Least squares sieve estimation of mixture distributions with boundary effects

Lee, Mihee; Shen, Haipeng; Hall, Peter; Guo, Guang; Marron, J. S.

doi:10.1016/j.jkss.2014.07.003

Cited by 5 publications

(12 citation statements)

References 15 publications

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“…Case 1 (𝛼 = 0): This model is different than that in Lee et al (2015) as here we consider the case with single known point mass location (i.e., U = 0) and random noise with unknown variance, 𝜎 2 V . To estimate the unknown parameters, the first step is to discretize the continuous component of U, which we denote by U c .…”

Section: Homoskedastic Frontier Estimationmentioning

confidence: 99%

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

Cai,

Horrace,

Parmeter

2023

J of Applied Econometrics

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SummaryStochastic frontier models for cross‐sectional data typically assume that the one‐sided distribution of firm‐level inefficiency is continuous. However, it may be reasonable to hypothesize that inefficiency is continuous except for a discrete mass at zero capturing fully efficient firms (zero‐inefficiency). We propose a sieve‐type density estimator for such a mixture distribution in a nonparametric stochastic frontier setting under a unimodality‐at‐zero assumption. Consistency, rates of convergence and asymptotic normality of the estimators are established, as well as a test of the zero‐inefficiency hypothesis. Simulations and two applications are provided to demonstrate the practicality of the method.

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Section: Homoskedastic Frontier Estimationmentioning

confidence: 99%

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

Cai,

Horrace,

Parmeter

2023

J of Applied Econometrics

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“…Optimization criteria like the quadratic objective function that we use in our QP estimator are much more amenable to incorporating shape constraints than other density deconvolution approaches. Optimization-based estimators using a regularized version of likelihood (Staudenmayer et al [2008]; Lee et al [2013]) or least squares (Lee et al [2015]) as the objective function were recently considered for density deconvolution, although these works did not investigate the effects of incorporating shape constraints, as we do in this work. Another difference between our work and Lee et al [2015] is that we derive a computationally efficient SURE-like approach for selecting the most appropriate value for the regularization parameter, whereas Lee et al [2015] used the simulation-based approach of Lee et al [2013].…”

Section: Introductionmentioning

confidence: 99%

“…Optimization-based estimators using a regularized version of likelihood (Staudenmayer et al [2008]; Lee et al [2013]) or least squares (Lee et al [2015]) as the objective function were recently considered for density deconvolution, although these works did not investigate the effects of incorporating shape constraints, as we do in this work. Another difference between our work and Lee et al [2015] is that we derive a computationally efficient SURE-like approach for selecting the most appropriate value for the regularization parameter, whereas Lee et al [2015] used the simulation-based approach of Lee et al [2013]. We also introduce a simple graphical method that serves as a check on the selected regularization parameter, and we demonstrate that it is effective at preventing poor estimation results in the small proportion of cases where the SURE-like method selects the regularization parameter that results in too little regularization.…”

Section: Introductionmentioning

confidence: 99%

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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“…For these cases, the traditional chart X is not convenient for reasons of costs and the normality assumption of the probability distribution. The development and application of finite mixture models distributions can be found in studies such as Everitt and Hand (1981), Titterington et al (1985), West and Smith (1992), and most recently, Yu (2011), Lee et al (2014), Kaffel and Prigent (2016), Kayid and Izadkhah (2015), Kim et al (2015), Masmoudi et al (2016).…”

Section: Introductionmentioning

confidence: 99%

“…The probability distribution of the mixture can be characterized as a probabilistic combination of two or more random variables (Yu, 2011;Dixon, 2012;Lee et al, 2014). A mixture of distribution models are formed from a weighted combination of two or more underlying distributions with g(x) ¼ ∑ j α j f (x) j ; ∑α j ¼ 1.…”

Section: Introductionmentioning

confidence: 99%

Monitoring process control chart with finite mixture probability distribution

Vicentin

Silva

Piccirillo

et al. 2018

IJQRM

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Purpose The purpose of this paper is to develop a monitoring multiple-stream processes control chart with a finite mixture of probability distributions in the manufacture industry. Design/methodology/approach Data were collected during production of a wheat-based dough in a food industry and the control charts were developed with these steps: to collect the master sample from different production batches; to verify, by graphical methods, the quantity and the characterization of the number of mixing probability distributions in the production batch; to adjust the theoretical model of probability distribution of each subpopulation in the production batch; to make a statistical model considering the mixture distribution of probability and assuming that the statistical parameters are unknown; to determine control limits; and to compare the mixture chart with traditional control chart. Findings A graph was developed for monitoring a multi-stream process composed by some parameters considered in its calculation with similar efficiency to the traditional control chart. Originality/value The control chart can be an efficient tool for customers that receive product batches continuously from a supplier and need to monitor statistically the critical quality parameters.

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Least squares sieve estimation of mixture distributions with boundary effects

Cited by 5 publications

References 15 publications

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Monitoring process control chart with finite mixture probability distribution

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