量适应未知光滑性与任意二次损失，实现多变量Sobolev椭球上的极小极大自适应估计。
We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations X(0),X(Δ),...,X(nΔ), we construct an estimator for the Lévy kernel's density. We prove its consistency (as nΔ->\infty and Δ->0) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator's rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of th…
Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain
针对连续时间系统辨识中窄带扰动导致的Whittle估计敏感问题，提出基于Gumbel位置变换和对称截断得分的鲁棒估计方法，实现局部渐近最小最大性，在b=1.5时方差仅增16%，偏差大幅降低。
We observe a sample of $n$ independent $p$-dimensional Gaussian vectors with Toeplitz covariance matrix $ Σ= [σ_{|i-j|}]_{1 \leq i,j \leq p}$ and $σ_0=1$. We consider the problem of testing the hypothesis that $Σ$ is the identity matrix asymptotically when $n \to \infty$ and $p \to \infty$. We suppose that the covariances $σ_k$ decrease either polynomially ($\sum_{k \geq 1} k^{2α} σ^2_{k} \leq L$ for $ α>1/4$ and $L>0$) or exponentially ($\sum_{k \geq 1} e^{2Ak} σ^2_{k} \leq L$ for $ A,L>0$). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen a…
Minimax-optimal estimation for sparse multi-reference alignment with collision-free signals
针对无碰撞稀疏松度信号的多参考对齐问题，证明极小极大最优估计速率为σ^2/√n，确立受限最大似然估计的统计最优性，并给出其偏差的集中不等式。
Minimax Binary Counting under Shuffle Privacy: Poisson Limits and Randomized-Response Optimality
本文研究了洗牌隐私的非高斯极限，通过泊松、斯凯拉姆和复合泊松模型，揭示了局部随机器集中度对隐私保护的影响。
We study the switch distribution, introduced by Van Erven et al. (2012), applied to model selection and subsequent estimation. While switching was known to be strongly consistent, here we show that it achieves minimax optimal parametric risk rates up to a $\log\log n$ factor when comparing two nested exponential families, partially confirming a conjecture by Lauritzen (2012) and Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn criterion. Moreover, like Bayes factor model selection but unlike standard significance testing, when one of the models represents a simple h…
Asymptotically efficient estimation under local constraint in Wicksell's problem
本文在Wicksell问题中针对分布函数局部为常数的情形，提出三种利用常数区间信息的投影型估计器，证明其渐近正态且达到$\sqrt{n}$最优速率，并通过局部渐近极小极大下界和卷积结果说明IIE虽自适应但非渐近有效。
Ising Models on Inhomogeneous Random Graphs: Inference, Local Asymptotic Minimaxity, and Limit of Experiments
针对亚临界参数下非均匀随机图上的伊辛模型，提出一种计算高效的闭合形式估计量，证明其与极大似然估计具有相同渐近分布和方差，并建立局部渐近极小极大最优性。
In the present paper we consider Laplace deconvolution for discrete noisy data observed on the interval whose length may increase with a sample size. Although this problem arises in a variety of applications, to the best of our knowledge, it has been given very little attention by the statistical community. Our objective is to fill this gap and provide statistical treatment of Laplace deconvolution problem with noisy discrete data. The main contribution of the paper is explicit construction of an asymptotically rate-optimal (in the minimax sense) Laplace deconvolution estimator which is adapti…
Asymmetry along with heteroscedasticity or contamination often occurs with the growth of data dimensionality. In ultra-high dimensional data analysis, such irregular settings are usually overlooked for both theoretical and computational convenience. In this paper, we establish a framework for estimation in high-dimensional regression models using Penalized Robust Approximated quadratic M-estimators (PRAM). This framework allows general settings such as random errors lack of symmetry and homogeneity, or the covariates are not sub-Gaussian. To reduce the possible bias caused by the data's irregu…
The problem of quickly diagnosing an unknown change in a stochastic process is studied. We establish novel bounds on the performance of misspecified diagnosis algorithms designed for changes that differ from those of the process, and pose and solve a new robust quickest change diagnosis problem in the asymptotic regime of few false alarms and false isolations. Simulations suggest that our asymptotically robust solution offers a computationally efficient alternative to generalised likelihood ratio algorithms.
We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of asymptotically valid confidence intervals for the functional, having asymptotically minimal width. One interesting consequence of our …
We consider classification in the presence of class-dependent asymmetric label noise with unknown noise probabilities. In this setting, identifiability conditions are known, but additional assumptions were shown to be required for finite sample rates, and so far only the parametric rate has been obtained. Assuming these identifiability conditions, together with a measure-smoothness condition on the regression function and Tsybakov's margin condition, we show that the Robust kNN classifier of Gao et al. attains, the minimax optimal rates of the noise-free setting, up to a log factor, even when …
Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph $\boldsymbolβ$-Model
本文研究了多层超图$\oldsymbolβ$-模型的推断问题，推导了极大似然估计的收敛速度与极限分布，并证明了拟合优度似然比检验的检测阈值具有极小化极大最优性，填补了图$\oldsymbolβ$-模型文献中的多项理论空白。
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