Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0605292

Linear and convex aggregation of density estimators

Philippe Rigollet·Alexandre Tsybakov
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the problem of linear and convex aggregation of $M$ estimators of a density with respect to the mean squared risk. We provide procedures for linear and convex aggregation and we prove oracle inequalities for their risks. We also obtain lower bounds showing that these procedures are rate optimal in a minimax sense. As an example, we apply general results to aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense for a large class of kernels including Gaussian, Silverman's and Pinsker's ones. We prove that, for Pinsker's kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.

Explore related subjects

math.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Philippe Rigollet, Alexandre Tsybakov. 2006-05-11. Linear and convex aggregation of density estimators. https://arxiv.org/abs/math/0605292

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gaussian Approximation for High-Dimensional Weighted and Two-Sample $U$-statistics with Size-Dependent Kernels

We develop Gaussian approximations for high-dimensional vectors formed by second-order $U$-and $V$-statistics whose kernels depend on sample size. To accommodate weighted and two-sample $U$-and $V$-statistics within a unified framework, we establish our results under independent but not identically distributed (i.n.i.d.) sampling. The resulting theorems apply to a broad range of problems, including estimation in linear regression models with many covariates or many-weak instruments, variance component analysis, smoothed two-sample testing, and inference for separately exchangeable arrays, among others. Our results do not require either component of the Hoeffding decomposition to dominate, thereby covering both non-degenerate and degenerate regimes. In addition, we develop maximal inequalities for high-dimensional $U$-statistics with size-dependent kernels under the i.n.i.d.~setting, in a form that remains sharp across a broad range of applications, which may be of independent interest. Finally, applications to estimation and testing with many weak instrumental variables demonstrate that, our results are sharp enough to recover the corresponding fixed-dimensional rate conditions up to logarithmic factors.

math.ST↗
arXiv2026

Asymmetric conformal prediction with penalized kernel sum-of-squares

Conformal prediction (CP) is a distribution-free method to construct reliable prediction intervals that has gained significant attention in recent years. Despite its success and various proposed extensions, a significant practical feature which has been overlooked in previous research is the potential skewed nature of the noise, or of the residuals when the predictive model exhibits bias. In this work, we leverage recent developments in CP to propose a new asymmetric procedure that bridges the gap between skewed and non-skewed noise distributions, while still maintaining adaptivity of the prediction intervals. We introduce a new statistical learning problem to construct adaptive and asymmetric prediction bands, with a unique feature based on a penalty which promotes symmetry: when its intensity varies, the intervals smoothly change from symmetric to asymmetric ones. This learning problem is based on reproducing kernel Hilbert spaces and the recently introduced kernel sum-of-squares framework. First, we establish representer theorems to make our problem tractable in practice, and derive dual formulations which are essential for scalability to larger datasets. Second, the intensity of the penalty is chosen using a novel data-driven method which automatically identifies the symmetric nature of the noise. We show that consenting to some asymmetry can let the learned prediction bands better adapt to small sample regimes or biased predictive models.

math.ST↗
arXiv2026

The Pivotal Information Criterion

The Bayesian and Akaike information criteria aim at finding a good balance between under- and over-fitting. They are extensively used every day by practitioners. Yet we contend they suffer from at least two afflictions: their penalty parameters $λ=\log n$ and $λ=2$ are too small, leading to many false discoveries, and their inherent (best subset) discrete optimization is infeasible in high dimension. We alleviate these issues with the pivotal information criterion: PIC is defined as a continuous optimization problem, and the PIC penalty parameter $λ$ is selected at the detection boundary (under pure noise). PIC's choice of $λ$ is an upper quantile of a statistic that we prove to be (asymptotically) pivotal, provided the loss function is appropriately transformed. As a result, simulations show a phase transition in the probability of exact support recovery with PIC, a phenomenon studied with no noise in compressed sensing. Applied to real data, for similar predictive performances, PIC selects the least complex model among state-of-the-art learners.

math.ST↗