Two-stage conditional density estimation based on Bernstein polynomials

Abstract

Two-stage conditional probability density function estimators are proposed and studied. Specifically, the Nadaraya-Watson (NW) and local linear (LL) conditional distribution function estimators have been smoothed using Bernstein polynomials in the first stage. Second, the proposed estimators are obtained by differentiating NW and LL estimators. The asymptotic properties of these estimators are established such as asymptotic bias, variance, and normality. Finally, a simulation study is carried out to assess the relative advantage of our estimators compared to other competitors. In addition, the well-known Old Faithful Geyser data is analyzed using the proposed estimators.

Keywords:

• Conditional density
• conditional distribution
• Bernstein polynomials
• asymptotic properties

2020 Mathematics Subject Classification:

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Acknowledgements

The authors wish to thank professor T. Bouezmarni from Université de Sherbrooke for his invaluable comments.

Additional information

Funding

M. Belalia gratefully acknowledge the research support of the Natural Sciences and Engineering Research Council of Canada.