Probability approximations and beyond : [on 25 and 26 June 2010, a conference, Probability Approximation and Beyond, was held ... Singapore to honor Louis Chen on his 70th birthday

In June 2010, a conference, __Probability Approximations and Beyond__, was held at the National University of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Chen made the first of several seminal contributions to the theory and application of Stein’s method. One of his most important contributions has been to turn Stein’s concentration inequality idea into an effective tool for providing error bounds for the normal approximation in many settings, and in particular for sums of random variables exhibiting only local dependence. This conference attracted a large audience that came to pay homage to Chen and to hear presentations by colleagues who have worked with him in special ways over the past 40+ years. The papers in this volume attest to how Louis Chen’s cutting-edge ideas influenced and continue to influence such areas as molecular biology and computer science. He has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Chen’s work alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today. Probability Approximations and Beyond 1 Title page 3 Preface 5 Contents 10 Contributors 12 Part I Stein’s Method 14 1 Couplings for Irregular Combinatorial Assemblies 15 1.1 Introduction 15 1.2 A Mineka---like Coupling 18 1.3 A Poisson-Based Coupling 21 References 24 2 Berry-Esseen Inequality for Unbounded Exchangeable Pairs 25 2.1 Introduction and Main Result 25 2.2 Proof of the Main Result 26 2.3 An Application to an Independence Test 30 References 41 3 Clubbed Binomial Approximation for the Lightbulb Process 43 3.1 Introduction 43 3.2 The Lightbulb Process 45 3.3 Stein Operator 46 3.4 Bounds on the Stein Equation 49 References 53 4 Coverage of Random Discs Driven by a Poisson Point Process 54 4.1 Introduction and Main Results 54 4.2 An Estimation via Compound Poisson Approximation 56 4.3 Proofs of the Main Results 59 4.4 Comparisons Between the Estimations ([Equ1]4.1) and ([Equ2]4.2) 68 References 69 5 On the Optimality of Stein Factors 71 5.1 Introduction 71 5.2 An Illustrative Example 72 5.3 Multivariate Poisson Distribution 75 5.4 Poisson Point Processes 78 References 82 Part II Related Topics 83 6 Basic Estimates of Stability Rate for One-Dimensional Diffusions 84 6.1 Introduction 84 6.2 The Main Result and Motivation 85 6.2.1 Two Types of Exponential Convergence 85 6.2.2 Statement of the Result 86 6.2.3 Short Review on the Known Results 89 6.2.4 Motivation and Application 90 6.3 Sketch of the Proof 91 6.3.1 Coupling Method 91 6.3.2 Dual Method 92 6.3.3 Capacitary Method 93 6.3.4 The Final Step 94 6.3.5 Summary of the Proof 94 6.4 Improvements 95 6.5 Examples 100 Appendix 106 References 107 7 Trend Analysis of Extreme Values 109 7.1 Catastrophes 109 7.2 The Insured Losses 111 7.2.1 No Time Component 111 7.2.2 Time Component 111 7.3 Conclusion 115 References 116 8 Renormalizations in White Noise Analysis 117 8.1 Introduction 117 8.2 White Noise and Its Generalized Functionals 118 8.2.1 Linear Functionals of White Noise 118 8.2.2 Generalized White Noise Functionals 119 8.3 Renormalizations 120 8.3.1 The Algebra A 120 8.3.2 Renormalizations, a General Theory 120 8.3.3 Exponentials of Quadratic Functionals 122 References 123 9 M-Dependence Approximation for Dependent Random Variables 125 9.1 Introduction 125 9.2 Strong Invariance Principle for Partial Sums 127 9.3 Strong Invariance Principle for Empirical Process 130 9.3.1 Linear Process 131 9.3.2 Nonlinear AR Model 132 9.3.3 GARCH Model 132 9.4 Kernel Density Estimation 132 9.5 The Maximum of the Periodogram 134 9.6 Spectral Density Estimation 136 References 138 10 Variable Selection for Classification and Regression in Large p, Small n Problems 142 10.1 Introduction 142 10.2 GUIDE Variable Selection 145 10.3 Expected Squared Error 147 10.4 Some Theory for Linear Models 155 10.5 Application to Real Data 162 10.6 Conclusion 165 References 166 In June 2010, a conference, Probability Approximations and Beyond, was held at the National University of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Louis is perhaps best known for his elegant Poisson approximation method, developed from Stein's original approach to normal approximation. Another of his important contributions has been to turn Stein's concentration inequality idea into an effective tool for providing error bounds for the normal approximation in many settings, and in particular for sums of random variables exhibiting only local dependence.¡ The conference attracted a large audience that came to pay homage to Louis, and to hear presentations by colleagues who have worked with him in special ways over the past 40 years. The papers in this volume attest to how Louis Chen's ideas have influenced and continue to influence such diverse areas as molecular biology and computer science. He himself has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Louis's work, alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today. The papers in this volume attest to how Louis Chen's ideas have influenced and continue to influence such diverse areas as molecular biology and computer science. He himself has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Louis's work, alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today In June 2010, a conference, Probability Approximations and Beyond, was held at the National University of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Chen made the first of several seminal contributions to the theory and application of Stein's method. One of his most important contributions has been to turn Stein's concentration inequality idea into an effective tool for providing error bounds for the normal approximation in many settings, and in particular for sums of random variables exhibiting only local dependence. This conference attracted a large audience that came to pay homage to Chen and to hear presentations by colleagues who have worked with him in special ways over the past 40+ years. The papers in this volume attest to how Louis Chen's cutting-edge ideas influenced and continue to influence such areas as molecular biology and computer science. He has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Chen's work alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today.-- Provided by publisher In June 2010, a conference, Probability Approximations and Beyond, was held at the National University of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Chen made the first of several seminal contributions to the theory and application of Stein s method. One of his most important contributions has been to turn Stein s concentration inequality idea into an effective tool for providing error bounds for the normal approximation in many settings, and in particular for sums of random variables exhibiting only local dependence. This conference attracted a large audience that came to pay homage to Chen and to hear presentations by colleagues who have worked with him in special ways over the past 40+ years. The papers in this volume attest to how Louis Chen s cutting-edge ideas influenced and continue to influence such areas as molecular biology and computer science. He has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Chen s work alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today. Stein's methods: Couplings for irregular combinatorial assemblies / Andrew D. Barbour and Anna Pósfai. Berry-Esseen inequality for unbounded exchangeable pairs / Yanchu Chen and Qi-Man Shao. Clubbed binomial approximation for the lightbulb process / LArry Goldstein and Aihua Xia. Coverage of random discs driven by a poisson point process / Guo-Lie Lan, Zhi-Ming Ma and Su-Yong Sun. On the optimality of Stein factors / Adrian Röllin Related topics: Basic estimates of stability rate for one-dimensional diffusions / Mu-Fa Chen. Trend analysis of extreme values / Goedele Dierckx and Jef Teugels. Renormalizations in white noise analysis / Takeyuki Hida. M-dependence approximation for dependent random variables / Zheng-Yan Lin and Weidong Liu. Variable selection for classification and regression in large p, small n problems / Wei-Lin Loh. Front Matter....Pages i-xiv Front Matter....Pages 1-1 Couplings for Irregular Combinatorial Assemblies....Pages 3-12 Berry-Esseen Inequality for Unbounded Exchangeable Pairs....Pages 13-30 Clubbed Binomial Approximation for the Lightbulb Process....Pages 31-41 Coverage of Random Discs Driven by a Poisson Point Process....Pages 43-59 On the Optimality of Stein Factors....Pages 61-72 Front Matter....Pages 73-73 Basic Estimates of Stability Rate for One-Dimensional Diffusions....Pages 75-99 Trend Analysis of Extreme Values....Pages 101-108 Renormalizations in White Noise Analysis....Pages 109-116 M-Dependence Approximation for Dependent Random Variables....Pages 117-133 Variable Selection for Classification and Regression in Large p , Small n Problems....Pages 135-159