Edgeworth Expansions: A Brief Review of Zhidong Bai's Contributions (original) (raw)

A unified approach to Edgeworth expansions for a general class of statistics

This paper is concerned with Edgeworth expansions for a general class of statistics under very weak conditions. Our approach unifies the treatment for both standardized and studentized statistics that have been traditionally studied separately under usually different conditions. These results are then applied to several special classes of well-known statistics: U-statistics, L-statistics, and functions of sample means. Special attention is paid to the studentized statistics. We establish Edgeworth expansions under very weak or minimal moment conditions.

Rearranging Edgeworth–Cornish–Fisher expansions

Economic Theory, 2010

This paper applies a regularization procedure called increasing rearrangement to monotonize Edgeworth and Cornish-Fisher expansions and any other related approximations of distribution and quantile functions of sample statistics. In addition to satisfying monotonicity, required of distribution and quantile functions, the procedure often delivers strikingly better approximations to the distribution and quantile functions of the sample mean than the original Edgeworth-Cornish-Fisher expansions.

Edgeworth expansions and rates of convergence for normalized sums: Chung's 1946 method revisited

2010

In this paper we revisit, correct and extend Chung's 1946 method for deriving higher order Edgeworth expansions with respect to t-statistics and generalized self-normalized sums. Thereby we provide a set of formulas which allows the computation of the approximation of any order and specify the first four polynomials in the Edgeworth expansion the first two of which are well known. It turns out that knowledge of the first four polynomials is necessary and sufficient for characterizing the rate of convergence of the Edgeworth expansion in terms of moments and the norming sequence appearing in generalized self-normalized sums. It will be shown that depending on moments and norming sequence the rate of convergence can be O(n −i/2 ), i = 1, . . . , 4. Finally, we study expansions and rates of convergence if the normal distribution is replaced by the t-distribution.

A Bayesian Edgeworth expansion by Stein's identity

Bayesian Analysis

The Edgeworth expansion is a series that approximates a probability distribution in terms of its cumulants. One can derive it by first expanding the probability distribution in Hermite orthogonal functions and then collecting terms in powers of the sample size. This paper derives an expansion for posterior distributions which possesses these features of an Edgeworth series. The techniques used are a version of Stein's Identity and properties of Hermite polynomials. Two examples are provided to illustrate the accuracy of our series.

Generalized Newman-Unti expansions

Physics Letters A, 1985

Newman-Unti type expansions for the metric, the spin coefficients and the Riemann tensor components are exhibited for spacetimes which are not necessarily vacuum and in a frame which is not necessarily a Bondi frame. These expansions are given for both the physical spacetime and for the corresponding conformally rescaled (" unphysical") space.

Connection relations and expansions

Pacific Journal of Mathematics, 2011

We give new proofs of the evaluation of the connection relation for the Askey-Wilson polynomials and for expressing the Askey-Wilson basis in those polynomials using q-Taylor series. This led to some inverse relations. We also evaluate the coefficients in the expansions of (x + b) n in various q-orthogonal polynomials, including the Askey-Wilson polynomials, which leads to explicit expressions for the moments of the Askey-Wilson weight function. We generalize the q-plane wave expansion by expanding Ᏹ q (x; α) in Askey-Wilson polynomials. Further, we prove a bibasic extension of the Nassrallah-Rahman integral and establish a recently conjectured identity of George Andrews. 5 j=1 (t j e iθ , t j e −iθ ; q) ∞ dθ = 2π(t 1 t 2 t 3 t 4 t 5 /t 6 ; q) ∞ 5 j=1 (t j t 6 ; q) ∞ (q, t 2 6 ; q) ∞ 1≤ j<k≤5 (t j t k ; q) ∞

Extrapolation of perturbation-theory expansions by self-similar approximants

European Journal of Applied Mathematics, 2014

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.

On the validity of the expansion

Physics Letters A, 1982

We consider possible effects that may invalidate the 1/N method in some statistical and field models. Discussion is in the framework of the model of charged, non-relativistic fermions with N-component spin. The static dielectric function e(q, 0) is predicted to have a pole in N. The real parameter of the 1/N expansion is shown to be an increasing function of the coupling constant.

Edgeworth expansion for U-statistics under minimal conditions

The Annals of Statistics, 2003

Berry-Esseen bounds for U-statistics under the optimal moment conditions were derived by Koroljuk and Borovskich and Friedrich. Under the same optimal moment assumptions with an additional nonlattice condition, we establish a one-term Edgeworth expansion with remainder o(n −1/2) for U-statistics.