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On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

Ouimet, Frédéric (2021) On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators. Journal of Mathematical Analysis and Applictions, 499 (1). Art. No. 125033. ISSN 0022-247X. doi:10.1016/j.jmaa.2021.125033. https://resolver.caltech.edu/CaltechAUTHORS:20210401-083400555

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Abstract

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard [41], who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc [32] is addressed in the Appendix.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jmaa.2021.125033DOIArticle
https://arxiv.org/abs/2010.05146arXivDiscussion Paper
ORCID:
AuthorORCID
Ouimet, Frédéric0000-0001-7933-5265
Alternate Title:A local limit theorem for the Poisson distribution and its application to the Le Cam distance between Poisson and Gaussian experiments and asymptotic properties of Szasz estimators
Additional Information:© 2021 Elsevier Inc. Received 16 October 2020, Available online 4 February 2021. F. O. is supported by a postdoctoral fellowship from the NSERC (PDF) and FRQNT (B3X supplement). The author would like to thank an anonymous referee for his valuable comments that led to improvements in the presentation of this paper.
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Fonds de recherche du Québec – Nature et technologies (FRQNT)UNSPECIFIED
Subject Keywords:Poisson distribution; Local limit theorem; Deficiency; Comparison of experiments; Bernstein estimator; Distribution function estimation
Issue or Number:1
DOI:10.1016/j.jmaa.2021.125033
Record Number:CaltechAUTHORS:20210401-083400555
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210401-083400555
Official Citation:Frédéric Ouimet, On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators, Journal of Mathematical Analysis and Applications, Volume 499, Issue 1, 2021, 125033, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2021.125033. (https://www.sciencedirect.com/science/article/pii/S0022247X21001128)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108599
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 Apr 2021 18:59
Last Modified:08 Apr 2021 18:59

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