Abstract
In this paper, we study some asymptotic properties for the Bernstein estimators of the limit distribution function and the limit density function under a triangular sample. Specifically, we obtain the uniform strong consistency, mean squared error (MSE) and mean integrated squared error (MISE) for the resulting estimators. In addition, we give the optimal choice of the bandwidth parameter m in terms of the sample size n, for both the MSE and MISE. Numerical simulations are presented to show that the Bernstein estimators outperform Gaussian kernel estimators in terms of MISE under a triangular sample.
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All data generated during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors wish to thank the anonymous referees for their remarks that contributed to improve the presentation.
Funding
Dawei Lu was supported by Dalian High-level Talent Innovation Program (Grant No. 2020RD09).
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Lina Wang and Dawei Lu wrote the main manuscript text and Lina Wang prepared Figures 1–2. All authors reviewed the manuscript.
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Wang, L., Lu, D. Application of Bernstein Polynomials on Estimating a Distribution and Density Function in a Triangular Array. Methodol Comput Appl Probab 25, 56 (2023). https://doi.org/10.1007/s11009-023-10032-3
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DOI: https://doi.org/10.1007/s11009-023-10032-3