Unimodal density estimation using Bernstein polynomials

Turnbull, Bradley; Ghosh, Sujit K.

doi:10.1016/j.csda.2013.10.021

Cited by 38 publications

(34 citation statements)

References 31 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]).…”

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confidence: 78%

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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confidence: 78%

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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“…Natanson, 1964). In fairly general situations, applying an asymptotic result of Babu et al (2002), Turnbull and Ghosh (2014) recommend setting the degree m = log n/n .…”

Section: Likelihood Functionmentioning

confidence: 99%

“…The first two methods were also investigated in Section 10.1 of Zhang (1997). For analyzing the angular parameters in the Bayesian approach, following the recommendation of Turnbull and Ghosh (2014), we set the degree of the Bernstein polynomial, m = n/ log n = 38.…”

Section: Simulation Studymentioning

confidence: 99%

Probabilistic Detection and Estimation of Conic Sections From Noisy Data

Guha

Ghosh

2020

Journal of Computational and Graphical Statistics

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Inferring unknown conic sections on the basis of noisy data is a challenging problem with applications in computer vision. A major limitation of the currently available methods for conic sections is that estimation methods rely on the underlying shape of the conics (being known to be ellipse, parabola or hyperbola). A general purpose Bayesian hierarchical model is proposed for conic sections and corresponding estimation method based on noisy data is shown to work even when the specific nature of the conic section is unknown. The model, thus, provides probabilistic detection of the underlying conic section and inference about the associated parameters of the conic section. Through extensive simulation studies where the true conics may not be known, the methodology is demonstrated to have practical and methodological advantages relative to many existing techniques. In addition, the proposed method provides probabilistic measures of uncertainty of the estimated parameters. Furthermore, we observe high fidelity to the true conics even in challenging situations, such as data arising from partial conics in arbitrarily rotated and non-standard form, and where a visual inspection is unable to correctly identify the type of conic section underlying the data.

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“…2) границы интервала [a,b] распределения случайной величины. Если границы не известны, то их можно заменить оценками [20]: ()…”

Section: расчет оптимальных весовых коэффициентов в программе Mathcadunclassified

“…The criteria minimizing the root-mean-square error of approximation (L 2 metric), the uniform metric L ∞ , the sigma-metric, the Kullback-Leibler divergence, the Anderson and Darling (AD) criterion, and the sum of the error squares are considered. Instead of AD statistics, which is used in a well-known work by Bradley C. Turnbull, Sujit K. Ghosh (2014), it is suggested to apply the criterion of the least squares method. This allowed to do without solving quadratic programming problems.…”

mentioning

confidence: 99%