Deconvolution for an atomic distribution: rates of convergence

Gugushvili, Shota; Es, Bert van; Spreij, Peter

doi:10.1080/10485252.2011.576763

Cited by 8 publications

(18 citation statements)

References 15 publications

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“…There are only two recent works (van Es et al, 2008; Lee et al, 2010) which consider mixtures of one discrete atom and one continuous component in the context of measurement error models. (The two estimators are essentially the same, whose convergence rates were recently derived by Gugushvili et al (2011).) However, they do not consider boundaries, hence the proposed estimators give poor performance near non-smooth boundaries.…”

Section: Introductionmentioning

confidence: 89%

Deconvolution estimation of mixture distributions with boundaries

Lee¹,

Hall²,

Shen³

et al. 2013

Electron. J. Statist.

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In this paper, motivated by an important problem in evolutionary biology, we develop two sieve type estimators for distributions that are mixtures of a finite number of discrete atoms and continuous distributions under the framework of measurement error models. While there is a large literature on deconvolution problems, only two articles have previously addressed the problem taken up in our article, and they use relatively standard Fourier deconvolution. As a result the estimators suggested in those two articles are degraded seriously by boundary effects and negativity. A major contribution of our article is correct handling of boundary effects; our method is asymptotically unbiased at the boundaries, and also is guaranteed to be nonnegative. We use roughness penalization to improve the smoothness of the resulting estimator and reduce the estimation variance. We illustrate the performance of the proposed estimators via our real driving application in evolutionary biology and two simulation studies. Furthermore, we establish asymptotic properties of the proposed estimators.

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Section: Introductionmentioning

confidence: 89%

Deconvolution estimation of mixture distributions with boundaries

Lee¹,

Hall²,

Shen³

et al. 2013

Electron. J. Statist.

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“…Recently there are two studies (Lee, Shen, Burch, & Marron, 2010;van Es, Gugushvili, & Spreij, 2008) which consider mixtures of one discrete atom and one continuous component in the context of measurement error models, and independently propose the same estimator. The convergence rate of the estimator is recently derived by Gugushvili, Van Es, and Spreij (2011).…”

Section: Introductionmentioning

confidence: 99%

Least squares sieve estimation of mixture distributions with boundary effects

Lee

Shen

Hall

et al. 2015

Journal of the Korean Statistical Society

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a b s t r a c tIn this study, we propose two types of sieve estimators, based on least squares (LS), for probability distributions that are mixtures of a finite number of discrete atoms and a continuous distribution under the framework of measurement error models. This research is motivated by the maximum likelihood (ML) sieve estimator developed in Lee et al. (2013). We obtain two types of LS sieve estimators through minimizing the distance between the empirical distribution/characteristic functions and the model distribution/characteristic functions. The LS estimators outperform the ML sieve estimator in several aspects: (1) they need much less computational time; (2) they give smaller integrated mean squared error;(3) the characteristic function based LS estimator is more robust against mis-specification of the error distribution. We also use roughness penalization to improve the smoothness of the resulting estimators and reduce the estimation variance. As an application of our proposed LS estimators, we use the Framingham Heart Study data to investigate the distribution of genetic effects on body mass index. Finally asymptotic properties of the LS estimators are investigated.

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“…We use the construction of [GvES11] and introduce a smooth real valued kernel Φ K in the Fourier domain that satisfies:…”

Section: Non Adaptive Estimation Of Pmentioning

confidence: 99%

“…We keep the same presentation and first briefly describe the estimator proposed by [GvES11]. We describe our adaptive procedure.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Cytometry inference through adaptive atomic deconvolution

Costa

Gadat

Gonnord

et al. 2019

Journal of Nonparametric Statistics

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In this paper we consider a statistical estimation problem known as atomic deconvolution. Introduced in reliability, this model has a direct application when considering biological data produced by flow cytometers. In these experiments, biologists measure the fluorescence emission of treated cells and compare them with their natural emission to study the presence of specific molecules on the cells' surface. They observe a signal which is composed of a noise (the natural fluorescence) plus some additional signal related to the quantity of molecule present on the surface if any. From a statistical point of view, we aim at inferring the percentage of cells expressing the selected molecule and the probability distribution function associated with its fluorescence emission. We propose here an adaptive estimation procedure based on a previous deconvolution procedure introduced by [vEGS08, GvES11]. For both estimating the mixing parameter and the mixing density automatically, we use the Lepskii method based on the optimal choice of a bandwidth using a bias-variance decomposition. We then derive some concentration inequalities for our estimators and obtain the convergence rates, that are shown to be minimax optimal (up to some log terms) in Sobolev classes. Finally, we apply our algorithm on simulated and real biological data.

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Deconvolution for an atomic distribution: rates of convergence

Cited by 8 publications

References 15 publications

Deconvolution estimation of mixture distributions with boundaries

Deconvolution estimation of mixture distributions with boundaries

Least squares sieve estimation of mixture distributions with boundary effects

Cytometry inference through adaptive atomic deconvolution

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