Wavelet estimations of a density function in two-class mixture model

Junke Kou; Xianmei Chen; Junke Kou; Xianmei Chen

doi:10.3934/math.20241000

AIMS Mathematics

2024, Volume 9, Issue 8: 20588-20611. doi: 10.3934/math.20241000

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Research article

Wavelet estimations of a density function in two-class mixture model

Junke Kou ^,,
Xianmei Chen

School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi, 541004, China

Received: 15 April 2024 Revised: 13 June 2024 Accepted: 18 June 2024 Published: 25 June 2024
MSC : 62G07, 62G20

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.
- nonparametric estimation,
- wavelet estimator,
- density function,
- convergence rate
Citation: Junke Kou, Xianmei Chen. Wavelet estimations of a density function in two-class mixture model[J]. AIMS Mathematics, 2024, 9(8): 20588-20611. doi: 10.3934/math.20241000

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Abstract

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.

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