### Theory and Practice

### Theory and Practice

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In 1946 Paul Halmos studied unbiased estimators of minimum variance, and planted the seed from which the subject matter of the present monograph sprang. The author has undertaken to provide experts and advanced students with a review of the present status of the evolved theory of U-statistics, including applications to indicate the range and scope of U-statistic methods. Complete with over 200 end-of-chapter references, this is an invaluable addition to the libraries of applied and theoretical statisticians and mathematicians.

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat

In 1946 Paul Halmos studied unbiased estimators of minimum variance, and planted the seed from which the subject matter of the present monograph sprang. The author has undertaken to provide experts and advanced students with a review of the present status of the evolved theory of U-statistics, including applications to indicate the range and scope of U-statistic methods. Complete with over 200 end-of-chapter references, this is an invaluable addition to the libraries of applied and theoretical statisticians and mathematicians.

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat

In 1946 Paul Halmos studied unbiased estimators of minimum variance, and planted the seed from which the subject matter of the present monograph sprang. The author has undertaken to provide experts and advanced students with a review of the present status of the evolved theory of U-statistics, including applications to indicate the range and scope of U-statistic methods. Complete with over 200 end-of-chapter references, this is an invaluable addition to the libraries of applied and theoretical statisticians and mathematicians.

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat

**Preface**

**Chapter 1 Basics**

Origins

*U*-statistics

The variance of a __U__-statistic

The covariance of two *U*-statistics

Higher moments of *U*-statistics

The *H*-decomposition

A geometric perspective on the *H*-decomposition

Bibliographic details

Chapter 2 Variations

Introduction

Generalised *U*-statistics

Dropping the identically distributed assumption

*U*-Statistics based on stationary random sequences

M-dependent stationary sequences

Weakly dependent stationary sequences

U-statistics based on sampling from finite populations

Weighted *U*-statistics

Generalised *L*-statistics

Bibliographic details

Chapter 3 Asymptotics

Introduction

Convergence in distribution of *U*-statistics

Asymptotic normality

First order degeneracy

The general case

Poisson convergence

Rates of convergence in the *U*-Statistic central limit theorem

Introduction

The Berry-Esseen Theorem for *U*-statistics

Asymptotic expansions

The strong law of large numbers for *U*-statistics

Martingales

*U*-statistics as martingales and SLLN

The law of the iterated logarithm for U-statistics

Invariance principles

Asymptotics for *U*-statistic variations

Asymptotics for generalised *U*-statistics

The independent, non-identically distributed case

Asymptotics for *U*-statistics based on stationary sequences

Asymptotics for weights and generlised *L*-statistics

Random* U*-statistics

Kernels with estimated parameters

Bibliographic details

Chapter 4 Related statistics

Introduction

Symmetric stat