Associated Sequences, Demimartingales and Nonparametric Inference

ISBN-13: 9783034807463 / Angielski / Miękka / 2014 / 272 str.

B.L.S. Prakasa Rao

Opis

This book gives a comprehensive review of results for associated sequences and demimartingales developed so far, with special emphasis on demimartingales and related processes. Probabilistic properties of associated sequences, demimartingales and related processes are discussed in the first six chapters. Applications of some of these results to some problems in nonparametric statistical inference for such processes are investigated in the last three chapters.

Recenzje

From the reviews:

"This book is the first one exclusively devoted to sequences of associated random variables and the closely related concept of demimartingales. ... The book is well written, with a good balance of theoretical results, applications and examples. Authored by one of the leading experts of the field, it contains the most important results on associated sequences and demimartingales and will become an extremely valuable reference book for researchers in the broad area of association." (Tasos C. Christofides, Mathematical Reviews, November, 2013)

Spis treści

Preface.- 1. Associated Random Variables and Related Concepts.- Introduction.- Some Probabilistic Properties of Associated Sequences.- Related Concepts of Association.- 2. Demimartingales.- Introduction.- Characteristic Function Inequalities.- Doob Type Maximal Inequality.- An Upcrossing Inequality.- Chow Type Maximal Inequality.- Whittle Type Maximal Inequality.- More on Maximal Inequalities.- Maximal phi-Inequalities for Nonnegative Demisubmartingales.- Maximal Inequalities for Functions of Demisubmartingales.- Central Limit Theorems.- Dominated Demisubmartingales.- 3. N-Demimartingales.- Introduction.- Maximal Inequalities.- More on Maximal Inequalities.- Downcrossing Inequality.- Chow Type Maximal Inequality.- Functions of N-Demimartingales.- Strong Law of Large Numbers.- Azuma Type Inequality.- Marcinkiewicz-Zygmund Type inequality.- Comparison Theorem on Moment Inequalities.- 4. Conditional Demimartingales.- Introduction.- Conditional Independence.- Conditional Association.- Conditional Demimartingales.- 5. Multidimensionally Indexed Demimartingales and Continuous Parameter Demimartingales.- Introduction.- Multidimensionally Indexed Demimartingales.- Chow Type Maximal Inequality for Two-parameter Demimartingales.- Continuous Parameter Demisubmartingales.- Maximal inequality for Continuous Parameter Demisubmartingales.- Upcrossing Inequality.- 6. Limit Theorems for Associated Random Variables.- Introduction.- Covariance Inequalities.- Hajek-Renyi Type Inequalities.- Exponential Inequality.- Non-uniform and Uniform Berry-Esseen Type Bounds.- Limit Theorems for U-statistics.- More Limit Theorems for U-statistics.- Application to Two-sample Problem.- Limit Theorems for V-statistics.- Limit Theorems for Associated Random Fields.- Remarks.- 7. Nonparametric Estimation for Associated Sequences.- Introduction.- Nonparametric Estimation for Survival Function.- Nonparametric Density Estimation.- Nonparametric Failure Rate Estimation.- Nonparametric Mean Residual Life Function Estimation.- Remarks.- 8. Nonparametric Tests for Associated Sequences.- Introduction.- More on Covariance Inequalities.- Tests for Location.- Mann-Whitney Test.- 9. Nonparametric Tests for Change in Marginal Density Function for Associated Sequences.- Introduction.- Tests for Change Point.- Test for Change in Marginal Density Function.- References.- Index.

Blurb

This book gives a comprehensive review of results for associated sequences and demimartingales developed so far, with special emphasis on demimartingales and related processes.

 

One of the basic aims of theory of probability and statistics is to build stochastic models which explain the phenomenon under investigation and explore the dependence among various covariates which influence this phenomenon. Classic examples are the concepts of Markov dependence or of mixing for random processes. Esary, Proschan and Walkup introduced the concept of association for random variables, and Newman and Wright studied properties of processes termed as demimartingales. It can be shown that the partial sums of mean zero associated random variables form a demimartingale.

 

Probabilistic properties of associated sequences, demimartingales and related processes are discussed in the first six chapters. Applications of some of these results to problems in nonparametric statistical inference for such processes are investigated in the last three chapters.

 

This book will appeal to graduate students and researchers interested in probabilistic aspects of various types of stochastic processes and their applications in reliability theory, statistical mechanics, percolation theory and other areas.