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Nonuniform Berry-Esseen bounds for Studentized U-statistics

Dennis Leung, Qi-Man Shao

Abstract

We establish nonuniform Berry-Esseen (B-E) bounds for Studentized U-statistics of the rate $1/\sqrt{n}$ under a third-moment assumption, which covers the t-statistic that corresponds to a kernel of degree $1$ as a special case. While an interesting data example raised by Novak (2005) can show that the form of the nonuniform bound for standardized U-statistics is actually invalid for their Studentized counterparts, our main results suggest that, the validity of such a bound can be restored by minimally augmenting it with an additive correction term that decays exponentially in $n$. To our best knowledge, this is the first time that valid nonuniform B-E bounds for Studentized U-statistics have appeared in the literature.

Nonuniform Berry-Esseen bounds for Studentized U-statistics

Abstract

We establish nonuniform Berry-Esseen (B-E) bounds for Studentized U-statistics of the rate under a third-moment assumption, which covers the t-statistic that corresponds to a kernel of degree as a special case. While an interesting data example raised by Novak (2005) can show that the form of the nonuniform bound for standardized U-statistics is actually invalid for their Studentized counterparts, our main results suggest that, the validity of such a bound can be restored by minimally augmenting it with an additive correction term that decays exponentially in . To our best knowledge, this is the first time that valid nonuniform B-E bounds for Studentized U-statistics have appeared in the literature.
Paper Structure (23 sections, 19 theorems, 306 equations)

This paper contains 23 sections, 19 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Studentized U-statistics and novak2005self's example
  3. Main results
  4. Proof of the nonuniform B-E bound for Studentized U-statistics
  5. How the constants scale in $m$
  6. The required sample size relative to $m$
  7. Proof of the nonuniform B-E bound for Student's t-statistic
  8. Proof for the self-normalized sum
  9. Proof for the t-statistic
  10. Bounding $|P(S_n/V_n > x a_n(x)) - \overline{\Phi}(x a_n(x))|$
  11. Bounding $|\overline{\Phi}(x a_n(x)) - \overline{\Phi}(x)|$
  12. Technical lemmas
  13. Properties of the solution to Stein's equation
  14. Bounds for the censored summands $\xi_{b,i}$'s and their sum $W_b$
  15. Bounds related to the components of the censored denominator remainder in Section \ref{['sec:mainpf']}
  16. ...and 8 more sections

Key Result

Theorem 2.1

Assume zero_mean_kernel_assumption-non_degeneracy, $2m < n$ and $\mathop{\mathrm{\mathbb{E}}}\nolimits[|h|^3] < \infty$. For a positive absolute constant $C(m) >0$ depending on $m$ only, the following Berry-Esseen bound holds: In particular, the bound above can be further simplified as

Theorems & Definitions (32)

  • Theorem 2.1: Uniform B-E bound for Studentized U-statistics, leungshaounif
  • Theorem 3.1: Nonuniform B-E bounds for Studentized U-statistics
  • Theorem 3.2: Nonuniform B-E bound for Student's t-statistic
  • Lemma 4.1: Nonuniform bound for $R_x$
  • Lemma 4.2: Intermediate nonuniform bound by Stein's method
  • Lemma 4.3: Exponential lower tail bound for U-statistics with non-negative kernels
  • Lemma 4.4: General moment bound of U-statistics
  • Lemma 5.1: Sub-Gaussian property for self-normalized sums
  • Lemma A.1: Uniform bounds
  • Lemma A.2: Nonuniform bounds when $x \geq 1$
  • ...and 22 more