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Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma Z_i$ and $Y_i$ and $Z_i$ are independent. Assume that unobservable $Y$'s are distributed as a random variable $UV,$ where $U$ and $V$ are independent, $U$ has a Bernoulli…

Statistics Theory · Mathematics 2008-04-30 Bert van Es , Shota Gugushvili , Peter Spreij

Let $X_1,..., X_n$ be i.i.d.\ copies of a random variable $X=Y+Z,$ where $ X_i=Y_i+Z_i,$ and $Y_i$ and $Z_i$ are independent and have the same distribution as $Y$ and $Z,$ respectively. Assume that the random variables $Y_i$'s are…

Statistics Theory · Mathematics 2018-04-17 Shota Gugushvili , Bert van Es , Peter Spreij

We study the recovery of the distribution function $F_X$ of a random variable $X$ that is subject to an independent additive random error $\varepsilon$. To be precise, it is assumed that the target variable $X$ is available only in the form…

Statistics Theory · Mathematics 2025-10-31 Henrik Kaiser

We consider estimation of the common probability density $f$ of i.i.d. random variables $X_i$ that are observed with an additive i.i.d. noise. We assume that the unknown density $f$ belongs to a class $\mathcal{A}$ of densities whose…

Statistics Theory · Mathematics 2007-06-13 Cristina Butucea , Alexandre B. Tsybakov

Uncertainty estimation is a key component in any deployed machine learning system. One way to evaluate uncertainty estimation is using "out-of-distribution" (OoD) detection, that is, distinguishing between the training data distribution and…

Machine Learning · Computer Science 2021-12-03 Haiwen Huang , Joost van Amersfoort , Yarin Gal

This paper addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the…

Statistics Theory · Mathematics 2023-04-12 David Kent , David Ruppert

In reliability theory and survival analysis, observed data are often weakly dependent and subject to additive measurement errors. Such contamination arises when the underlying data are neither independent nor strongly mixed but instead…

Statistics Theory · Mathematics 2025-03-20 Benjrada Mohammed Essalih

This paper considers the deconvolution problem in the case where the target signal is multidimensional and no information is known about the noise distribution. More precisely, no assumption is made on the noise distribution and no samples…

Statistics Theory · Mathematics 2021-02-18 Elisabeth Gassiat , Sylvain Le Corff , Luc Lehéricy

We consider nonparametric measurement error density deconvolution subject to heteroscedastic measurement errors as well as symmetry about zero and shape constraints, in particular unimodality. The problem is motivated by applications where…

Methodology · Statistics 2020-02-19 Ya Su , Anirban Bhattacharya , Yan Zhang , Nilanjan Chatterjee , Raymond J. Carroll

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative.…

Statistics Theory · Mathematics 2011-01-06 Bert van Es

Deconvolution is the important problem of estimating the distribution of a quantity of interest from a sample with additive measurement error. Nearly all methods in the literature are based on Fourier transformation because it is…

Methodology · Statistics 2026-03-03 Yun Cai , Hong Gu , Toby Kenney

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing…

Machine Learning · Statistics 2026-05-06 Arnaud Vadeboncoeur , Mark Girolami , Andrew M. Stuart

In this paper, we address the problem of estimating a multidimensional density $f$ by using indirect observations from the statistical model $Y=X+\varepsilon$. Here, $\varepsilon$ is a measurement error independent of the random vector $X$…

Statistics Theory · Mathematics 2015-05-15 Gilles Rebelles

Conditional density estimation (CDE) is the task of estimating the probability of an event conditioned on some inputs. A neural network (NN) can also be used to compute the output distribution for continuous-domain, which can be viewed as…

Machine Learning · Computer Science 2021-12-30 Bing Chen , Mazharul Islam , Jisuo Gao , Lin Wang

We consider the problem of estimating the density $g$ of identically distributed variables $X\_i$, from a sample $Z\_1, ..., Z\_n$ where $Z\_i=X\_i+\sigma\epsilon\_i$, $i=1, ..., n$ and $\sigma \epsilon\_i$ is a noise independent of $X\_i$…

Statistics Theory · Mathematics 2008-02-11 Fabienne Comte , Yves Rozenholc , Marie-Luce Taupin

Density deconvolution deals with the estimation of the probability density function $f$ of a random signal from $n\geq1$ data observed with independent and known additive random noise. This is a classical problem in statistics, for which…

Methodology · Statistics 2024-12-16 Stefano Favaro , Sandra Fortini

Nonparametric density estimation is an unsupervised learning problem. In this work we propose a two-step procedure that casts the density estimation problem in the first step into a supervised regression problem. The advantage is that we…

Statistics Theory · Mathematics 2024-06-04 Thijs Bos , Johannes Schmidt-Hieber

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax…

Statistics Theory · Mathematics 2016-01-18 Itai Dattner , Markus Reiß , Mathias Trabs

In the convolution model $Z\_i=X\_i+ \epsilon\_i$, we give a model selection procedure to estimate the density of the unobserved variables $(X\_i)\_{1 \leq i \leq n}$, when the sequence $(X\_i)\_{i \geq 1}$ is strictly stationary but not…

Statistics Theory · Mathematics 2016-08-16 Fabienne Comte , Jérôme Dedecker , Marie-Luce Taupin

A new maximum likelihood method for deconvoluting a continuous density with a positive lower bound on a known compact support in additive measurement error models with known error distribution using the approximate Bernstein type polynomial…

Methodology · Statistics 2018-01-30 Zhong Guan