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Indirect inference estimators (i.e., simulation-based minimum distance estimators) in a parametric model that are based on auxiliary non-parametric maximum likelihood density estimators are shown to be asymptotically normal. If the…

Statistics Theory · Mathematics 2012-01-24 Florian Gach , Benedikt M. Pötscher

We propose a new threshold selection method for the nonparametric estimation of the extremal index of stochastic processes. The so-called discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability…

Statistics Theory · Mathematics 2020-09-07 Natalia M. Markovich , Igor V. Rodionov

We consider divergence-based high order discretizations of an $L^2$-based first order system least squares formulation of a second order elliptic equation with Robin boundary conditions. For smooth geometries, we show optimal convergence…

Numerical Analysis · Mathematics 2024-08-08 Maximilian Bernkopf , Jens Markus Melenk

Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge2$ where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb{T}$ is a…

Probability · Mathematics 2009-12-14 Vidmantas Bentkus , Bing-Yi Jing , Wang Zhou

We present a new non-parametric statistic, called the weighed $\ell_2$ divergence, based on empirical distributions for sequential change detection. We start by constructing the weighed $\ell_2$ divergence as a fundamental building block…

Statistics Theory · Mathematics 2021-02-25 Liyan Xie , Yao Xie

Multivariate elliptically-contoured distributions are widely used for modeling correlated and non-Gaussian data. In this work, we study the kurtosis of the elliptical model, which is an important parameter in many statistical analysis.…

Statistics Theory · Mathematics 2024-08-23 Bowen Zhou , Peirong Xu , Cheng Wang

We focus on the nonparametric density estimation problem with directional data. We propose a new rule for bandwidth selection for kernel density estimation. Our procedure is automatic, fully data-driven and adaptive to the smoothness degree…

Statistics Theory · Mathematics 2018-08-08 Thanh Mai Pham Ngoc

Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a…

Machine Learning · Computer Science 2024-08-06 Gen Li , Yuting Wei , Yuejie Chi , Yuxin Chen

Nonparametric maximum likelihood estimators (MLEs) in inverse problems often have non-normal limit distributions, like Chernoff's distribution. However, if one considers smooth functionals of the model, with corresponding functionals of the…

Statistics Theory · Mathematics 2023-10-24 Piet Groeneboom

We consider a recurrent Markov process which is an It\^o semi-martingale. The L\'evy kernel describes the law of its jumps. Based on observations X(0),X({\Delta}),...,X(n{\Delta}), we construct an estimator for the L\'evy kernel's density.…

Statistics Theory · Mathematics 2013-05-14 Florian A. J. Ueltzhöfer

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…

Statistics Theory · Mathematics 2021-03-15 Michael Li , Matey Neykov , Sivaraman Balakrishnan

We consider the problem of estimating a $d$-dimensional discrete distribution from its samples observed under a $b$-bit communication constraint. In contrast to most previous results that largely focus on the global minimax error, we study…

Information Theory · Computer Science 2021-11-02 Wei-Ning Chen , Peter Kairouz , Ayfer Özgür

Consider nonparametric function estimation under $L^p$-loss. The minimax rate for estimation of the regression function over a H\"older ball with smoothness index $\beta$ is $n^{-\beta/(2\beta+1)}$ if $1\leq p<\infty$ and $(n/\log…

Statistics Theory · Mathematics 2015-02-10 Johannes Schmidt-Hieber

We address the problem of learning an unknown smooth function and its derivatives from noisy pointwise evaluations under the supremum norm. While classical nonparametric regression provides a strong theoretical foundation, traditional…

Machine Learning · Computer Science 2026-03-10 Davide Maran , Marcello Restelli

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They…

Statistics Theory · Mathematics 2016-07-25 Shashank Singh , Simon S. Du , Barnabás Póczos

We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on $(0,\infty)$. For the maximum likelihood (ML) estimator and…

Statistics Theory · Mathematics 2009-04-02 Geurt Jongbloed , Frank H. van der Meulen

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown…

Methodology · Statistics 2017-08-21 Catia Scricciolo

We study the performances of an adaptive procedure based on a convex combination, with data-driven weights, of term-by-term thresholded wavelet estimators. For the bounded regression model, with random uniform design, and the nonparametric…

Statistics Theory · Mathematics 2016-08-16 Christophe Chesneau , Guillaume Lecué

We develop a finite-sample optimal estimator for regression discontinuity design when the outcomes are bounded, including binary outcomes as the leading case. Our estimator achieves minimax mean squared error among linear shrinkage…

Econometrics · Economics 2025-12-29 Takuya Ishihara , Masayuki Sawada , Kohei Yata

We obtain an error estimate between viscosity solutions and \delta-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is…

Analysis of PDEs · Mathematics 2016-03-07 Olga Turanova