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Published July 1, 2021 | Accepted Version
Journal Article Open

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

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Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard [41], who uses a different approach.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.

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.

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Created:
August 22, 2023
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October 23, 2023