Multivariate Statistics and Probability (eBook)

Essays in Memory of Paruchuri R. Krishnaiah

C. R. Rao, M. M. Rao (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
582 Seiten
Elsevier Science (Verlag)
978-1-4832-6383-0 (ISBN)

Infos

Multivariate Statistics and Probability: Essays in Memory of Paruchuri R. Krishnaiah is a collection of essays on multivariate statistics and probability in memory of Paruchuri R. Krishnaiah (1932-1987), who made significant contributions to the fields of multivariate statistical analysis and stochastic theory. The papers cover the main areas of multivariate statistical theory and its applications, as well as aspects of probability and stochastic analysis. Topics range from finite sampling and asymptotic results, including aspects of decision theory, Bayesian analysis, classical estimation, regression, and time-series problems. Comprised of 35 chapters, this book begins with a discussion on the joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population. The reader is then introduced to kernel estimators of density function of directional data; moment conditions for valid formal edgeworth expansions; and ergodicity and central limit theorems for a class of Markov processes. Subsequent chapters focus on minimal complete classes of invariant tests for equality of normal covariance matrices and sphericity; normed likelihood as saddlepoint approximation; generalized Gaussian random fields; and smoothness properties of the conditional expectation in finitely additive white noise filtering. This monograph should be of considerable interest to researchers as well as to graduate students working in theoretical and applied statistics, multivariate analysis, and random processes.

Inhalt

Front Cover 1
Multivariate Statistics and Probability 4
Copyright Page 5
Table of Contents 6
Contributors 10
Preface 14
In Memoriam 16
Chapter 1. Joint Asymptotic Distribution of Marginal Quantiles and Quantile Functions in Samples from a Multivariate Population 30
1. INTRODUCTION 30
2. DISTRIBUTION OF THE MARGINAL SAMPLE QUANTILES 32
3. TESTS OF SIGNIFICANCE BASED ON MEDIANS 34
4. JOINT DISTRIBUTION OF THE MARGINAL QUANTILE PROCESSES 35
REFERENCES 38
Chapter 2. Kernel Estimators of Density Function of Directional Data 39
1. INTRODUCTION 39
2. POINTWISE STRONG CONSISTENCY 43
3. UNIFORM STRONG CONSISTENCY 46
4. STRONG L1-NORM CONSISTENCY 48
REFERENCES 53
Chapter 3. On Determination of the Order of an Autoregressive Model 55
1. INTRODUCTION 55
2. DETERMINATION OF THE ORDER p 56
3. PROOF OF THE THEOREMS 58
4. SOME REMARKS 65
REFERENCES 67
Chapter 4. Admissible Linear Estimation in a General Gauss-Markov Model with an Incorrectly Specified Dispersion Matrix 68
1. INTRODUCTION AND PRELIMINARIES 68
2. CHARACTERIZATION OF ADMISSIBLE LINEAR ESTIMATORS 70
3. VALIDITY OF ADMISSIBLE LINEAR ESTIMATORS 77
ACKNOWLEDGMENTS 81
REFERENCES 81
Chapter 5. On Moment Conditions for Valid Formal Edgeworth Expansions 83
INTRODUCTION 83
1. THE MAIN RESULT 84
2. EXAMPLES 91
ACKNOWLEDGMENTS 93
REFERENCES 93
Chapter 6. Ergodicity and Central Limit Theorems for a Class of Markov Processes 95
1. INTRODUCTION 95
2. MAIN RESULTS 96
REFERENCES 105
Chapter 7. Conditionally Ordered Distributions 106
1. INTRODUCTION 106
2. CONDITIONALLY MORE POSITIVELY QUADRANT DEPENDENT 107
3. CONDITIONALLY MORE DISPERSED 112
4. CONDITIONAL POSITIVE AND NEGATIVE DEPENDENCE 115
5. FGM DISTRIBUTIONS 117
REFERENCES 118
Chapter 8. A Discounted Cost Relationship 120
1. INTRODUCTION 120
2. REVIEW OF THE BASIC MODEL 120
3. DISCOUNTED COST RELATIONSHIP 123
4. OTHER COST RELATIONSHIPS 129
REFERENCES 130
Chapter 9. Strong Consistency of M-Estimates in Linear Models 131
1. INTRODUCTION 131
2. FORMULATION OF RESULTS 133
3. PROOF OF THEOREMS 1-3 135
4. PROOF OF THEOREM 4 141
REFERENCES 145
Chapter 10. Minimal Complete Classes of Invariant Tests for Equality of Normal Covariance Matrices and Sphericity 146
INTRODUCTION AND SUMMARY 146
2. TESTING .1 = .2 151
3. TESTING SPHERICITY 157
4. PROOFS OF THEOREMS 2.1, 2.2, AND 3.1 159
REFERENCES 165
Chapter 11. Invariance Principles for Changepoint Problems 166
1. INTRODUCTION 166
2. ASYMPTOTICS UNDER Ho 167
3. ASYMPTOTICS UNDER H1 178
4. ANTISYMMETRIC KERNEL 179
ACKNOWLEDGMENTS 182
REFERENCES 182
Chapter 12. On the Area of the Circles Covered by a Random Walk 184
1. INTRODUCTION 184
2. A LOWER ESTIMATE OF R(n) 185
3. CIRCLES COVERED WITH POSITIVE DENSITY 191
4. SOME FURTHER PROBLEMS 193
REFERENCES 195
Chapter 13. Normed Likelihood as Saddlepoint Approximation 196
1. INTRODUCTION 196
2. BARNDORFF-NIELSEN'S FORMULA 198
3. MAXIMUM LIKELIHOOD ESTIMATE: LOCAL DISTRIBUTION FORM 200
4. SADDLEPOINT APPROXIMATIONS 203
5. NORMED LIKELIHOOD AS SADDLEPOINT APPROXIMATION 204
6. REMARKS 207
ACKNOWLEDGMENT 208
REFERENCES 208
Chapter 14. Non-uniform Error Bounds for Asymptotic Expansions of Scale Mixtures of Distributions 209
1. INTRODUCTION 209
2. SCALE MIXTURE OF A GENERAL DISTRIBUTION 210
3. SCALE MIXTURES OF A SYMMETRIC DISTRIBUTION 213
4. APPLICATIONS 215
ACKNOWLEDGMENTS 219
REFERENCES 219
Chapter 15. Empirical and Hierarchical Bayes Competitors of Preliminary Test Estimators in Two Sample Problems 221
1. INTRODUCTION 221
2. THE EB ESTIMATOR AND ITS BAYESIAN PROPERTIES 223
3. MINIMAX ESTIMATION 228
4. HIERARCHICAL BAYES ESTIMATION 233
REFERENCES 241
Chapter 16. On Confidence Bands in Nonparametric Density Estimation and Regression 243
1. INTRODUCTION 243
2. NONPARAMETRIC DENSITY ESTIMATION 244
3. NONPARAMETRIC REGRESSION 250
4. WIDTHS OF CONFIDENCE BANDS 255
5. ILLUSTRATIVE EXAMPLES 258
6. PROOFS 263
REFERENCES 269
Chapter 17. A Note on Generalized Gaussian Random Fields 270
0. INTRODUCTION 270
1. WHITE NOISE AND GAUSSIAN RANDOM FIELDS ON D 270
2. RESTRICTION OF PARAMETER 272
3. GAUSSIAN RANDOM FIELDS DEPENDING ON A CURVE 273
4. CONCLUDING REMARKS 274
REFERENCES 274
Chapter 18. Smoothness Properties of the Conditional Expectation in Finitely Additive White Noise Filtering 276
1. INTRODUCTION 276
2. NOTATION AND TERMINOLOGY 277
3. MAIN RESULTS 280
REFERENCES 284
Chapter 19. Equivariant Estimation of a Mean Vector µ of N(µ, .) with µ'.-1µ=1 or .-1/2µ=c or .=s2µ'µl 285
1. INTRODUCTION 285
2. PROBLEM WITH . KNOWN 287
3. PROBLEM WITH v KNOWN 292
4. THE CASE . = s2µ'µl 295
REFERENCES 297
Chapter 20. A Generalized Cauchy-Binet Formula and Applications to Total Positivity and Majorization 299
1. INTRODUCTION 299
2. A GENERALIZED CAUCHY-BINET FORMULA FOR THE SYMMETRIC GROUP 300
3. GENERALIZED TOTAL POSITIVITY 304
4. SEMIGROUP OF GENERALIZED TOTALLY POSITIVE KERNELS 308
5. COMPLEMENTS 310
REFERENCES 313
Chapter 21. Isotonic M-Estimation of Location: Union-Intersection Principle and Preliminary Test Versions 315
1. INTRODUCTION 315
2. M-ESTIMATORS OF LOCATION AND REGULARITY CONDITIONS 316
3. THE UI-PRELIMINARY M-TEST 318
4. ISOTONIC M-ESTIMATION OF LOCATION 323
5. THE PRELIMINARY TEST ISOTONIC M-ESTIMATOR (PTIME) 324
6. SOME SIMULATION STUDIES 329
ACKNOWLEDGMENTS 333
REFERENCES 333
Chapter 22. Some Asymptotic Inferential Problems Connected with Elliptical Distributions 334
1. INTRODUCTION 334
2. ASYMPTOTIC DISTRIBUTION OF Z AND S 336
3. ASYMPTOTIC CONFIDENCE BOUNDS ON . 339
4. ASYMPTOTIC DISTRIBUTION OF CANONICAL CORRELATIONS 340
5. ASYMPTOTIC CONFIDENCE BOUNDS ON DISCRIMINATORY VALUES 342
REFERENCES 348
Chapter 23. Stochastic Integrals of Empirical-Type Processes with Applications to Censored Regression 349
1. INTRODUCTION 349
2. METRIC ENTROPY AND CONVERGENCE PROPERTIES OF EMPIRICAL-TYPE PROCESSES 352
3. STOCHASTIC INTEGRALS OF EMPIRICAL-TYPE PROCESSES 359
4. APPLICATIONS TO CENSORED RANK ESTIMATORS 363
5. APPLICATIONS TO THE BUCKLEY-JAMES ESTIMATOR 367
REFERENCES 372
Chapter 24. Nonminimum Phase Non-Gaussian Deconvolution 374
INTRODUCTION 374
COMPUTATION 378
CONCLUSIONS 388
ACKNOWLEDGMENTS 389
REFERENCES 389
Chapter 25. Inference in a Model with at Most One Slope-Change Point 390
1. INTRODUCTION 390
2. NORMAL ERROR WITH KNOWN VARIANCE 391
3. NORMAL ERROR WITH UNKNOWN VARIANCE 396
4. NONNORMAL ERROR 401
5. ESTIMATION OF THE SLOPE CHANGE ß1—ß2 403
REFERENCES 406
Chapter 26. Maximum Likelihood Principle and Model Selection when the True Model Is Unspecified 407
1. INTRODUCTION 407
2. OBSERVATIONS AND A FAMILY OF DENSITIES 408
3. MODEL SELECTION 413
4. DISCUSSION 417
ACKNOWLEDGMENTS 417
REFERENCES 418
Chapter 27. An Asymptotic Minimax Theorem of Order n–1/2 419
1. THE RESULTS 419
2. PROOF OF THE THEOREM 423
3. CONSTRUCTION OF THE ESTIMATOR-SEQUENCE 431
4. LEMMAS 434
ACKNOWLEDGMENTS 435
REFERENCES 435
Chapter 28. An Improved Estimation Method for Univariate Autoregressive Models 437
1. INTRODUCTION 437
2. SIMULATION RESULTS 439
3. THE YULE-WALKER AND BURG METHODS 440
4. IMPROVED ESTIMATION OF AUTOREGRESSIONS 443
5. CONCLUDING REMARKS 446
ACKNOWLEDGMENTS 447
REFERENCES 447
Chapter 29. Paradoxes in Conditional Probability 449
1. INTRODUCTION 449
2. THE FRAMEWORK 450
3. CONDITIONAL PROBABILITY AS AN INTEGRATOR 452
4. TWO TYPES OF PARADOXES 455
5. ANOTHER APPROACH AND COMPLEMENTS 459
ACKNOWLEDGMENTS 460
REFERENCES 461
Chapter 30. Inference Properties of a One-Parameter Curved Exponential Family of Distributions with Given Marginals 462
1. INTRODUCTION 462
2. ONE-PARAMETER SYSTEM 464
3. SOME STATISTICAL PROPERTIES 466
4. MAXIMUM LIKELIHOOD ESTIMATION OF . 469
REFERENCES 470
Chapter 31. Asymptotically Precise Estimate of the Accuracy of Gaussian Approximation in Hubert Space 472
1. INTRODUCTION 472
2. THE MAIN RESULT 473
3. AUXILIARY LEMMAS 473
REFERENCES 497
Chapter 32. The Estimation of the Bispectral Density Function and the Detection of Periodicities in a Signal 499
1. INTRODUCTION 499
2. SPECTRAL AND BISPECTRAL DENSITY FUNCTIONS 500
3. TRUNCATED BISPECTRUM 501
4. ESTIMATION OF THE TRUNCATED BISPECTRAL DENSITY FUNCTION hn(.1, .2) 503
5. NUMERICAL ILLUSTRATIONS 504
6. THE RETRIEVAL OF HARMONICS VIA SPECTRUM AND BISPECTRUM 511
7. THE PERIODICITY OF THE EARTH'S MAGNETIC REVERSALS 516
ACKNOWLEDGMENT 519
REFERENCES 519
Chapter 33. Analysis of Odds Ratios in 2×n Ordinal Contingency Tables 520
1. INTRODUCTION 520
2. CONVEXITY PROPERTIES 522
3. AN APPLICATION 525
4. SOME GENERALIZATIONS 532
5. CONCLUDING REMARKS 534
ACKNOWLEDGMENT 534
REFERENCES 534
Chapter 34. Asymptotic Expansions of the Distributions of Some Test Statistics for Gaussian ARMA Processes 536
1. INTRODUCTION 536
2. PRELIMINARIES 537
3. ASYMPTOTIC EXPANSIONS FOR THE NULL DISTRIBUTIONS 540
4. BARTLETT'S ADJUSTMENT 545
5. ASYMPTOTIC EXPANSIONS FOR THE NONNULL DISTRIBUTIONS 546
6. POWER COMPARISONS BETWEEN THE TEST CRITERIA 551
REFERENCES 553
Chapter 35. Estimating Multiple Rater Agreement for a Rare Diagnosis 554
1. INTRODUCTION 554
2. MULTIPLICATIVE MODEL 560
3. MIXING MODELS 566
4. EXAMPLE 572
5. CONCLUDING REMARKS 575
APPENDIX: PROOF OF THEOREM 3.2 575
REFERENCES 576
Author Index 578
Subject Index 580

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 1-4832-6383-5 / 1483263835
• ISBN-13: 978-1-4832-6383-0 / 9781483263830