Associated Sequences, Demimartingales and Nonparametric Inference

285 Pages·2012·3.246 MB·English

by  B.L.S. Prakasa Rao (auth.)

Probability and its Applications
Series Editors
CharlesNewman
SidneyResnick
Forfurthervolumes:
www.springer.com/series/4893
B.L.S. Prakasa Rao
Associated Sequences,
Demimartingales
and Nonparametric
Inference
B.L.S.PrakasaRao
DepartmentofMathematicsandStatistics
UniversityofHyderabad
Hyderabad,India
ISBN978-3-0348-0239-0 e-ISBN978-3-0348-0240-6
DOI10.1007/978-3-0348-0240-6
SpringerBaselDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2011941605
MathematicsSubjectClassification(2010):60G48
©SpringerBaselAG2012
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SpringerBaselAGispartofSpringerScience+BusinessMedia(www.birkhauser-science.com)
In memory of my mother
Bhagavatula Saradamba
(1926–2002)
for her abiding trust in me
and for her love and affection
Contents
Preface xi
1 Associated Random Variables and Related Concepts 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Some Probabilistic Properties of Associated Sequences . . . . . . . 5
1.3 Related Concepts of Association . . . . . . . . . . . . . . . . . . . 23
2 Demimartingales 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Characteristic Function Inequalities. . . . . . . . . . . . . . . . . . 37
2.3 Doob Type Maximal Inequality . . . . . . . . . . . . . . . . . . . . 39
2.4 An Upcrossing Inequality . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Chow Type Maximal Inequality . . . . . . . . . . . . . . . . . . . . 44
2.6 Whittle Type Maximal Inequality. . . . . . . . . . . . . . . . . . . 50
2.7 More on Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . 54
2.8 Maximal φ-Inequalities for Nonnegative Demisubmartingales . . . 58
2.9 Maximal Inequalities for Functions of Demisubmartingales . . . . . 65
2.10 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . 69
2.11 Dominated Demisubmartingales. . . . . . . . . . . . . . . . . . . . 70
3 N-Demimartingales 73
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 More on Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . 81
3.4 Downcrossing Inequality . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Chow Type Maximal Inequality . . . . . . . . . . . . . . . . . . . . 89
3.6 Functions of N-Demimartingales . . . . . . . . . . . . . . . . . . . 92
3.7 Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . 94
3.8 Azuma Type Inequality . . . . . . . . . . . . . . . . . . . . . . . . 95
3.9 Marcinkiewicz-Zygmund Type inequality . . . . . . . . . . . . . . . 98
3.10 Comparison Theorem on Moment Inequalities . . . . . . . . . . . . 102
vii
viii Contents
4 Conditional Demimartingales 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Conditional Independence . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Conditional Association . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Conditional Demimartingales . . . . . . . . . . . . . . . . . . . . . 109
5 Multidimensionally Indexed Demimartingales and Continuous
Parameter Demimartingales 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Multidimensionally Indexed Demimartingales . . . . . . . . . . . . 113
5.3 Chow Type Maximal Inequality for Two-Parameter
Demimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Continuous Parameter Demisubmartingales . . . . . . . . . . . . . 118
5.5 Maximal Inequality for Continuous Parameter Demisubmartingales 119
5.6 Upcrossing Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 Limit Theorems for Associated Random Variables 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Covariance Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Hajek-Renyi Type Inequalities . . . . . . . . . . . . . . . . . . . . 137
6.4 Exponential Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5 Non-uniform and Uniform Berry-Esseen Type Bounds . . . . . . . 145
6.6 Limit Theorems for U-Statistics . . . . . . . . . . . . . . . . . . . . 150
6.7 More Limit Theorems for U-Statistics . . . . . . . . . . . . . . . . 157
6.8 Application to a Two-Sample Problem . . . . . . . . . . . . . . . . 172
6.9 Limit Theorems for V-Statistics . . . . . . . . . . . . . . . . . . . . 180
6.10 Limit Theorems for Associated Random Fields . . . . . . . . . . . 182
6.11 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7 Nonparametric Estimation for Associated Sequences 187
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 Nonparametric Estimation for Survival Function . . . . . . . . . . 188
7.3 Nonparametric Density Estimation . . . . . . . . . . . . . . . . . . 194
7.4 Nonparametric Failure Rate Estimation . . . . . . . . . . . . . . . 213
7.5 Nonparametric Mean Residual Life Function Estimation . . . . . . 214
7.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8 Nonparametric Tests for Associated Sequences 217
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.2 More on Covariance Inequalities . . . . . . . . . . . . . . . . . . . 217
8.3 Tests for Location . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.4 Mann-Whitney Test . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Contents ix
9 Nonparametric Tests for Change in Marginal Density Function
for Associated Sequences 235
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.2 Tests for Change Point . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.3 Test for Change in Marginal Density Function. . . . . . . . . . . . 242
Bibliography 253
Index 271