Electronic Journal of Statistics

Deconvolution estimation of mixture distributions with boundaries

Mihee Lee, Peter Hall, Haipeng Shen, J. S. Marron, Jon Tolle, and Christina Burch

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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.

Article information

Electron. J. Statist., Volume 7 (2013), 323-341.

First available in Project Euclid: 28 January 2013

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 65F30: Other matrix algorithms

Boundary effect maximum likelihood measurement error mixture distribution penalization sieve method


Lee, Mihee; Hall, Peter; Shen, Haipeng; Marron, J. S.; Tolle, Jon; Burch, Christina. Deconvolution estimation of mixture distributions with boundaries. Electron. J. Statist. 7 (2013), 323--341. doi:10.1214/13-EJS774. https://projecteuclid.org/euclid.ejs/1359382682

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