Rao Multivariate Statistics and Probability

Essays in Memory of Paruchuri R. Krishnaiah
1. Auflage 2014
ISBN: 978-1-4832-6383-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 582 Seiten, Web PDF

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.

Autoren/Hrsg.

Rao, C. R.

Inhaltsverzeichnis

1;Front Cover;1
2;Multivariate Statistics and Probability;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Preface;14
7;In Memoriam;16
8;Chapter 1. Joint Asymptotic Distribution of Marginal Quantiles and Quantile Functions in Samples from a Multivariate Population;30
8.1;1. INTRODUCTION;30
8.2;2. DISTRIBUTION OF THE MARGINAL SAMPLE QUANTILES;32
8.3;3. TESTS OF SIGNIFICANCE BASED ON MEDIANS;34
8.4;4. JOINT DISTRIBUTION OF THE MARGINAL QUANTILE PROCESSES;35
8.5;REFERENCES;38
9;Chapter 2. Kernel Estimators of Density Function of Directional Data;39
9.1;1. INTRODUCTION;39
9.2;2. POINTWISE STRONG CONSISTENCY;43
9.3;3. UNIFORM STRONG CONSISTENCY;46
9.4;4. STRONG L1-NORM CONSISTENCY;48
9.5;REFERENCES;53
10;Chapter 3. On Determination of the Order of an Autoregressive Model;55
10.1;1. INTRODUCTION;55
10.2;2. DETERMINATION OF THE ORDER p;56
10.3;3. PROOF OF THE THEOREMS;58
10.4;4. SOME REMARKS;65
10.5;REFERENCES;67
11;Chapter 4. Admissible Linear Estimation in a General Gauss-Markov Model with an Incorrectly Specified Dispersion Matrix;68
11.1;1. INTRODUCTION AND PRELIMINARIES;68
11.2;2. CHARACTERIZATION OF ADMISSIBLE LINEAR ESTIMATORS;70
11.3;3. VALIDITY OF ADMISSIBLE LINEAR ESTIMATORS;77
11.4;ACKNOWLEDGMENTS;81
11.5;REFERENCES;81
12;Chapter 5. On Moment Conditions for Valid Formal Edgeworth Expansions;83
12.1;INTRODUCTION;83
12.2;1. THE MAIN RESULT;84
12.3;2. EXAMPLES;91
12.4;ACKNOWLEDGMENTS;93
12.5;REFERENCES;93
13;Chapter 6. Ergodicity and Central Limit Theorems for a Class of Markov Processes;95
13.1;1. INTRODUCTION;95
13.2;2. MAIN RESULTS;96
13.3;REFERENCES;105
14;Chapter 7. Conditionally Ordered Distributions;106
14.1;1. INTRODUCTION;106
14.2;2. CONDITIONALLY MORE POSITIVELY QUADRANT DEPENDENT;107
14.3;3. CONDITIONALLY MORE DISPERSED;112
14.4;4. CONDITIONAL POSITIVE AND NEGATIVE DEPENDENCE;115
14.5;5. FGM DISTRIBUTIONS;117
14.6;REFERENCES;118
15;Chapter 8. A Discounted Cost Relationship;120
15.1;1. INTRODUCTION;120
15.2;2. REVIEW OF THE BASIC MODEL;120
15.3;3. DISCOUNTED COST RELATIONSHIP;123
15.4;4. OTHER COST RELATIONSHIPS;129
15.5;REFERENCES;130
16;Chapter 9. Strong Consistency of M-Estimates in Linear Models;131
16.1;1. INTRODUCTION;131
16.2;2. FORMULATION OF RESULTS;133
16.3;3. PROOF OF THEOREMS 1-3;135
16.4;4. PROOF OF THEOREM 4;141
16.5;REFERENCES;145
17;Chapter 10. Minimal Complete Classes of Invariant Tests for Equality of Normal Covariance Matrices and Sphericity;146
17.1;INTRODUCTION AND SUMMARY;146
17.2;2. TESTING .1 = .2;151
17.3;3. TESTING SPHERICITY;157
17.4;4. PROOFS OF THEOREMS 2.1, 2.2, AND 3.1;159
17.5;REFERENCES;165
18;Chapter 11. Invariance Principles for Changepoint Problems;166
18.1;1. INTRODUCTION;166
18.2;2. ASYMPTOTICS UNDER Ho;167
18.3;3. ASYMPTOTICS UNDER H1;178
18.4;4. ANTISYMMETRIC KERNEL;179
18.5;ACKNOWLEDGMENTS;182
18.6;REFERENCES;182
19;Chapter 12. On the Area of the Circles Covered by a Random Walk;184
19.1;1. INTRODUCTION;184
19.2;2. A LOWER ESTIMATE OF R(n);185
19.3;3. CIRCLES COVERED WITH POSITIVE DENSITY;191
19.4;4. SOME FURTHER PROBLEMS;193
19.5;REFERENCES;195
20;Chapter 13. Normed Likelihood as Saddlepoint Approximation;196
20.1;1. INTRODUCTION;196
20.2;2. BARNDORFF-NIELSEN'S FORMULA;198
20.3;3. MAXIMUM LIKELIHOOD ESTIMATE: LOCAL DISTRIBUTION FORM;200
20.4;4. SADDLEPOINT APPROXIMATIONS;203
20.5;5. NORMED LIKELIHOOD AS SADDLEPOINT APPROXIMATION;204
20.6;6. REMARKS;207
20.7;ACKNOWLEDGMENT;208
20.8;REFERENCES;208
21;Chapter 14. Non-uniform Error Bounds for Asymptotic Expansions of Scale Mixtures of Distributions;209
21.1;1. INTRODUCTION;209
21.2;2. SCALE MIXTURE OF A GENERAL DISTRIBUTION;210
21.3;3. SCALE MIXTURES OF A SYMMETRIC DISTRIBUTION;213
21.4;4. APPLICATIONS;215
21.5;ACKNOWLEDGMENTS;219
21.6;REFERENCES;219
22;Chapter 15. Empirical and Hierarchical Bayes Competitors of Preliminary Test Estimators in Two Sample Problems;221
22.1;1. INTRODUCTION;221
22.2;2. THE EB ESTIMATOR AND ITS BAYESIAN PROPERTIES;223
22.3;3. MINIMAX ESTIMATION;228
22.4;4. HIERARCHICAL BAYES ESTIMATION;233
22.5;REFERENCES;241
23;Chapter 16. On Confidence Bands in Nonparametric Density Estimation and Regression;243
23.1;1. INTRODUCTION;243
23.2;2. NONPARAMETRIC DENSITY ESTIMATION;244
23.3;3. NONPARAMETRIC REGRESSION;250
23.4;4. WIDTHS OF CONFIDENCE BANDS;255
23.5;5. ILLUSTRATIVE EXAMPLES;258
23.6;6. PROOFS;263
23.7;REFERENCES;269
24;Chapter 17. A Note on Generalized Gaussian Random Fields;270
24.1;0. INTRODUCTION;270
24.2;1. WHITE NOISE AND GAUSSIAN RANDOM FIELDS ON D;270
24.3;2. RESTRICTION OF PARAMETER;272
24.4;3. GAUSSIAN RANDOM FIELDS DEPENDING ON A CURVE;273
24.5;4. CONCLUDING REMARKS;274
24.6;REFERENCES;274
25;Chapter 18. Smoothness Properties of the Conditional Expectation in Finitely Additive White Noise Filtering;276
25.1;1. INTRODUCTION;276
25.2;2. NOTATION AND TERMINOLOGY;277
25.3;3. MAIN RESULTS;280
25.4;REFERENCES;284
26;Chapter 19. Equivariant Estimation of a Mean Vector µ of N(µ, .) with µ'.-1µ=1 or .-1/2µ=c or .=s2µ'µl;285
26.1;1. INTRODUCTION;285
26.2;2. PROBLEM WITH . KNOWN;287
26.3;3. PROBLEM WITH v KNOWN;292
26.4;4. THE CASE . = s2µ'µl;295
26.5;REFERENCES;297
27;Chapter 20. A Generalized Cauchy-Binet Formula and Applications to Total Positivity and Majorization;299
27.1;1. INTRODUCTION;299
27.2;2. A GENERALIZED CAUCHY-BINET FORMULA FOR THE SYMMETRIC GROUP;300
27.3;3. GENERALIZED TOTAL POSITIVITY;304
27.4;4. SEMIGROUP OF GENERALIZED TOTALLY POSITIVE KERNELS;308
27.5;5. COMPLEMENTS;310
27.6;REFERENCES;313
28;Chapter 21. Isotonic M-Estimation of Location: Union-Intersection Principle and Preliminary Test Versions;315
28.1;1. INTRODUCTION;315
28.2;2. M-ESTIMATORS OF LOCATION AND REGULARITY CONDITIONS;316
28.3;3. THE UI-PRELIMINARY M-TEST;318
28.4;4. ISOTONIC M-ESTIMATION OF LOCATION;323
28.5;5. THE PRELIMINARY TEST ISOTONIC M-ESTIMATOR (PTIME);324
28.6;6. SOME SIMULATION STUDIES;329
28.7;ACKNOWLEDGMENTS;333
28.8;REFERENCES;333
29;Chapter 22. Some Asymptotic Inferential Problems Connected with Elliptical Distributions;334
29.1;1. INTRODUCTION;334
29.2;2. ASYMPTOTIC DISTRIBUTION OF Z AND S;336
29.3;3. ASYMPTOTIC CONFIDENCE BOUNDS ON .;339
29.4;4. ASYMPTOTIC DISTRIBUTION OF CANONICAL CORRELATIONS;340
29.5;5. ASYMPTOTIC CONFIDENCE BOUNDS ON DISCRIMINATORY VALUES;342
29.6;REFERENCES;348
30;Chapter 23. Stochastic Integrals of Empirical-Type Processes with Applications to Censored Regression;349
30.1;1. INTRODUCTION;349
30.2;2. METRIC ENTROPY AND CONVERGENCE PROPERTIES OF EMPIRICAL-TYPE PROCESSES;352
30.3;3. STOCHASTIC INTEGRALS OF EMPIRICAL-TYPE PROCESSES;359
30.4;4. APPLICATIONS TO CENSORED RANK ESTIMATORS;363
30.5;5. APPLICATIONS TO THE BUCKLEY-JAMES ESTIMATOR;367
30.6;REFERENCES;372
31;Chapter 24. Nonminimum Phase Non-Gaussian Deconvolution;374
31.1;INTRODUCTION;374
31.2;COMPUTATION;378
31.3;CONCLUSIONS;388
31.4;ACKNOWLEDGMENTS;389
31.5;REFERENCES;389
32;Chapter 25. Inference in a Model with at Most One Slope-Change Point;390
32.1;1. INTRODUCTION;390
32.2;2. NORMAL ERROR WITH KNOWN VARIANCE;391
32.3;3. NORMAL ERROR WITH UNKNOWN VARIANCE;396
32.4;4. NONNORMAL ERROR;401
32.5;5. ESTIMATION OF THE SLOPE CHANGE ß1—ß2;403
32.6;REFERENCES;406
33;Chapter 26. Maximum Likelihood Principle and Model Selection when the True Model Is Unspecified;407
33.1;1. INTRODUCTION;407
33.2;2. OBSERVATIONS AND A FAMILY OF DENSITIES;408
33.3;3. MODEL SELECTION;413
33.4;4. DISCUSSION;417
33.5;ACKNOWLEDGMENTS;417
33.6;REFERENCES;418
34;Chapter 27. An Asymptotic Minimax Theorem of Order n–1/2;419
34.1;1. THE RESULTS;419
34.2;2. PROOF OF THE THEOREM;423
34.3;3. CONSTRUCTION OF THE ESTIMATOR-SEQUENCE;431
34.4;4. LEMMAS;434
34.5;ACKNOWLEDGMENTS;435
34.6;REFERENCES;435
35;Chapter 28. An Improved Estimation Method for Univariate Autoregressive Models;437
35.1;1. INTRODUCTION;437
35.2;2. SIMULATION RESULTS;439
35.3;3. THE YULE-WALKER AND BURG METHODS;440
35.4;4. IMPROVED ESTIMATION OF AUTOREGRESSIONS;443
35.5;5. CONCLUDING REMARKS;446
35.6;ACKNOWLEDGMENTS;447
35.7;REFERENCES;447
36;Chapter 29. Paradoxes in Conditional Probability;449
36.1;1. INTRODUCTION;449
36.2;2. THE FRAMEWORK;450
36.3;3. CONDITIONAL PROBABILITY AS AN INTEGRATOR;452
36.4;4. TWO TYPES OF PARADOXES;455
36.5;5. ANOTHER APPROACH AND COMPLEMENTS;459
36.6;ACKNOWLEDGMENTS;460
36.7;REFERENCES;461
37;Chapter 30. Inference Properties of a One-Parameter Curved Exponential Family of Distributions with Given Marginals;462
37.1;1. INTRODUCTION;462
37.2;2. ONE-PARAMETER SYSTEM;464
37.3;3. SOME STATISTICAL PROPERTIES;466
37.4;4. MAXIMUM LIKELIHOOD ESTIMATION OF .;469
37.5;REFERENCES;470
38;Chapter 31. Asymptotically Precise Estimate of the Accuracy of Gaussian Approximation in Hubert Space;472
38.1;1. INTRODUCTION;472
38.2;2. THE MAIN RESULT;473
38.3;3. AUXILIARY LEMMAS;473
38.4;REFERENCES;497
39;Chapter 32. The Estimation of the Bispectral Density Function and the Detection of Periodicities in a Signal;499
39.1;1. INTRODUCTION;499
39.2;2. SPECTRAL AND BISPECTRAL DENSITY FUNCTIONS;500
39.3;3. TRUNCATED BISPECTRUM;501
39.4;4. ESTIMATION OF THE TRUNCATED BISPECTRAL DENSITY FUNCTION hn(.1, .2);503
39.5;5. NUMERICAL ILLUSTRATIONS;504
39.6;6. THE RETRIEVAL OF HARMONICS VIA SPECTRUM AND BISPECTRUM;511
39.7;7. THE PERIODICITY OF THE EARTH'S MAGNETIC REVERSALS;516
39.8;ACKNOWLEDGMENT;519
39.9;REFERENCES;519
40;Chapter 33. Analysis of Odds Ratios in 2×n Ordinal Contingency Tables;520
40.1;1. INTRODUCTION;520
40.2;2. CONVEXITY PROPERTIES;522
40.3;3. AN APPLICATION;525
40.4;4. SOME GENERALIZATIONS;532
40.5;5. CONCLUDING REMARKS;534
40.6;ACKNOWLEDGMENT;534
40.7;REFERENCES;534
41;Chapter 34. Asymptotic Expansions of the Distributions of Some Test Statistics for Gaussian ARMA Processes;536
41.1;1. INTRODUCTION;536
41.2;2. PRELIMINARIES;537
41.3;3. ASYMPTOTIC EXPANSIONS FOR THE NULL DISTRIBUTIONS;540
41.4;4. BARTLETT'S ADJUSTMENT;545
41.5;5. ASYMPTOTIC EXPANSIONS FOR THE NONNULL DISTRIBUTIONS;546
41.6;6. POWER COMPARISONS BETWEEN THE TEST CRITERIA;551
41.7;REFERENCES;553
42;Chapter 35. Estimating Multiple Rater Agreement for a Rare Diagnosis;554
42.1;1. INTRODUCTION;554
42.2;2. MULTIPLICATIVE MODEL;560
42.3;3. MIXING MODELS;566
42.4;4. EXAMPLE;572
42.5;5. CONCLUDING REMARKS;575
42.6;APPENDIX: PROOF OF THEOREM 3.2;575
42.7;REFERENCES;576
43;Author Index;578
44;Subject Index;580