A precise local limit theorem for the multinomial distribution and some applications

Abstract

In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N −1 . In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R d . Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter ( 2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

On the Le Cam Distance Between Multivariate Hypergeometric and Multivariate Normal Experiments

“…Not only Brown et al [3] streamlined the proof of the asymptotic equivalence originally shown by Nussbaum [13], but their results hold for a larger class of densities and the asymptotic equivalence was also extended to Poisson processes. Our third main result in the present paper extends the multinomial/multivariate normal comparison from [4] (revisited and improved by Ouimet [14], who removed the inductive part of the argument) to the multivariate hypergeometric/multivariate normal comparison (recall from (1.2) that the multinomial distribution is just the limiting case N = ∞ of the multivariate hypergeometric distribution). For an excellent and concise review on Le Cam's theory for the comparison of statistical models, we refer the reader to Mariucci [10].…”

“…The local limit theorem above together with the total variation bound in [4,14] between jittered multinomials and the corresponding multivariate normals allow us to derive an upper bound on the total variation between the probability measure on R d induced by a multivariate hypergeometric random vector jittered by a uniform random vector on (−1/2, 1/2) d and the probability measure on R d induced by a multivariate normal random vector with the same mean and covariances as the multinomial distribution in (1.2).…”

“…Also, by Lemma 3.1 in[14] (a slightly weaker bound can be found in Lemma 2 of[4]), we already know thatQ n, p − Q n, p = O         d √ n max{p 1 , . .…”

General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

“…over the d-dimensional simplex. Some of their asymptotic properties were investigated by Vitale (1975), Stadtmüller (1986), Tenbusch (1997), , Ghosal (2001) Lu (2015), Guan (2016Guan ( , 2017 and Belalia et al (2017Belalia et al ( , 2019 when d = 1, by Tenbusch (1994) [31] when d = 2, and by Ouimet (2020) [32,33] for all d ≥ 1, using a local limit theorem from Ouimet (2020) [34] for the multinomial distribution (see also Arenbaev (1976) [35]). The estimator ( 5) is a discrete analogue of the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) [36] and studied theoretically in Brown and Chen (1999), Chen (1999Chen ( , 2000 and Bouezmarni and Rolin (2003) [37][38][39][40] when d = 1 (among others), and in Ouimet (2020) [41] for all d ≥ 1.…”

Selecting a number of voters for a voting ensemble

“…To identify a suitable superset of the likely set, perhaps we can apply bounds on the difference between X 2 (or similar statistics) and the chi-squared distribution (Matsunawa, 1977;Siotani and Fujikoshi, 1984;Bickel and Ghosh, 1990;Taneichi et al, 2002;Gaunt et al, 2016;Ouimet, 2021) to expand an approximate likely set, for example by increasing the constraint C δ on X 2 to ensure that all w in the likely set are enclosed in the region of the approximate set while keeping the set small enough to yield effective error bounds. It may also be possible to extend bounds on L 1 distance between ŵ and w (Valiant and Valiant, 2017;Balakrishnan and Wasserman, 2018) to derive a superset of the likely set that gives linear constraints instead of a feasible region with curved boundaries.…”