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Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach.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 (2012) is addressed in the Appendix.

19 pages, 1 figure

Cited by in corpus (4)

• A study of seven asymmetric kernels for the estimation of cumulative distribution functions
• A refined continuity correction for the negative binomial distribution and asymptotics of the median
• Refined normal approximations for the central and noncentral chi-square distributions and some applications
• Asymptotic properties of the multivariate Szász-Mirakyan estimator for cumulative distribution functions on the nonnegative orthant