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Tail Bounds for Canonical $U$-Statistics and $U$-Processes with Unbounded Kernels

Abhishek Chakrabortty, Arun K. Kuchibhotla

Abstract

In this paper, we prove exponential tail bounds for canonical (or degenerate) $U$-statistics and $U$-processes under exponential-type tail assumptions on the kernels. Most of the existing results in the relevant literature often assume bounded kernels or obtain sub-optimal tail behavior under unbounded kernels. We obtain sharp rates and optimal tail behavior under sub-Weibull kernel functions. Some examples from nonparametric and semiparametric statistics literature are considered.

Tail Bounds for Canonical $U$-Statistics and $U$-Processes with Unbounded Kernels

Abstract

In this paper, we prove exponential tail bounds for canonical (or degenerate) -statistics and -processes under exponential-type tail assumptions on the kernels. Most of the existing results in the relevant literature often assume bounded kernels or obtain sub-optimal tail behavior under unbounded kernels. We obtain sharp rates and optimal tail behavior under sub-Weibull kernel functions. Some examples from nonparametric and semiparametric statistics literature are considered.

Paper Structure

This paper contains 15 sections, 13 theorems, 205 equations.

Table of Contents

  1. Introduction and Motivation
  2. Degenerate or Canonical $U$-statistics.
  3. Related Literature
  4. Tail Bounds for Degenerate $U$--Statistics
  5. Tail Bounds for Degenerate $U$--Processes
  6. Maximal Inequality for Bounded Degenerate U-Processes
  7. Proofs of Results in Section \ref{['sec:UStat']}
  8. Proof of Theorem \ref{['thm:MainUStat-bounded-kernel']}
  9. Proof of Lemma \ref{['lem:ReductionSymmetrization']}
  10. Proof of Theorem \ref{['thm:MainUStat']}
  11. Auxiliary Lemmas Used in Theorem \ref{['thm:MainUStat']}
  12. Proofs of Results in Section \ref{['sec:UProc']}
  13. Proof of Theorem \ref{['thm:MainUProc']}
  14. Auxiliary Lemmas Used in Theorem \ref{['thm:MainUProc']}
  15. Proof of the Maximal Inequality (Theorem \ref{['thm:UniformEntropyUProcesses']})

Key Result

Theorem 1

For any collection of degenerate kernels $\{f_{i,j}:1\le i\neq j\le n\}$ satisfying assumption assump:product-upper-bnd-kernel, for every $p\ge1$, where $\alpha^* = \min\{\alpha, 1\}$ and $\beta^* = \min\{\beta, 1\}$. Consequently, for every $\delta\in[0,1]$, with probability at least $1-\delta$,

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 2
  • ...and 15 more