Advances in Directional and Linear Statistics (eBook)

Professor Sreenivasa Rao Jammalamadaka, formerly known as J. S. Rao and affectionately known to most of us as JS, was born on December 7, 1944, at Munipalle, India. Being under-aged for engineering studies was a blessing in disguise, and he was among the first batch of students selected for the Bachelor of Statistics (B. Stat.) degree at the Indian Statistical Institute (ISI), Kolkata. He received his Masters and Ph.D. degrees also from the ISI, and has the distinction of being the first B.Stat.-M.Stat.-Ph.D. of the ISI. He received his education from such legendary figures as Professors P. C. Mahalanobis, J. B. S. Haldane, C. R. Rao, and D. Basu among others, and worked with Professor C.R. Rao for his Ph.D. (1969) on a then newly emerging area, Directional Data Analysis. JS moved to the USA in 1969 and was a faculty member at the Indiana University and then at the University of Wisconsin, Madison, before he moved to the University of California, Santa Barbara (UCSB) in 1976, where he has remained since then. At UCSB he played a leading role in creating the new Department of Statistics and Applied Probability and was its first Chairman. He has been a prodigious mentor to the graduate students in that department, having provided guidance to as many of 35 Ph.D. students, at the last count. Throughout his career, JS has been generous to his colleagues in India, inviting them to the U.S. and spending many of his sabbaticals helping Universities in India as well as in other countries. JS has published extensively in leading international journals. His research work spans a wide spectrum which includes: Goodness-of-Fit tests, Linear Models, Non-parametric and Semi-parametric inference, Bayesian analysis, Reliability, Spacings statistics, and most notably, Directional Data Analysis. He has written several books, both for undergraduate students as well as for advanced researchers. He has collaborated with a large number of researchers from around the world in general, and from India in particular. A Fellow of both the ASA and the IMS among other professional organizations, he has served the cause of statistics at many national and international levels, including that of the President of the International Indian Statistical Association.

Preface 
8 
Contents 
10 
Contributors 
12 
Chapter 1: 
15 
1.1 Introduction 15
1.2 Some Well-Known Circular Models 16
1.3 Introducing Asymmetry 17
1.4 Axial Models 18
1.5 Asymmetric Axial Models 21
1.6 Bivariate Axial Distributions 22
References 23
Chapter 2: 
24 
2.1 Introduction 24
2.2 Preliminaries 26
2.3 Assumptions and Main Results 29
2.4 Proof of Theorems and Corollary 2.1 32
References 38
Chapter 3: 
39 
3.1 Introduction 39
3.2 Data 40
3.3 Multivariate Regression Models with Space-Time ARMA Errors 42
3.3.1 The Model 42
3.3.2 Model Estimation and Model Selection 43
3.4 Asymptotic Properties and Simulations 45
3.5 Simulation Study 54
3.5.1 Model 1 54
3.5.2 Model 2 54
3.5.3 Model 3 55
3.6 Real Data Analysis 55
3.7 Conclusion 61
References 61
Chapter 4: 
63 
4.1 Introduction 63
4.2 Information Theoretic Results for Directional Distributions 65
4.3 Other Properties 69
References 79
Chapter 5: 
81 
5.1 Introduction 81
5.2 Theory 82
5.2.1 Estimation of the Asymptotic Variance 85
5.3 Simulation Studies 86
5.4 An Example 89
5.5 Conclusion 90
References 93
Chapter 6: 
96 
6.1 Introduction 96
6.2 Model Description 97
6.2.1 Heteroscedastic Model 97
6.2.2 Homoscedastic Model 99
6.3 Model Fitting and Estimation 99
6.3.1 Heteroscedastic Model 100
6.3.2 Homoscedastic Model 101
6.4 A Simulation Study 102
6.5 Application to Spellman Data 104
6.6 Conclusion 106
References 106
Chapter 7: 
108 
7.1 Introduction 108
7.2 Markov Chain Monte Carlo Stochastic Approximation Algorithms 109
7.3 Simulations 111
7.4 A Hybrid Algorithm 116
7.5 Conclusions 121
References 121
Chapter 8: 
123 
8.1 Introduction 123
8.2 Stochastic Orders 124
8.3 Spacings 127
8.3.1 One-Sample Problem 128
8.3.2 Two-Samples Problem 129
8.4 Sample Range 134
8.5 Applications 135
8.5.1 Type-II Censoring 135
8.5.2 Reliability 136
8.5.3 Dependence Orderings Among Order Statistics 137
References 137
Chapter 9: 
140 
9.1 Introduction 140
9.2 Models with Deterministic Number of Terms 142
9.2.1 Peak to Sum Ratio 142
9.2.2 Peak to Average Ratio 144
9.3 Models with Random Number of Terms 144
9.3.1 Peak to Sum Ratio 144
9.3.2 Peak to Average Ratio 145
9.4 Geometric Example 146
9.5 An Illustrative Data Example 148
References 150
Chapter 10: 
152 
10.1 Introduction 152
10.2 Main Results 153
10.3 Almost Sure Limits 156
10.4 Asymptotic Normality 158
10.5 Partial Loss of Association 160
10.6 Conclusion 161
References 163
Chapter 11: 
164 
11.1 Introduction 164
11.2 Main Result 170
11.3 Large Deviation Results 174
Appendix 178
References 178
Chapter 12: 
181 
12.1 Introduction 181
12.2 Time Series Models 182
12.3 Innovations 183
12.4 Market Setting and Data 187
12.5 SARIMA Model of Balancing Energy Demand 189
12.6 Innovation Distribution 192
12.7 Conclusion 193
References 195
Chapter 13: 
197 
13.1 Introduction 197
13.2 Results 198
13.3 Proofs 201
References 207
Chapter 14: 
208 
14.1 Introduction 208
14.2 Classification 209
14.2.1 Conventional Classification Methods 210
14.2.1.1 Parametric Model 210
14.2.1.2 Nonparametric Model 211
14.2.2 Nonparametric Classification: New Developments 212
14.2.3 Probabilistic Classifier 215
14.2.4 Applications of the Probabilistic Classifier 217
14.3 k-NN Estimation in Natural Resources 219
14.4 Final Remarks 221
References 222
Chapter 15: 
224 
15.1 Introduction 224
15.2 An Illustrative Example 226
15.3 Dermal Patch Problem 226
15.4 Patterns in Coin Tossing 228
15.5 Chemical Bonding Problem 232
15.6 Yell Game 234
15.7 Noodles Problem 235
15.7.1 Introduction 235
15.7.2 Expected Value 236
15.7.3 Variance 237
15.7.4 Distribution 237
15.7.5 Asymptotic Normality 238
15.8 Conclusions 239
References 239
Chapter 16: 
241 
16.1 Introduction 241
16.1.1 Hardy's Inequalities with Weights 242
16.1.2 Wirtinger Inequality 243
16.1.3 Weighted Wirtinger Type Inequality 243
16.2 Borovkov–Utev Inequality 244
16.3 Circular Random Variables 248
16.4 Chernoff Type Inequality for Wrapped Normal Distribution 251
16.5 Chernoff Type Inequality for von-Mises Distribution 253
References 256
Chapter 17: 
258 
17.1 Introduction, Notation, and Assumptions 258
17.2 Some Basic Results 260
17.3 Some Statistical Applications of Theorems 17.1–17.7 263
17.4 A Convolution Representation Theorem 264
17.5 Some Applications of Theorem 17.8 in the Quest for Asymptotically Efficient Estimates 265
17.5.1 The Weiss–Wolfowitz Approach 265
17.5.2 The Classical Approach 267
17.6 Some Comments 268
17.7 Some Generalizations of Results Stated in Sects.17.2 and 17.4 269
17.8 The Local Asymptotic Mixed Normal and Local Asymptotic Quadratic Experiments 270
17.9 Some Basic Results 273
17.10 Examples Pertaining to the LAMN and LAQ Cases 276
17.11 Outline of Proofs of Some of the Basic Results 279
References 283
Chapter 18: 
286 
18.1 Introduction 286
18.2 Long-Range Dependence 287
18.2.1 Bispectrum and Cumulants 289
18.2.2 Long-Range Dependence in Third Order 292
18.3 Non-Gaussian LRD Models 294
18.3.1 Fractionally Integrated Noise 294
18.3.1.1 Third Order Properties 295
18.3.1.2 Marginals 296
18.3.2 Linear Fractional Noise 297
18.3.2.1 Third Order Properties 298
18.3.2.2 Marginals 299
18.3.3 H2-Process 299
18.3.3.1 Third Order Properties 300
18.3.3.2 Marginals 302
18.3.4 Rosenblatt Process 302
18.3.4.1 Third Order Properties 303
18.3.4.2 Marginals 303
18.3.5 LISDLG Process 304
18.3.5.1 Marginals 306
18.4 Conclusions 307
References 308
Chapter 19: 
310 
19.1 Introduction 310
19.2 The Model 312
19.3 Full-Information Likelihood Inference 313
19.3.1 The Generalized EM Algorithm 314
19.3.2 Standard Errors 315
19.4 The Developmental Toxicity Study 316
19.5 Simulation Study 317
19.6 Discussion 321
References 325