U-Statistics : 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.  Read more... Abstract: 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 Content: Cover Half Title Title Page Copyright Page Contents Preface Chapter 1. Basics 1.1 Origins 1.2 U-statistics 1.3 The variance of a U-statistic 1.4 The covariance of two U-statistics 1.5 Higher moments of U-statistics 1.6 The H -decomposition 1.7 A geometric perspective on the H -decomposition 1.8 Bibliographic details Chapter 2. Variations 2.1 Introduction 2.2 Generalised U-statistics 2.3 Dropping the identically distributed assumption 2.4 U-statistics based on stationary random sequences 2.4.1 M-dependent stationary sequences 2.4.2 Weakly dependent stationary sequences 2.5 U-statistics based on sampling from finite populations2.6 Weighted U-statistics 2.7 Generalised L-statistics 2.8 Bibliographic details Chapter 3. Asymptotics 3.1 Introduction 3.2 Convergence in distribution of U -statistics 3.2.1 Asymptotic normality 3.2.2 First order degeneracy 3.2.3 The general case 3.2.4 Poisson convergence 3.3 Rates of convergence in the U -statistic central limit theorem 3.3.1 Introduction 3.3.2 The Berry-Esseen Theorem for U-statistics 3.3.3 Asymptotic expansions 3.4 The strong law of large numbers for U -statistics, 3.4.1 Martingales 3.4.2 U-statistics as martingales and the SLLN3.5 The law of the iterated logarithm for U -statistics 3.6 Invariance principles 3.7 Asymptotics for U -statistic variations 3.7.1 Asymptotics for generalised U-statistics 3.7.2 The independent, non-identically distributed case 3.7.3 Asymptotics for U -statistics based on stationary sequences 3.7.4 Asymptotics for U -statistics based on finite population sampling 3.7.5 Asymptotics for weights and generalised L-statistics 3.7.6 Random U -statistics 3.8 Kernels with estimated parameters 3.9 Bibliographic details Chapter 4. Related statistics4.1 Introduction 4.1.1 Symmetric statistics: basics 4.1.2 Asymptotic behaviour of symmetric statistics 4.2 V-statistics 4.3 Incomplete U-statistics 4.3.1 Basics 4.3.2 Minimum variance designs 4.3.3 Asymptotics for random subset selection 4.3.4 Asymptotics for balanced designs 4.4 Bibliographic details Chapter 5. Estimating standard errors 5.1 Standard errors via the jackknife 5.1.1 The jackknife estimate of variance 5.1.2 Jackknifing functions of U-statistics 5.1.3 Extension to functions of several U-statistics 5.1.3 Additional results 5.2 Bootstrapping U-statistics5.3 Variance estimation for incomplete U-statistics 5.3.1 The balanced case, 5.3.2 Incomplete U-statistics based on random choice 5.4 Bibliographic details Chapter 6. Applications 6.1 Introduction 6.2 Applications to the estimation of statistical parameters 6.2.1 Circular and spherical correlation 6.2.2 Testing for symmetry 6.2.3 Testing for normality 6.2.4 A test for independence 6.2.5 Applications to the several-sample problem 6.2.6 A test for ""New better than used 6.3 Applications of Poisson convergence 6.3.1 Comparing correlations