Lepski O. V., Rebelles G
This paper deals with non-parametric density estimation on $\bR^2$ from i.i.d observations. It is assumed that after unknown rotation of the coordinate system the coordinates of the observations are independent random variables whose densities belong to a Hölder class with unknown parameters. The minimax and adaptive minimax theories for this structural statistical model are developed.
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The paper deals with the density estimation on Rd under sup- norm loss. We provide with fully data-driven estimation procedure and establish for it so called sup-norm oracle inequality. The pro- posed estimator allows to take into account not only approximation properties of the underlying density but eventual independence struc- ture as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in multivariate density model. Usefulness of the developed approach is illustrated by appli- cation to adaptive estimation over anisotropic Nikolskii clas…
Besov-Laplace priors in density estimation: optimal posterior contraction rates and adaptation
本文研究了密度估计模型中 Besov-Laplace 先验的理论恢复表现，证明其经调参可达最优后验收缩率，且通过在正则性参数上引入超先验的分层过程能适应任意平滑度水平。
The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, e.g., squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e.\ made up of $k$ affine pieces), provided $k$ is not too large. Our results use two different techniques: the first relies on a new Marshall's inequality for log-concave dens…
A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Hölder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a density-dependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that…
The Cost of Adaptation under Differential Privacy: Optimal Adaptive Federated Density Estimation
研究探讨了在差分隐私约束下密度估计的适应成本，提出了一种新的FDP估计器，揭示了隐私对适应性性能的影响。
This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\in\mathbb{R}$ with i.i.d. observations, $$ X_1,\dots,X_n\sim (1-ε)f+εg, $$ where $g$ stands for a contamination distribution. In the context of multiple testing, this can be interpreted as estimating the null density at a point. We carefully study the effect of contamination on estimation through the following model indices: contamination proportion $ε$, smoothness of target density $β_0$, smoothness of contamination density $β_1$, and level of contami…
We consider the estimation of the global mode of a density under some decay rate condition around the global mode. We show that the maximum of a histogram, with proper choice of bandwidth, achieves the minimax rate that we establish for the setting that we consider. This is based on knowledge of the decay rate. Addressing the situation where the decay rate is unknown, we propose a multiscale variant consisting in the recursive refinement of a histogram, which is shown to be minimax adaptive. These methods run in linear time, and we prove in an appendix that this is best possible: There is no e…
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piecewise affine. The complexity of such densities~$f$ can be measured in terms of the sum $Γ(f)$ of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by $f$. Given $n$ independent observations from a $d$-dimensional log-concave density with $d \in \{2,3\}$, we prove a sharp oracle inequality, which in particular implies that the Kull…
A multivariate adaptation of direct kernel estimator of density ratio
本研究将Ćwik和Mielniczuk（1989）提出的无需估计目标密度的单变量核密度比直接估计量，推广至多元情况。
We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $α$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $B^s_{pq}$ under mean integrated $\mathbb L^r$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfil the privacy requirement, we suggest…
In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary $\rL_p$--losses. These results are applied for adaptive estimation in the additive multi--index model.
Double Debiased Covariate Shift Adaptation Robust to Density-Ratio Estimation
针对协变量偏移下重要性加权依赖密度比估计并引入偏差的问题，提出结合回归函数与 double machine learning 的双重稳健估计量，证明其渐近性质与一致性，并通过模拟验证有效性。
Our article addresses the problem of flexibly estimating a multivariate density while also attempting to estimate its marginals correctly. We do so by proposing two new estimators that try to capture the best features of mixture of normals and copula estimators while avoiding some of their weaknesses. The first estimator we propose is a mixture of normals copula model that is a flexible alternative to parametric copula models such as the normal and t copula. The second is a marginally adapted mixture of normals estimator that improves on the standard mixture of normals by using information con…
Consider the problem when $X_1,X_2,..., X_n$ are distributed on a circle following an unknown distribution $F$ on $S^1$. In this article we have consider the absolute general set-up where the density can have local features such as discontinuities and edges. Furthermore, there can be outlying data which can follow some discrete distributions. The traditional Kernel Density Estimation methods fail to identify such local features in the data. Here we device a non-parametric density estimate on $S^1$, by the use of a novel technique which we term as Fourier Spline. We have also tried to identify …
We present a new discriminative technique for the multiple-source adaptation, MSA, problem. Unlike previous work, which relies on density estimation for each source domain, our solution only requires conditional probabilities that can easily be accurately estimated from unlabeled data from the source domains. We give a detailed analysis of our new technique, including general guarantees based on Rényi divergences, and learning bounds when conditional Maxent is used for estimating conditional probabilities for a point to belong to a source domain. We show that these guarantees compare favorably…
Domain adaptation (DA) arises as an important problem in statistical machine learning when the source data used to train a model is different from the target data used to test the model. Recent advances in DA have mainly been application-driven and have largely relied on the idea of a common subspace for source and target data. To understand the empirical successes and failures of DA methods, we propose a theoretical framework via structural causal models that enables analysis and comparison of the prediction performance of DA methods. This framework also allows us to itemize the assumptions n…
Structural adaptation via directional regularity: rate accelerated estimation in multivariate functional data
本文提出方向正则性定义多元函数数据的各向异性，通过自适应基变换加速收敛，并构建了估计基变换矩阵的算法，提供非渐近界和模拟验证，建议作为标准预处理步骤。
This paper continues the research started in \cite{LW16}. In the framework of the convolution structure density model on $\bR^d$, we address the problem of adaptive minimax estimation with $\bL_p$--loss over the scale of anisotropic Nikol'skii classes. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. In particular, we show that the boundedness of the function to be estimated leads to an essential improvement of the asymptotic of the minimax risk. We prove…
A novel block prior is proposed for adaptive Bayesian estimation. The prior does not depend on the smoothness of the function or the sample size. It puts sufficient prior mass near the true signal and automatically concentrates on its effective dimension. A rate-optimal posterior contraction is obtained in a general framework, which includes density estimation, white noise model, Gaussian sequence model, Gaussian regression and spectral density estimation.
We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density. These confidence sets are adaptive in $L^2$ loss over a continuous class of Sobolev type spaces. Adaptation holds in the smoothness of the regression function, over the maximal parameter spaces where adaptation is possible, provided the design density is smooth enough. We identify two key regimes --- one where adaptation is possible, and one where some critical regions must be removed. We address related questions about goodness of fit testing and adaptive estimation of r…
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