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Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach.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

Refined normal approximations for the Student distribution

“…For the interested reader, local approximations in the same vein as Lemma 2.1 were derived for the Poisson, binomial, negative binomial, multinomial, Dirichlet, Wishart and multivariate hypergeometric distributions in Ouimet (2021aOuimet ( , 2022aOuimet ( , 2021cOuimet ( ,b, 2022b, respectively. See also earlier references such as Govindarajulu (1965) (based on results from Esseen (1945)) for the Poisson, binomial and negative binomial distributions, and Cressie (1978) for the binomial distribution.…”

Refined normal approximations for the central and noncentral chi-square distributions and some applications

“…For the interested reader, local approximations akin to Lemma 3.1 were derived for the Poisson, binomial, negative binomial, multinomial, Dirichlet, Wishart and multivariate hypergeometric distributions in (Ouimet, 2021a, Lemma 2.1), (Ouimet, 2022a, Lemma 3.1), (Ouimet, 2021c, Lemma 2.1), (Ouimet, 2021b, Theorem 2.1), (Ouimet, 2022b, Theorem 1), (Ouimet, 2022d, Theorem 1), (Ouimet, 2022c, Theorem 1), respectively. See also earlier references such as Govindarajulu (1965) (based on Fourier analysis results from Esseen (1945)) for the Poisson, binomial and negative binomial distributions, and Cressie (1978) for the binomial distribution.…”

A Compressive Receding Horizon Approach for Smart Home Energy Management

“…The Poisson probability distribution expresses the probability of observing a given number of events in a time period. Let X be a discrete random Poisson variable with probability density fX (k), which is given by (1) [46].…”