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全凹回归AI 解读

Shape constraints in nonparametric regression provide a powerful framework for estimating regression functions under realistic assumptions without tuning parameters. However, most existing methods—except additive models—impose too weak restrictions, often leading to overfitting in high dimensions. Conversely, additive models can be too rigid, failing to capture covariate interactions. This paper introduces a novel multivariate shape-constrained regression approach based on…

通过非参数极大似然实现多元异方差经验贝叶斯估计AI 解读

Abstract Multivariate, heteroscedastic errors complicate statistical inference in many large-scale denoizing problems. Empirical Bayes is attractive in such settings, but standard parametric approaches rest on assumptions about the form of the prior distribution which can be hard to justify and which introduce unnecessary tuning parameters. We extend the nonparametric maximum-likelihood estimator (NPMLE) for Gaussian location mixture densities to allow for multivariate,…

多维凸回归：最小二乘估计量的次优性AI 解读

Under the usual nonparametric regression model with Gaussian errors, Least Squares Estimators (LSEs) over natural subclasses of convex functions are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random…

含协变量的双成分混合模型AI 解读

In this article, we study a generalization of the two-groups model in the presence of covariates—a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study…

通过完全单调性与Hardy–Krause变差推广多元保序回归和总变差去噪AI 解读

We consider the problem of nonparametric regression when the covariate is d dimensional, where d≥1. In this paper, we introduce and study two nonparametric least squares estimators (LSEs) in this setting—the entirely monotonic LSE and the constrained Hardy–Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the…

多元对数凹密度估计的自适应性AI 解读

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 $\Gamma(f)$ of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by $f$. Given $n$ independent observations from a $d$-dimens…

高斯位置混合密度的非参数极大似然估计及其高斯去噪应用AI 解读

We study the nonparametric maximum likelihood estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true…

一元全变差去噪与趋势滤波的自适应风险界AI 解读

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer $r\geq1$, the $r$th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and…

凸锥假设检验的几何：广义似然比检验与极小极大半径AI 解读

We consider a compound testing problem within the Gaussian sequence model in which the null and alternative are specified by a pair of closed, convex cones. Such cone testing problem arises in various applications, including detection of treatment effects, trend detection in econometrics, signal detection in radar processing and shape-constrained inference in nonparametric statistics. We provide a sharp characterization of the GLRT testing radius up to a universal…

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, for example, 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), or close to $k$…

平面凸集的自适应估计AI 解读

In this paper, we consider adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function. Both the problem of estimating the support function at a point and that of estimating the whole convex set are studied. For pointwise estimation, we consider the problem in a general nonasymptotic framework, which evaluates the performance of a procedure at each individual set, instead of the worst case performance over a large parameter…

保序及其他形状约束回归中的风险界AI 解读

We consider the problem of estimating an unknown $\theta\in\mathbb{R}^{n}$ from noisy observations under the constraint that $\theta$ belongs to certain convex polyhedral cones in $\mathbb{R}^{n}$. Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence $\theta$ which highlights the adaptive behavior of $\theta$. As special cases of our general result, we derive risk…

由支持函数估计凸集的最优收敛率AI 解读

We present a minimax optimal solution to the problem of estimating a compact, convex set from finitely many noisy measurements of its support function. The solution is based on appropriate regularizations of the least squares estimator. Both fixed and random designs are considered.