Probability Models, Volume 51 in the Handbook of Statistics series, highlights new advances in the field, with this new volume presenting interesting chapters on Stein’s methods, Probabilities and thermodynamics third law, Random Matrix Theory, General tools for understanding fluctuations of random variables, An approximation scheme to compute the Fisher-Rao distance between multivariate normal distributions, Probability Models Applied to Reliability and Availability Engineering, Backward stochastic differential equation– Stochastic optimization theory and viscous solution of HJB equation, and much more.
Additional chapters cover Probability Models in Machine Learning, The recursive stochastic algorithm, randomized urn models and response-adaptive randomization in clinical trials, Random matrix theory: local laws and applications, KOO methods and their high-dimensional consistencies in some multivariate models, Fourteen Lectures on Inference for Stochastic Processes, and A multivariate cumulative damage model and some applications.
1. 1 Introduction - p. 3 2. 2 Stein identity and Stein equation - p. 8 3. 3 Normal approximation - p. 6 4. 4 Nonnormal approximation - p. 18 5. 5 Poisson approximation by Stein's method - p. 22 6. References - p. 25 7. Chapter 2: Probabilities and thermodynamics’ third law - p. 27 8. Chapter 3: KOO methods and their high-dimensional consistencies in some multivariate models - p. 41 9. Chapter 4: Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models - p. 67 10. Abstract - p. 67 11. 1 Introduction - p. 68 12. 2 The Fisher-Rao distance - p. 70 13. 3 Generic closed-form cases and a canonical Fisher-Rao upper bound - p. 75 14. 4 Generic approximation schemes for the Fisher-Rao distances - p. 79 15. 5 Tight upper bounds on Fisher-Rao distances: The case of Hessian metrics - p. 89 16. 6 Tight lower bounds on Fisher-Rao distances from isometric embeddings - p. 93 17. 7 Approximating Fisher-Rao distances on elliptical distribution manifolds - p. 96 18. 8 Maximal invariant and Fisher-Rao distances - p. 104 19. 9 Conclusion: Summary and some open problems - p. 108 20. References - p. 112 21. Part II: Random matrix theory - p. 116 22. Chapter 5: Random matrix theory and high-dimensional statistics - p. 119 23. Abstract - p. 119 24. 1 An introductory example - p. 121 25. 2 Sample covariance matrix - p. 123 26. 3 Main theorems - p. 127 27. 4 Proof of - p. 126 28. 5 Proof of - p. 126 29. References - p. 140 30. Chapter 6: Random matrix theory: Local laws and applications - p. 143 31. Abstract - p. 143 32. 1 Introduction - p. 143 33. 2 Local law - p. 147 34. 3 Proof of local laws - p. 152 35. 4 Tracy-Widom law - p. 157 36. 5 Spiked model - p. 165 37. 6 Conclusion - p. 169 38. Acknowledgments - p. 171 39. References - p. 171 40. Part III: Models and applications - p. 173 41. Chapter 7: Multiplier bootstrap inference for generalized functional linear models - p. 177 42. Abstract - p. 177 43. 1 Introduction - p. 177 44. 2 Regularized estimation - p. 179 45. 3 Bootstrap inference - p. 180 46. 4 Simulation studies - p. 191 47. 5 Concluding remarks - p. 193 48. References - p. 193 49. Chapter 8: Probability models applied to reliability and availability engineering - p. 195 50. Chapter 9: The recursive stochastic algorithm, randomized urn models, and response-adaptive randomization in clinical trials - p. 215 51. Abstract - p. 215 52. 1 Recursive stochastic approximation algorithms - p. 216 53. 2 Randomized urn models - p. 234 54. 3 Response-adaptive randomization - p. 252 55. 4 Conclusions remarks - p. 270 56. References - p. 272 57. Chapter 10: Challenges of statistical machine learning when encountered deep neural network - p. 275 58. Chapter 11: Functional law of large numbers for an epidemic model with random effects - p. 297 59. Part IV: Higher dimensional models - p. 313 60. Chapter 12: Fourteen lectures on inference for stochastic processes - p. 317 61. Abstract - p. 317 62. 1 Introduction - p. 317 63. 2 Lecture 1. Stochastic processes - p. 319 64. 3 Lecture 2. Discrete time martingales - p. 322 65. 4 Lecture 3. Some limit theorems and inequalities for martingales - p. 324 66. 5 Lecture 4. Likelihood ratio - p. 329 67. 6 Lecture 5. Conditional exponential family - p. 337 68. 7 Lecture 6. Inference for branching processes (,; ; ) - p. 342 69. 8 Lecture 7. Estimation by the method of conditional least squares - p. 349 70. 9 Lecture 8. Neyman-Pearson Lemma - p. 357 71. 10 Lecture 9. Representation of a second order stochastic process - p. 360 72. 11 Lecture 10. Kakutani's theorem - p. 363 73. 12 Lecture 11. Stochastic integrals and stochastic differential equations - p. 369 74. 13 Lecture 12. Ito's Lemma - p. 375 75. 14 Lecture 13. Estimation for diffusion-type processes - p. 379 76. 15 Lecture 14. Fractional Brownian motion and long range dependence - p. 405 77. References - p. 404 78. Further reading - p. 407 79. Chapter 13: Backward stochastic differential equations: Stochastic optimization theory of backward stochastic differential equations and viscosity solutions of Hamilton–Jacobi–Bellman equations - p. 413 80. Abstract - p. 413 81. 1 Introduction - p. 413 82. 2 Backward stochastic differential equations - p. 418 83. 3 1-Dimensional case: Comparison theorem and semigroup property - p. 424 84. 4 BSDE Coupled with Forward SDE of Itô’s type - p. 429 85. 5 Stochastic optimal control and generalized dynamic programming principle - p. 435 86. 6 Dynamic programming principle of non-Markov system - p. 442 87. 7 Generalized Hamilton–Jacobi–Bellman equations and viscosity solutions - p. 445 88. 8 G–expectation - p. 450 89. Acknowledgments - p. 454 90. References - p. 454 91. Further reading - p. 455 92. Chapter 14: A multivariate cumulative damage model and some applications - p. 481 93. Abstract - p. 481 94. 1 Introduction - p. 461 95. 2 Preliminaries and some notations - p. 460 96. 3 Asymptotic results - p. 462 97. 4 An application to neurophysiology - p. 471 98. 5 A multivariate failure stopping time - p. 473 99. 6 Parametric inference for failure stopping times - p. 477 100. 7 Simulations - p. 480 101. 8 An application to infectious diseases - p. 487 102. 9 Conclusions and possible future work - p. 494 103. Acknowledgments - p. 494 104. References - p. 494 105. Index - p. 497
Probability Models, Volume 51 in the Handbook of Statistics series, highlights new advances in the field, with this new volume presenting interesting chapters on Stein’s methods, Probabilities and thermodynamics third law, Random Matrix Theory, General tools for understanding fluctuations of random variables, An approximation scheme to compute the Fisher-Rao distance between multivariate normal distributions, Probability Models Applied to Reliability and Availability Engineering, Backward stochastic differential equation– Stochastic optimization theory and viscous solution of HJB equation, and much more.
Additional chapters cover Probability Models in Machine Learning, The recursive stochastic algorithm, randomized urn models and response-adaptive randomization in clinical trials, Random matrix theory: local laws and applications, KOO methods and their high-dimensional consistencies in some multivariate models, Fourteen Lectures on Inference for Stochastic Processes, and A multivariate cumulative damage model and some applications.
1. 1 Introduction - p. 3 2. 2 Stein identity and Stein equation - p. 8 3. 3 Normal approximation - p. 6 4. 4 Nonnormal approximation - p. 18 5. 5 Poisson approximation by Stein's method - p. 22 6. References - p. 25 7. Chapter 2: Probabilities and thermodynamics’ third law - p. 27 8. Chapter 3: KOO methods and their high-dimensional consistencies in some multivariate models - p. 41 9. Chapter 4: Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models - p. 67 10. Abstract - p. 67 11. 1 Introduction - p. 68 12. 2 The Fisher-Rao distance - p. 70 13. 3 Generic closed-form cases and a canonical Fisher-Rao upper bound - p. 75 14. 4 Generic approximation schemes for the Fisher-Rao distances - p. 79 15. 5 Tight upper bounds on Fisher-Rao distances: The case of Hessian metrics - p. 89 16. 6 Tight lower bounds on Fisher-Rao distances from isometric embeddings - p. 93 17. 7 Approximating Fisher-Rao distances on elliptical distribution manifolds - p. 96 18. 8 Maximal invariant and Fisher-Rao distances - p. 104 19. 9 Conclusion: Summary and some open problems - p. 108 20. References - p. 112 21. Part II: Random matrix theory - p. 116 22. Chapter 5: Random matrix theory and high-dimensional statistics - p. 119 23. Abstract - p. 119 24. 1 An introductory example - p. 121 25. 2 Sample covariance matrix - p. 123 26. 3 Main theorems - p. 127 27. 4 Proof of - p. 126 28. 5 Proof of - p. 126 29. References - p. 140 30. Chapter 6: Random matrix theory: Local laws and applications - p. 143 31. Abstract - p. 143 32. 1 Introduction - p. 143 33. 2 Local law - p. 147 34. 3 Proof of local laws - p. 152 35. 4 Tracy-Widom law - p. 157 36. 5 Spiked model - p. 165 37. 6 Conclusion - p. 169 38. Acknowledgments - p. 171 39. References - p. 171 40. Part III: Models and applications - p. 173 41. Chapter 7: Multiplier bootstrap inference for generalized functional linear models - p. 177 42. Abstract - p. 177 43. 1 Introduction - p. 177 44. 2 Regularized estimation - p. 179 45. 3 Bootstrap inference - p. 180 46. 4 Simulation studies - p. 191 47. 5 Concluding remarks - p. 193 48. References - p. 193 49. Chapter 8: Probability models applied to reliability and availability engineering - p. 195 50. Chapter 9: The recursive stochastic algorithm, randomized urn models, and response-adaptive randomization in clinical trials - p. 215 51. Abstract - p. 215 52. 1 Recursive stochastic approximation algorithms - p. 216 53. 2 Randomized urn models - p. 234 54. 3 Response-adaptive randomization - p. 252 55. 4 Conclusions remarks - p. 270 56. References - p. 272 57. Chapter 10: Challenges of statistical machine learning when encountered deep neural network - p. 275 58. Chapter 11: Functional law of large numbers for an epidemic model with random effects - p. 297 59. Part IV: Higher dimensional models - p. 313 60. Chapter 12: Fourteen lectures on inference for stochastic processes - p. 317 61. Abstract - p. 317 62. 1 Introduction - p. 317 63. 2 Lecture 1. Stochastic processes - p. 319 64. 3 Lecture 2. Discrete time martingales - p. 322 65. 4 Lecture 3. Some limit theorems and inequalities for martingales - p. 324 66. 5 Lecture 4. Likelihood ratio - p. 329 67. 6 Lecture 5. Conditional exponential family - p. 337 68. 7 Lecture 6. Inference for branching processes (,; ; ) - p. 342 69. 8 Lecture 7. Estimation by the method of conditional least squares - p. 349 70. 9 Lecture 8. Neyman-Pearson Lemma - p. 357 71. 10 Lecture 9. Representation of a second order stochastic process - p. 360 72. 11 Lecture 10. Kakutani's theorem - p. 363 73. 12 Lecture 11. Stochastic integrals and stochastic differential equations - p. 369 74. 13 Lecture 12. Ito's Lemma - p. 375 75. 14 Lecture 13. Estimation for diffusion-type processes - p. 379 76. 15 Lecture 14. Fractional Brownian motion and long range dependence - p. 405 77. References - p. 404 78. Further reading - p. 407 79. Chapter 13: Backward stochastic differential equations: Stochastic optimization theory of backward stochastic differential equations and viscosity solutions of Hamilton–Jacobi–Bellman equations - p. 413 80. Abstract - p. 413 81. 1 Introduction - p. 413 82. 2 Backward stochastic differential equations - p. 418 83. 3 1-Dimensional case: Comparison theorem and semigroup property - p. 424 84. 4 BSDE Coupled with Forward SDE of Itô’s type - p. 429 85. 5 Stochastic optimal control and generalized dynamic programming principle - p. 435 86. 6 Dynamic programming principle of non-Markov system - p. 442 87. 7 Generalized Hamilton–Jacobi–Bellman equations and viscosity solutions - p. 445 88. 8 G–expectation - p. 450 89. Acknowledgments - p. 454 90. References - p. 454 91. Further reading - p. 455 92. Chapter 14: A multivariate cumulative damage model and some applications - p. 481 93. Abstract - p. 481 94. 1 Introduction - p. 461 95. 2 Preliminaries and some notations - p. 460 96. 3 Asymptotic results - p. 462 97. 4 An application to neurophysiology - p. 471 98. 5 A multivariate failure stopping time - p. 473 99. 6 Parametric inference for failure stopping times - p. 477 100. 7 Simulations - p. 480 101. 8 An application to infectious diseases - p. 487 102. 9 Conclusions and possible future work - p. 494 103. Acknowledgments - p. 494 104. References - p. 494 105. Index - p. 497
