Edgeworth Expansions: a Brief Review of Zhidong Bai’s Contributions

Professor Bai's contributions to Edgeworth Expansions are reviewed. Author's collaborations with Professor Bai on the topic are also discussed. I have the pleasure of collaborating with Professor Bai Zhidong on many papers including three [2–4] on Edgeworth expansions. The earliest work of Bai on Edgeworth expansions that I came across is the English translation [10] of his joint work with Lin Cheng, which was first published in Chinese. They investigate expansions for the distribution of sums of independent but not necessarily identically distributed random variables. The expansions are obtained in terms of truncated moments and characteristic functions. From this, they derive an ideal result for non-uniform estimates of the residual term in the expansion. In addition they also derive the non-uniform rate of the asymptotic normality of the distribution of the sum of independent but identically distributed random variables, extending some of the earlier work by A. Bikyalis [13] and L. V. Osipov [17]. Few years later Bai [7] obtains Edgeworth expansions for convolutions by providing bounds for the approximation of ψ * F n by ψ * U kn , where F n denotes the distribution function of the sum of n independent random variables, ψ is a function of bounded variation and U kn denotes the " formal " Edgeworth expansion of F n up to the kth order. Many important statistics can be written as functions of sample means of random vectors. Bhattacharya and Ghosh [11] made fundamental contributions to the theory of Edgeworth expansions for functions of sample means of random vectors. Their results are derived under Cramér's condition on the joint distribution of all the components of the vector variable. However, in many practical situations, such as ratio statistics [6] and survival analysis, only one or a few of the components satisfy Cramér's condition while the rest do not. Bai along with Rao [8] estab