Optimal-order bounds on the rate of convergence to normality in the multivariate delta method

Abstract

Uniform and nonuniform Berry-Esseen (BE) bounds of optimal orders on the closeness to normality for general abstract nonlinear statistics are given, which are then used to obtain optimal bounds on the rate of convergence in the delta method for vector statistics. Specific applications to Pearson's, non-central Student's and Hotelling's statistics, sphericity test statistics, a regularized canonical correlation, and maximum likelihood estimators (MLEs) are given; all these uniform and nonuniform BE bounds appear to be the first known results of these kinds, except for uniform BE bounds for MLEs. When applied to the well-studied case of the central Student statistic, our general results compare well with known ones in that case, obtained previously by specialized methods. The proofs use a Stein-type method developed by Chen and Shao, a Cramér-type of tilt transform, exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis, Sakhanenko, and Utev, as well as a number of other, quite recent results motivated by this study. The method allows one to obtain bounds with explicit and rather moderate-size constants, at least as far as the uniform bounds are concerned. For instance, one has the uniform BE bound 3.61 E(Y 6 1 + Z 6 1 ) (1 + σ −3 )/ √ n for the Pearson sample correlation coefficient based on independent identically distributed random pairs

Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

“…Z one has d Ko (Y, Z) 2 d bW (Y, Z), where d Ko denotes the Kolmogorov distance. This bound on d Ko in terms of d bW is the best possible one, up a constant factor, as shown in [20]. Therefore, even though the bounds on the bounded Wasserstein distance d bW obtained in [2,1] are of the optimal order O(1/ √ n), the resulting bounds on the Kolmogorov distance are only of the order O(1/n 1/4 ).…”

“…In [20], optimal-order bounds of the form O(1/ √ n) on the rate of convergence to normality in the general multivariate delta method were given. Those results are applicable when the statistic of interest can be expressed as a smooth enough function of the sum of independent random vectors.…”

“…The crucial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of independent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to M -estimators.AMS 2010 subject classifications: 62F10, 62F12, 60F05, 60E15.…”

User-Friendly Covariance Estimation for Heavy-Tailed Distributions

“…where Z ∼ N (0, 1). To this end, we will use Theorem 2.9 in Pinelis and Molzon (2016). Using the notation therein, we take V = (X − EX , X m − EX m , X X m − E(X X m )) , f (x 1 , x 2 , x 3 ) = x 3 − x 1 · x 2 , and deduce that…”

An Asymptotically Optimal Transform of Pearson’s Correlation Statistic

“…As noted in [9], the condition σ = 0 is equivalent to the following exceptional situation: there exists some κ ∈ R such that the random point (Y, Z) lies almost surely on the union of the two straight lines through the origin with slopes κ and 1/κ (for κ = 0, these two lines should be understood as the two coordinate axes in the plane R 2 ).…”