Advances in Statistics: Proceedings of the Conference in Honor of Professor Zhidong Bai on His 65th Birthday, National University of Singapore, 20 July 2008

CONTENTS......Page 8
Preface......Page 6
Part A Professor Bai’s Life and His Contributions......Page 12
A Conversation with Zhidong Bai - A. Biswas......Page 14
Professor Z. D. Bai: My Friend, Philosopher and Guide - D. Kundu......Page 22
References......Page 24
Collaboration with a Dear Friend and Colleague - J . W. Silverstein......Page 25
References......Page 26
Edgeworth Expansions: A Brief Review of Zhidong Bai’s Contributions - G. J. Babu......Page 27
References......Page 29
Bai’s Contribution to M-estimation and Relevant Tests in Linear Models - L. Zhao......Page 30
References......Page 36
1. Theoretical Properties of Urn Models......Page 38
2. Accommodating Heterogeneity......Page 39
References......Page 40
2. The case that the discrepancy function is a difference of two convex functions......Page 42
3. Minimum &-norm estimation......Page 43
4. Recursive algorithm......Page 44
5. Solvability of an equation arising in the study of M-estimates......Page 45
6. General M-estimation......Page 46
References......Page 47
a). Limiting spectral distributions......Page 48
b). Limits of extreme eigenvalues......Page 49
d). CLT's for smooth integrals of F,......Page 50
References......Page 51
Part B Selected Papers of Professor Bai......Page 54
1. INTBODUOTION......Page 56
2. PBl4LIWNABIES AND TFIB STATEMENTS OB THHl MAIN BIOEIULTB......Page 57
3. PROOFS OF THE MAIN THEOREMS......Page 59
Appendix......Page 66
REFERENCES......Page 70
1. Introduction.......Page 71
2. Inequalities of distance between distributions in terms of their Stieltjes transforms.......Page 73
3. Preliminaries.......Page 77
4. Convergence rates of expected spectral distributions of Wigner matrices.......Page 81
REFERENCES......Page 93
2. Preliminaries.......Page 95
3. Convergence rates of spectral distributions of sample covari- ance matrices.......Page 102
REFERENCES......Page 118
1. Introduction.......Page 119
2. Some lemmas.......Page 122
APPENDIX......Page 133
REFERENCES......Page 137
1. Introduction.......Page 139
2. List of notations.......Page 142
3. Integral range reduction.......Page 143
4. Convergence of v,(x, z) and the limiting spectrum v ( x , z).......Page 147
5. Proof of (1.4).......Page 159
6. Proof of Theorem 1.1.......Page 162
7. Comments and extensions.......Page 163
A l . Lemmas on densities or expectations of functions of random variables.......Page 167
A2. A result known in the literature.......Page 172
REFERENCES......Page 173
1. Introduction.......Page 175
2. Proof of the theorem.......Page 178
REFERENCES......Page 183
1. Introduction......Page 185
2.1.1. Wigner matrix......Page 187
2.1.2. Sample covariance matrix......Page 194
2.1.3. Product of two random matrices......Page 199
2.2.1. Limits of extreme eigenvalues of the Wigner matrix......Page 204
2.2.2. Limits of extreme eigenvalues of sample covariance matrices......Page 208
2.3. Limiting behavior of eigenvectors......Page 210
2.4. Miscellanea......Page 212
3. Stieltjes Transform......Page 213
3.1.1. Wigner matrix......Page 215
3.1.2. General sample covariance matrix......Page 217
3.2. Convergence rates of spectral distributions......Page 221
3.2.1. Wigner matrix......Page 222
3.2.2. Sample covariance matrix......Page 224
4. Circular Law - Non-Hermitian Matrices......Page 225
5 . Applications......Page 227
5.1. Non-exact test for the two-sample problem......Page 228
5.2. Multivariate discrimination analysis......Page 229
5.3. Detection of the number of signals......Page 230
6.1.1. Existence of the LSD......Page 231
6.4.1. Second order convergence of spectral distributions......Page 232
References......Page 233
1. Introduction......Page 236
2. Illustration......Page 237
3. Summary......Page 240
COMMENT: COMPLEMENTS AND NEW DEVELOPMENTS......Page 241
1. Transform Understanding the Limiting Distribution Through Its Stieltjes......Page 242
2. Separation of Eigenvalues......Page 243
3. Results and Conjectures on the Rate of Convergence......Page 244
1. Spectrum Separation of Large Sample Covariance Matrices......Page 245
1.1. Spectrum separation of large sample covariance matrices......Page 246
1.2. Description of exact separation......Page 247
1.3. Strategy of the proof of exact spectrum separation......Page 248
2. On Replica Method......Page 249
Additional References......Page 250
1. Introduction.......Page 252
2. Consistency of maximal depth regression.......Page 254
3. Limiting distribution of the maximal depth regression.......Page 256
4. Asymptotics of the deepest point in RP.......Page 261
APPENDIX......Page 263
REFERENCES......Page 273
1.1. Introduction.......Page 274
1.2. Motivating example.......Page 276
2. Asymptotic properties of the urn composition.......Page 277
3. Likelihood results.......Page 281
4. Conclusions.......Page 283
APPENDIX......Page 284
REFERENCES......Page 290
1. Introduction.......Page 292
2. Convergence of finite-dimensional distributions.......Page 305
3. Tightness of M j ( z ) .......Page 318
4. Convergence of M i ( z ) .......Page 323
5. Some derivations and calculations.......Page 332
APPENDIX......Page 339
REFERENCES......Page 343
1. Introduction.......Page 345
2. Strong consistency of Y, and N,.......Page 348
3. Asymptotic normality of Y,.......Page 359
4. Asymptotic normality of N,.......Page 362
5.1. Adaptive allocation rules associated with covariates.......Page 367
5.3. Urn models for multi-arm clinical trials.......Page 369
REFERENCES......Page 370
1. Introduction......Page 372
2. Main results and examples......Page 374
3. Proofs......Page 377
References......Page 396