This is an introductory text on a broad class of statistical estimators that are minimizers of convex functions. It covers the basics of U-statistics and Mm-estimators and develops their asymptotic properties. It also provides an elementary introduction to resampling, particularly in the context of these estimators. The last chapter is on practical implementation of the methods presented in other chapters, using the free software R.
Arvustused
The aim of the book under review is to provide statistics graduate students, particularly those who work on nonparametric and semiparametric methods, a concise introduction to the techniques with adequate references for further reading. the book under review is a good introduction to U-statistics and resampling methods for graduate students. (Zhongwen Liang, Mathematical Reviews, November, 2019)
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xi | |
| About the Authors |
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xv | |
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1 Introduction to U-statistics |
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1 | (34) |
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1.1 Definition and examples |
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1 | (5) |
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1.2 Some finite sample properties |
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6 | (2) |
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6 | (1) |
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7 | (1) |
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1.3 Law of large numbers and asymptotic normality |
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8 | (4) |
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12 | (5) |
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1.5 Degenerate U-statistics |
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17 | (13) |
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30 | (5) |
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2 Mm-estimators and U-statistics |
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35 | (34) |
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2.1 Basic definitions and examples |
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35 | (4) |
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39 | (2) |
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41 | (2) |
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43 | (2) |
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2.5 Weak representation, asymptotic normality |
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45 | (10) |
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55 | (3) |
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2.7 Strong representation theorem |
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58 | (9) |
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2.7.1 Comments on the exact rate |
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66 | (1) |
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67 | (2) |
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3 Introduction to resampling |
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69 | (34) |
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69 | (2) |
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3.2 Three standard examples |
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71 | (6) |
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3.3 Resampling methods: the jackknife and the bootstrap |
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77 | (6) |
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3.3.1 Jackknife: bias and variance estimation |
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78 | (3) |
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3.3.2 Bootstrap: bias, variance and distribution estimation |
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81 | (2) |
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3.4 Bootstrapping the mean and the median |
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83 | (10) |
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3.4.1 Classical bootstrap for the mean |
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83 | (4) |
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3.4.2 Consistency and Singh property |
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87 | (5) |
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3.4.3 Classical bootstrap for the median |
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92 | (1) |
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3.5 Resampling in simple linear regression |
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93 | (7) |
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94 | (2) |
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96 | (1) |
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3.5.3 Wild or external bootstrap |
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97 | (1) |
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3.5.4 Parametric bootstrap |
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97 | (1) |
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3.5.5 Generalized bootstrap |
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98 | (2) |
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100 | (3) |
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4 Resampling U-statistics and M-estimators |
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103 | (24) |
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103 | (2) |
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4.2 Classical bootstrap for U-statistics |
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105 | (2) |
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4.3 Generalized bootstrap for U-statistics |
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107 | (2) |
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4.4 GBS with additive weights |
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109 | (4) |
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4.4.1 Computational aspects for additive weights |
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112 | (1) |
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4.5 Generalized bootstrap for Mm-estimators |
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113 | (10) |
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4.5.1 Resampling representation results for m = 1 |
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115 | (4) |
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4.5.2 Results for general m |
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119 | (4) |
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123 | (4) |
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127 | (23) |
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5.1 Introduction, installation, basics |
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127 | (4) |
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5.1.1 Conventions and rules |
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130 | (1) |
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5.2 The first steps of R programming |
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131 | (2) |
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5.3 Initial steps of data analysis |
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133 | (12) |
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134 | (3) |
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137 | (5) |
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142 | (1) |
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5.3.4 Computing multivariate medians |
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143 | (2) |
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5.4 Multivariate median regression |
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145 | (4) |
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149 | (1) |
| Bibliography |
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150 | (13) |
| Author Index |
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163 | (2) |
| Subject Index |
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165 | |
Arup Bose is Professor at the Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India. He is a Fellow of the Institute of Mathematical Statistics and of all the three national science academies of India. He has significant research contributions in the areas of statistics, probability, economics and econometrics. He is a recipient of the Shanti Swarup Bhatnagar Prize and the C R Rao National Award in Statistics. His current research interests are in large dimensional random matrices, free probability, high dimensional data, and resampling. He has authored three books: Patterned Random Matrices, Large Covariance Autocovariance Matrices (with Monika Bhattacharjee) and (with Koushik Saha), published by Chapman & Hall.
Snigdhansu Chatterjee is Professor at the School of Statistics, University of Minnesota, USA. He is also the Director of the Institute for Research in Statistics and its Applications. His research interests are in resampling methods, high-dimensional and big data statistical methods, small area methods, and application of statistics in climate science, neuroscience and social sciences. He has written over 45 research articles.
