U-Statistics Theory and Practice

Cover......Page 1
Half Title......Page 2
Title Page......Page 8
Copyright Page......Page 9
Contents......Page 16
Preface......Page 12
Chapter 1. Basics......Page 20
1.1 Origins......Page 20
1.2 U-statistics......Page 27
1.3 The variance of a U-statistic......Page 29
1.4 The covariance of two U-statistics......Page 35
1.5 Higher moments of U-statistics......Page 40
1.6 The H -decomposition......Page 44
1.7 A geometric perspective on the H -decomposition......Page 53
1.8 Bibliographic details......Page 54
Chapter 2. Variations......Page 56
2.1 Introduction......Page 56
2.2 Generalised U-statistics......Page 56
2.3 Dropping the identically distributed assumption......Page 61
2.4 U-statistics based on stationary random sequences......Page 62
2.4.1 M-dependent stationary sequences......Page 62
2.4.2 Weakly dependent stationary sequences......Page 68
2.5 U-statistics based on sampling from finite populations......Page 76
2.6 Weighted U-statistics......Page 83
2.7 Generalised L-statistics......Page 86
2.8 Bibliographic details......Page 93
Chapter 3. Asymptotics......Page 94
3.1 Introduction......Page 94
3.2 Convergence in distribution of U -statistics......Page 94
3.2.1 Asymptotic normality......Page 94
3.2.2 First order degeneracy......Page 97
3.2.3 The general case......Page 102
3.2.4 Poisson convergence......Page 109
3.3 Rates of convergence in the U -statistic central limit theorem......Page 115
3.3.1 Introduction......Page 115
3.3.2 The Berry-Esseen Theorem for U-statistics......Page 116
3.3.3 Asymptotic expansions......Page 125
3.4 The strong law of large numbers for U -statistics,......Page 130
3.4.1 Martingales......Page 130
3.4.2 U-statistics as martingales and the SLLN......Page 137
3.5 The law of the iterated logarithm for U -statistics......Page 151
3.6 Invariance principles......Page 153
3.7 Asymptotics for U -statistic variations......Page 159
3.7.1 Asymptotics for generalised U-statistics......Page 159
3.7.2 The independent, non-identically distributed case......Page 162
3.7.3 Asymptotics for U -statistics based on stationary sequences......Page 164
3.7.4 Asymptotics for U -statistics based on finite population sampling......Page 167
3.7.5 Asymptotics for weights and generalised L-statistics......Page 172
3.7.6 Random U -statistics......Page 175
3.8 Kernels with estimated parameters......Page 176
3.9 Bibliographic details......Page 180
Chapter 4. Related statistics......Page 182
4.1 Introduction......Page 182
4.1.1 Symmetric statistics: basics......Page 182
4.1.2 Asymptotic behaviour of symmetric statistics......Page 189
4.2 V-statistics......Page 202
4.3 Incomplete U-statistics......Page 206
4.3.1 Basics......Page 206
4.3.2 Minimum variance designs......Page 213
4.3.3 Asymptotics for random subset selection......Page 219
4.3.4 Asymptotics for balanced designs......Page 222
4.4 Bibliographic details......Page 234
Chapter 5. Estimating standard errors......Page 236
5.1 Standard errors via the jackknife......Page 236
5.1.1 The jackknife estimate of variance......Page 236
5.1.2 Jackknifing functions of U-statistics......Page 243
5.1.3 Extension to functions of several U-statistics......Page 246
5.1.3 Additional results......Page 248
5.2 Bootstrapping U-statistics......Page 249
5.3 Variance estimation for incomplete U-statistics......Page 257
5.3.1 The balanced case,......Page 257
5.3.2 Incomplete U-statistics based on random choice......Page 262
5.4 Bibliographic details......Page 266
Chapter 6. Applications......Page 268
6.1 Introduction......Page 268
6.2 Applications to the estimation of statistical parameters......Page 268
6.2.1 Circular and spherical correlation......Page 269
6.2.2 Testing for symmetry......Page 277
6.2.3 Testing for normality......Page 278
6.2.4 A test for independence......Page 280
6.2.5 Applications to the several-sample problem......Page 281
6.2.6 A test for \"New better than used\"......Page 287
6.3 Applications of Poisson convergence......Page 288
6.3.1 Comparing correlations......Page 288
6.3.2 Applications to spatial statistics......Page 292
6.4 Sequential estimation......Page 293
6.5 Other applications......Page 295
References......Page 298
Index......Page 316