Shape constrained kernel density estimation

Birke, Melanie

doi:10.1016/j.jspi.2009.01.007

Cited by 7 publications

(4 citation statements)

References 26 publications

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“…The conclusion that performance can be improved when appropriate shape constraints are incorporated is consistent with findings in the large body of prior work that has incorporated shape constraints in density estimation with error-free data, e.g., Turnbull and Ghosh [2014], Zhang [1990], Dupačová [1992], Papp and Alizadeh [2014], Royset and Wets [2013], and in the limited prior work that has incorporated shape constraints in KD (Carroll et al [2011]; Birke [2009]).…”

Section: Introductionsupporting

confidence: 79%

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Yang

Apley

Staum

et al. 2020

Journal of Computational and Graphical Statistics

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Distribution estimation for noisy data via density deconvolution is a notoriously difficult problem for typical noise distributions like Gaussian. We develop a density deconvolution estimator based on quadratic programming (QP) that can achieve better estimation than kernel density deconvolution methods. The QP approach appears to have a more favorable regularization tradeoff between oversmoothing vs. oscillation, especially at the tails of the distribution. An additional advantage is that it is straightforward to incorporate a number of common density constraints such as nonnegativity, integration-to-one, unimodality, tail convexity, tail monotonicity, and support constraints. We demonstrate that the QP approach has outstanding estimation performance relative to existing methods. Its performance is superior when only the universally applicable nonnegativity and integration-to-one constraints are incorporated, and incorporating additional common constraints when applicable (e.g., nonnegative support, unimodality, tail monotonicity or convexity, etc.) can further substantially improve the estimation.We consider the following statistical problem. Suppose a random variable (r.v.) X and its probability density function (pdf) f X (·), cumulative distribution function (cdf) F X (·), and various quantiles are of interest, but only a random sample of noisy observations {Y 1 , · · · , Y n } are available with which to estimate the pdf, cdf, and quantiles. The underlying model is Y i = X i +Z i , i ∈ {1, 2, · · · , n}, where the Z i 's represent observation errors and are independent of the X i 's. As is typical in the extensive literature on density estimation with noisy observations, (e.g., Carroll and Hall [1988], Stefanski [1990], Fan [1991], Diggle and Hall [1993], Delaigle and Gijbels [2004], Hall and Meister [2007], Meister [2009]), the pdf f Z of Z is assumed to be known. Existing estimators for this problem have slow convergence rates and poor finite-sample accuracy. Although their asymptotic convergence rates are optimal and thus cannot be improved, in this paper we propose new estimates based on quadratic programming whose finite-sample performance improves over existing estimators substantially. We note that most of the prior work on this topic casts the problem directly in terms of pdf estimation and refers to it as density deconvolution, recognizing that estimates of the cdf and the quantiles can be obtained in the obvious manner from an estimate of the pdf. We adopt the same convention in this paper, although we are interested in cdf and quantile estimation, in addition to pdf estimation.For the additive measurement error model, the pdf f Y of Y is the convolutionThis convolution in the spatial domain corresponds to multiplication φ Y (ω) = φ X (ω) · φ Z (ω) in the Fourier domain, where φ Y denotes the Fourier transform of f Y (likewise for φ Z and φ X ), and ω denotes frequency. In light of this, one classic and popular method is the Fourier-based kernel deconvolution (KD) (e.g., Carroll and Hall [1988], St...

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Section: Introductionsupporting

confidence: 79%

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Yang

Apley

Staum

et al. 2020

Journal of Computational and Graphical Statistics

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“…See for example the relatively recent contributions by Groeneboom (2001), Hall and Huang (2001, 2002), Hall and Kang (2005), Dette et al (2006), Antoniadis et al (2007), Neumeyer (2007), Pal and Woodroofe (2007), Birke and Dette (2007), Birke (2009), Dümbgen and Rufibach (2009), Cule et al (2010) and the references therein. Earlier work includes that of Friedman and Tibshirani (1984), Mukerjee (1988), Kelly and Rice (1990), Mammen (1991, 1995) and Sun and Woodroofe (1996).…”

Section: Introductionmentioning

confidence: 99%

Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

Carroll

Delaigle

Hall

2011

Journal of the American Statistical Association

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In many applications we can expect that, or are interested to know if, a density function or a regression curve satisfies some specific shape constraints. For example, when the explanatory variable, X, represents the value taken by a treatment or dosage, the conditional mean of the response, Y , is often anticipated to be a monotone function of X. Indeed, if this regression mean is not monotone (in the appropriate direction) then the medical or commercial value of the treatment is likely to be significantly curtailed, at least for values of X that lie beyond the point at which monotonicity fails. In the case of a density, common shape constraints include log-concavity and unimodality. If we can correctly guess the shape of a curve, then nonparametric estimators can be improved by taking this information into account. Addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint, and, if the conclusion of the test is positive, a technique for estimating the curve subject to the constraint. Nonparametric methodology for solving these problems already exists, but only in cases where the covariates are observed precisely. However in many problems, data can only be observed with measurement errors, and the methods employed in the error-free case typically do not carry over to this error context. In this paper we develop a novel approach to hypothesis testing and function estimation under shape constraints, which is valid in the context of measurement errors. Our method is based on tilting an estimator of the density or the regression mean until it satisfies the shape constraint, and we take as our test statistic the distance through which it is tilted. Bootstrap methods are used to calibrate the test. The constrained curve estimators that we develop are also based on tilting, and in that context our work has points of contact with methodology in the errorfree case.

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“…A variety of approaches to obtaining shape‐restricted estimates have been attempted, and a variety of different constraints have been considered. Important progress has been made in recent years, particularly on the constraint of log‐concavity (Dümbgen & Rufibach, ; Birke, ; Cule et al ; Samworth, ), but also in other directions, including the use of exponential epi‐splines to handle several different types of constraints (Royset & Wets, ), or Bernstein polynomials to handle unimodality (Turnbull & Ghosh, ; Papp & Alizadeh, ).…”

Section: Introductionmentioning

confidence: 99%

A practical implementation of weighted kernel density estimation for handling shape constraints

Wolters

Braun

2018

Stat

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The weighted kernel density estimator is an attractive option for shape-restricted density estimation, because it is simple, familiar, and potentially applicable to many different shape constraints. Despite this, no reliable software implementation has appeared since the method's original proposal in 2002. We found that serious numerical and practical difficulties arise when attempting to implement the method. We overcame these difficulties and in the process discovered that the weighted method and our own recently proposed method-controlling the shape of a kernel density using an adjustment curve-can be unified in a single computational framework. This article describes our findings and introduces the R package scdensity, which can be used to easily obtain density estimates that are unimodal, bimodal, symmetric, and more.

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Shape constrained kernel density estimation

Cited by 7 publications

References 26 publications

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

A practical implementation of weighted kernel density estimation for handling shape constraints

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