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Statistica Sinica 31 (2021), 891-908

FAST NONPARAMETRIC MAXIMUM LIKELIHOOD DENSITY DECONVOLUTION
USING BERNSTEIN POLYNOMIALS

Zhong Guan

Indiana University South Bend

Abstract: We proposed a new maximum approximate likelihood method for deconvoluting a continuous density on a finite interval in additive measurement error models with a known error distribution. The proposed method uses an approximate Bernstein polynomial model, that is, a finite mixture of specific beta distributions. The change-point detection method is used to choose an optimal model degree. Based on a contaminated sample of size n, under an assumption satisfied by, among others, the generalized normal error distribution, the optimal rate of convergence of the mean integrated squared error is proved of the mean integrated squared error is proved to be 𝒪(𝓃− 1+5/k log 𝓃) if the underlying unknown density admits an approximate Bernstein polynomial model of degree m within x²-divergence of order 𝒪(mk), with k > 5. Simulations show that the small-sample performance of the proposed estimator is better than that of the deconvolution kernel density estimator. The proposed method is llustrated using a real-data application.

Key words and phrases: Bernstein polynomial model, beta mixture model, deconvolution, density estimation, kernel density, measurement error model, model selection.

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