Exploration of a Nonlinear World: An Appreciation of Howell Tong's Contribution to Statistics

Contents......Page 12
Foreword P. Whittle......Page 6
Preface K.-S. Chan......Page 8
Acknowledgments......Page 10
Publications of Howell Tong......Page 16
Photograph Sets 1 and 2......Page 30
The Year 1977......Page 32
Non-linear Oscillations......Page 33
Pride and Prejudice......Page 34
What Next?......Page 36
References......Page 37
1. INTRODUCTION......Page 40
2. NON-LINEAR DIFFERENTIAL EQUATIONS......Page 41
3. A LIMIT CYCLE IN DISCRETE TIME......Page 42
4. THRESHOLD AUTOREGRESSIVE MODELS IN DISCRETE TIME......Page 43
5. SOME PERSPECTIVES......Page 44
6. TAR MODELS AND NON-LINEAR VIBRATIONS......Page 47
7. LIMIT CYCLES, LIMIT POINTS AND SETAR......Page 50
8. STATISTICAL IDENTIFICATION......Page 52
9. TAR MODELS FOR REAL DATA......Page 54
10. SOME DISCUSSION......Page 61
REFERENCES......Page 62
REFERENCES IN THE DISCUSSION......Page 86
1. Model Switching......Page 88
3. Partly Linear Parametric Form......Page 89
4. SETAR Models and Conditional Variance......Page 90
References......Page 91
Reflections on Threshold Autoregression P. J. Brockwell......Page 94
References......Page 98
1. Introduction......Page 100
2. The Seed Corn......Page 103
3. Meeting the Market Test......Page 104
4.1. Threshold Cointegration......Page 107
4.2. Threshold GARCH......Page 109
4.3. Spillover Citation Results for Two Progenitors of the TL Paper: Threshold Cointegration and Threshold GARCH......Page 110
5. The ìSignatureî of a Seminal Paper......Page 111
References......Page 112
The SETAR Model of Tong and Lim and Advances in Computation J. Geweke......Page 116
1. The SETAR of TL and Its Application......Page 117
2. A Bayesian treatment of SETAR......Page 118
3. Generalizing SETAR......Page 121
4. Conclusion......Page 124
References......Page 125
2. ARCH Models with a Threshold Structure......Page 126
3. Bayesian Inference for Threshold Volatility Models......Page 128
Conclusion......Page 129
References......Page 130
Dependence and Nonlinearity M. Rosenblatt......Page 132
References......Page 135
2. Comment......Page 138
3. The Simple Idea that Works......Page 139
References......Page 140
Photograph Sets 3 and 4......Page 142
1. INTRODUCTION......Page 144
2. CROSS-VALIDATORY ESTIMATE......Page 147
3. FINAL PREDICTION ERROR APPROACH......Page 150
4.1. Example 1......Page 151
4.2. Example 2......Page 154
4.3. Example 3......Page 157
4.4. Example 4......Page 158
5. DISCUSSION......Page 161
ACKNOWLEDGEMENTS......Page 162
A.l. Conditions for Theorem 1......Page 163
A.2. Proof of Theorem 2......Page 164
REFERENCES......Page 165
1. Introduction......Page 168
2. Cross–Validation Model Selection in Semiparametric Regression......Page 169
2.1. Cross–validation criterion for nonparametric regressors......Page 170
2.2. CV criterion for the selection of parametric regressors......Page 173
3. Discussion......Page 176
n......Page 177
An Introduction to a Paper by Bing Cheng and Howell Tong: On Consistent Nonparametric Order Determination and Chaos (with Discussion) D. Tj stheim......Page 178
References......Page 180
1. Introduction......Page 182
2. A simple criterion for non-null compact sets to be small......Page 184
3. From deterministic stability to ergodicity: a precursor......Page 185
4. Some general results......Page 187
5. Some extensions and examples......Page 189
References......Page 193
1. Introduction......Page 196
2. Noisy Dynamical Systems......Page 197
3. Relevant Notions of Stability......Page 198
4. The Lyapunov Exponent of Drift......Page 202
5. Limit Cycles and Chaos......Page 204
6. Bilinear and GARCH Time Series......Page 209
References......Page 211
1. Introduction......Page 214
2.1. Attractors......Page 216
2.2. Lyapunov exponents......Page 217
2.3. Stochastic difference equations......Page 218
3.1. Identical noise realizations......Page 220
3.2.1. The conditional distribution approach......Page 221
3.2.2. An example......Page 222
3.2.3. The conditional mean approach......Page 223
3.3. Noise amplification and prediction......Page 224
3.4. A decomposition theorem......Page 225
4. 1. Locally linear non -parame tric regression......Page 226
4.2. Order determination......Page 227
4.3. Attractor dimension......Page 229
4.4.1. Substantive modelling and black-box modelling......Page 230
4.4.3. Local function approximations......Page 231
5. Some recent applications......Page 232
Acknowledgements......Page 233
References......Page 234
DISCUSSION AND COMMENTS......Page 236
1. Introduction......Page 262
3. \"Some Comments On A Bridge : : : \" (Physica D, 1992)......Page 263
4. \"A Personal Overview of \" (Scand. J. Stat, 1995)......Page 264
References......Page 265
1. Introduction......Page 268
2. Contrasting Views on Chaotic Time Series Modelling......Page 269
3. Statistical Aspects of Chaotic Sequences......Page 270
4. Communications and Chaos......Page 272
References......Page 277
2.1. Deterministic chaos......Page 280
2.2. Sensitivity to initial values......Page 281
3. Ergodicity of Stochastic Di erence Equations......Page 283
References......Page 285
Photograph Sets 5 and 6......Page 286
1. INTRODUCTION......Page 288
2. DEFINITION OF THE SPECTRUM OF A NON-STATIONARY PROCESS......Page 289
3. UNIVARIATE NON-STATIONARY PROCESSES......Page 290
4. DEFINITION OF EVOLUTIONARY CROSS-SPECTRA......Page 292
5. ESTIMATION OF EVOLUTIONARY CROSS-SPECTRA......Page 294
7. EsTIMATION OF THE TRANSFER FUNCTION OF A TIME-DEPENDENT OPEN LooP SYSTEM......Page 297
8. NUMERICAL ILLUSTRATIONS......Page 298
REFERENCES......Page 300
2. LIKELIHOOD RATIO TEST FOR THRESHOLD AUTOREGRESSION......Page 302
4. FIRST-PASSAGE PROBABILITY......Page 304
APPENDIX A: OUTLINE PROOF OF BASIC RESULT......Page 306
APPENDIX C: RESULTS FOR SPECIAL CASE (b)......Page 307
REFERENCES......Page 309
1. Introduction......Page 310
2. Strong Consistency of the LSE......Page 311
3. Some Open Problems......Page 314
Appendix: Proof of Lemmas 2.1 and 2.2......Page 315
Acknowledgement......Page 317
References......Page 318
1. On \"Strong Consistency of the Least Squares Estimator for a Non-Ergodic Threshold Autoregressive Model\" by Pham, Chan and Tong (1991)......Page 320
2. On \"On Likelihood Ratio Tests for Threshold Autoregression\" by Chan and Tong (1990)......Page 323
References......Page 325
Photograph Sets 7 and 8......Page 328
1. Introduction......Page 330
2. 1. The estimation of effective dimension reduction directions......Page 333
2.1.1. Multidimensional kernel weight......Page 334
2.2. Dimension of effective dimension reduction space......Page 335
2.3. Bandwidth and algorithm......Page 337
3. 1. Outer product of gradients estimation......Page 338
3.2. Inverse regression minimum average (conditional) variance estimation We start with......Page 339
3.3. Semiparametric multi-index models......Page 340
4. Simulations......Page 341
4.1. Example 1......Page 342
4.3. Example 3......Page 343
5. 1. Circulatory and respiratory problems in Hong Kong......Page 344
5.2. Hitters\' salary data......Page 348
6. Conclusions......Page 351
Acknowledgements......Page 352
Appendix A......Page 353
Discussion on the paper by Xia, Tong, Li and Zhu......Page 355
References in the discussion......Page 375
1. Introduction......Page 378
2. Estimating Semiparametric Regression Models with Nonsmooth Loss Functions......Page 379
2.1. Semiparametric estimation with Lasso......Page 380
2.2. Quantile regression......Page 381
3.1. Dimension reduction of complete data......Page 383
3.2. Dimension reduction for the censored data......Page 384
4. Some Numerical Results......Page 387
References......Page 390
Common Dynamic Structure of Canada Lynx Populations within Three Climatic Regions N. Chr. Stenseth et al.......Page 392
References and Notes......Page 394
Introduction......Page 396
The TAR Lynx Models and Phase Dependence......Page 397
The Geographic Structuring of the Canadian Lynx: The Importance of Statistical Modelling......Page 399
Conclusions......Page 402
What Next?......Page 403
References......Page 404
1. Introduction......Page 406
2. Tong\'s Publications on Reliability Estimation......Page 407
References......Page 409
A Chinese Poem, with Translation H. Tong......Page 412