Rizvi / Rustagi / Siegmund Recent Advances in Statistics

Papers in Honor of Herman Chernoff on His Sixtieth Birthday
1. Auflage 2014
ISBN: 978-1-4832-6660-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 626 Seiten, Web PDF

Recent Advances in Statistics: Papers in Honor of Herman Chernoff on His Sixtieth Birthday is a collection of papers on statistics in honor of Herman Chernoff on the occasion of his 60th birthday. Topics covered range from sequential analysis (including designs) to optimization (including control theory), nonparametrics (including large sample theory), and statistical graphics. Comprised of 27 chapters, this book begins with a discussion on optimal stopping of Brownian motion, followed by an analysis of sequential design of comparative clinical trials. A two-sample sequential test for shift with one sample size fixed in advance is then presented. Subsequent chapters focus on set-valued parameters and set-valued statistics; large deviations of the maximum likelihood estimate in the Markov chain case; the limiting behavior of multiple roots of the likelihood equation; and optimal uniform rate of convergence for nonparametric estimators of a density function and its derivatives. The book concludes by considering significance and confidence levels, closed regions and models, and discrete distributions. This monograph should be of interest to students, researchers, and specialists in the fields of mathematics and statistics.

Autoren/Hrsg.

Rizvi, M. Haseeb

Rustagi, Jagdish S.

Siegmund, David

Inhaltsverzeichnis

1;Front Cover;1
2;Recent Advances in Statistics: Papers in Honor of Herman Chernoff on His Sixtieth Birthday;4
3;Copyright Page;5
4;Table of Contents;6
5;CONTRIBUTORS;10
6;PREFACE;14
7;HERMAN CHERNOFF: AN APPRECIATION;16
8;PUBLICATIONS OF HERMAN CHERNOFF;23
9;PART I: SEQUENTIAL ANALYSIS INCLUDING DESIGNS;32
9.1;CHAPTER 1. OPTIMAL STOPPING OF BROWNIAN MOTION: A COMPARISON TECHNIQUE;34
9.1.1;I. INTRODUCTION AND SUMMARY;34
9.1.2;II. STOPPING PROBLEMS FOR BROWNIAN MOTION;36
9.1.3;III. BANDIT PROBLEMS;43
9.1.4;IV. GITTINS INDICES FOR NORMAL PROCESSES;47
9.1.5;V. THE BUYER'S PROBLEM;56
9.1.6;VI. REFERENCES;63
9.2;CHAPTER 2.
SEQUENTIAL DESIGN OF COMPARATIVE CLINICAL TRIALS;66
9.2.1;I. INTRODUCTION;66
9.2.2;II. THE CONTINUOUS-TIME PROBLEM AND A CLASS OF STOPPING RULES;69
9.2.3;III. ASYMPTOTIC OPTIMALITY OF THE CLASS C OF STOPPING BOUNDARIES;74
9.2.4;IV. SOME ASYMPTOTICALLY SUBOPTIMAL STOPPING RULES;78
9.2.5;V. REFERENCES;82
9.3;CHAPTER 3. THE CRAMÉR-RAO INEQUALITY HOLDS ALMOST EVERYWHERE;84
9.3.1;I. INTRODUCTION;84
9.3.2;II. THE FIXED SAMPLE SIZE CASE;86
9.3.3;III. THE SEQUENTIAL CASE;93
9.3.4;IV. REMARKS;99
9.3.5;V. ACKNOWLEDGMENTS;101
9.3.6;VI. APPENDIX A;101
9.3.7;VII. APPENDIX B;103
9.3.8;VIII. REFERENCES;107
9.4;CHAPTER 4. A TWO-SAMPLE SEQUENTIAL TEST FOR SHIFT WITH ONE SAMPLE SIZE FIXED IN ADVANCE;110
9.4.1;I. INTRODUCTION;110
9.4.2;II. PROBLEM SPECIFICATION AND THE SUGGESTED PROCEDURE;112
9.4.3;III. CALCULATION OF ERROR PROBABILITIES;113
9.4.4;IV. AVERAGE NEW DATA SAMPLE-SIZE;119
9.4.5;V. COMPARISON WITH A FIXED NUMBER OF NEW DATA OBSERVATIONS;123
9.4.6;VI. OPTIMIZATION OF THE SEQUENTIAL PROCEDURE WHEN THE "OLD" DATA IS STILL TO BE COLLECTED; COMPARISON WITH BEST FIXED SAMPLE-SIZE PROCEDURES AND PROCEDURES FOR SAMPLING IN PAIRS;125
9.4.7;VII. APPENDIX;127
9.4.8;VIII. REFERENCES;128
9.5;CHAPTER 5. ON SEQUENTIAL RANK TESTS;130
9.5.1;I. INTRODUCTION;130
9.5.2;II. PRELIMINARIES;132
9.5.3;III. THE MAIN THEOREM;134
9.5.4;IV. A SEQUENTIAL PROBABILITY RATIO TEST;150
9.5.5;V. REFERENCES;154
10;PART II: OPTIMIZATION INCLUDING CONTROL THEORY;156
10.1;CHAPTER 6. A NON-HOMOGENEOUS MARKOV MODEL OF A CHAIN-LETTER SCHEME;158
10.1.1;I. INTRODUCTION;158
10.1.2;II. A MARKOV MODEL OF THE RECRUITMENT PROCESS AND ITS DIFFUSION APPROXIMATION;161
10.1.3;III. TERMINATION OF THE RECRUITMENT PROCESS;167
10.1.4;IV. A BOUND FOR THE SIZE OF THE SUB-CHAIN STARTED BY THE kth PARTICIPANT;170
10.1.5;V. MONTE CARLO SIMULATION OF THE PROCESS;174
10.1.6;VI. ACKNOWLEDGMENT;178
10.1.7;REFERENCES;188
10.2;CHAPTER 7. SET-VALUED PARAMETERS AND SET-VALUED STATISTICS;190
10.2.1;I. INTRODUCTION;190
10.2.2;II. PRELIMINARIES;192
10.2.3;III. SUFFICIENCY;198
10.2.4;IV. MAXIMUM LIKELIHOOD ESTIMATORS;200
10.2.5;V. BAYESIAN METHODS;204
10.2.6;VI. EXAMPLES;206
10.2.7;REFERENCES;210
10.3;CHAPTER 8. OPTIMAL SEQUENTIAL DECISIONS IN PROBLEMS INVOLVING MORE THAN ONE DECISION MAKER;212
10.3.1;I. INTRODUCTION;212
10.3.2;II. A SIMPLE GAME;213
10.3.3;III. THE STOCHASTIC GAME;217
10.3.4;IV. A MORE GENERAL GAME;222
10.3.5;V. DISCUSSION;224
10.3.6;VI. REFERENCES;225
10.4;CHAPTER 9. AN AVERAGING METHOD FOR STOCHASTIC APPROXIMATIONS WITH DISCONTINUOUS DYNAMICS, CONSTRAINTS, AND STATE DEPENDENT NOISE;226
10.4.1;I. INTRODUCTION;226
10.4.2;III. THE PROJECTION METHOD;236
10.4.3;IV. STATE DEPENDENT NOISE;240
10.4.4;V. EXAMPLES;243
10.4.5;REFERENCES;249
10.5;CHAPTER 10. ENLIGHTENED APPROXIMATIONS IN OPTIMAL STOPPING;252
10.5.1;I. CHARACTERISATIONS;252
10.5.2;II. THE MAXIMISATION OF A SAMPLE MEAN;255
10.5.3;III. EVALUATION OF THE GITTINS INDEX;257
10.5.4;IV. REFERENCES;258
10.6;CHAPTER 11. SURVEY OF CLASSICAL AND BAYESIAN APPROACHES TO THE CHANGE-POINT PROBLEM: FIXED SAMPLE AND SEQUENTIAL PROCEDURES OF TESTING AND ESTIMATION;260
10.6.1;I. INTRODUCTION;260
10.6.2;II. TESTING HYPOTHESES CONCERNING CHANGE POINTS;262
10.6.3;III. ESTIMATING THE LOCATION OF THE SHIFT POINT;269
10.6.4;IV. DYNAMIC CONTROL PROCEDURES;273
10.6.5;REFERENCES;280
11;PART III: NONPARAMETRICS INCLUDING LARGE SAMPLE THEORY;286
11.1;CHAPTER 12. LARGE DEVIATIONS OF THE MAXIMUM LIKELIHOOD ESTIMATE IN THE MARKOV CHAIN CASE;288
11.1.1;I. INTRODUCTION;288
11.1.2;II. PROOF;292
11.1.3;REFERENCES;300
11.2;CHAPTER 13. BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS;302
11.2.1;I. INTRODUCTION;302
11.2.2;II. THE PRIOR;304
11.2.3;III. CONNECTION WITH THE DIRICHLET PROCESS;307
11.2.4;IV. COMPUTATIONS;310
11.2.5;V. IMPROVING THE MONTE CARLO;314
11.2.6;VI. SUMMARY;316
11.2.7;REFERENCES;316
11.3;CHAPTER 14. AN EXTENSION OF A THEOREM OF H. CHERNOFF AND E. L. LEHMANN;318
11.3.1;I. INTRODUCTION;318
11.3.2;II. TEST CRITERIA FOR THE GAUSSIAN CASE;321
11.3.3;III. GAUSSIAN THEORY UNDER THE NULL HYPOTHESIS;326
11.3.4;IV. ASYMPTOTICS UNDER THE NULL HYPOTHESIS;330
11.3.5;V. ASYMPTOTIC BEHAVIOR UNDER ALTERNATIVES;344
11.3.6;VI. REFERENCES;351
11.4;CHAPTER 15. THE LIMITING BEHAVIOR OF MULTIPLE ROOTS OF THE LIKELIHOOD EQUATION;354
11.4.1;I. INTRODUCTION;354
11.4.2;II. NOTATION AND PRELIMINARY LEMMAS;357
11.4.3;III. GLOBAL LIMITING BEHAVIOR OF Sn;361
11.4.4;IV. LOCAL LIMITING BEHAVIOR OF Sn;364
11.4.5;V. SOME EXAMPLES OF K-L ORDERED FAMILIES;368
11.4.6;VI. TWO SPECIAL FAMILIES;372
11.4.7;VII. IMPLICATIONS FOR THE ESTIMATION OF .0;376
11.4.8;VIII. CONCLUDING REMARKS;379
11.4.9;REFERENCES;384
11.5;CHAPTER 16. ON SOME RECURSIVE RESIDUAL RANK TESTS FOR CHANGE-POINTS;386
11.5.1;I. INTRODUCTION;386
11.5.2;II. RECURSIVE RESIDUAL RANK TESTS FOR THE LOCATION MODEL;388
11.5.3;III. RECURSIVE RESIDUAL RANK TESTS FOR THE REGRESSION MODEL;391
11.5.4;IV. ASYMPTOTIC PROPERTIES OF Dn+ AND Dn;394
11.5.5;V. ASYMPTOTIC RELATIVE EFFICIENCY RESULTS;403
11.5.6;VI. APPENDIX;404
11.5.7;VII. ACKNOWLEDGMENT;405
11.5.8;VIII. REFERENCES;405
11.6;CHAPTER 17. OPTIMAL UNIFORM RATE OF CONVERGENCE FOR NONPARAMETRIC ESTIMATORS OF A DENSITY FUNCTION OR ITS DERIVATIVES;408
11.6.1;I. INTRODUCTION;408
11.6.2;II. PROOF OF THEOREM 2;411
11.6.3;REFERENCES;421
11.7;CHAPTER 18. RANKS AND ORDER STATISTICS;422
11.7.1;I. INTRODUCTION;422
11.7.2;II. THE RESULT;425
11.7.3;III. PROOF;427
11.7.4;IV. APPENDIX;435
11.7.5;V. REFERENCES;437
12;PART IV: STATISTICAL GRAPHICS;438
12.1;CHAPTER 19. M AND N PLOTS;440
12.1.1;I. INTRODUCTION;440
12.1.2;II. EXAMPLES OF M AND N PLOTS;445
12.1.3;III. SOME PRACTICAL DETAILS;454
12.1.4;IV. ACKNOWLEDGMENT;462
12.1.5;V. REFERENCES;462
12.2;CHAPTER 20. INVESTIGATING THE SPACE OF CHERNOFF FACES;464
12.2.1;I. AN EMPIRICAL METHOD FOR EVALUATING REGIONS OF THE SPACE OF FACE PARAMETER VECTORS;467
12.2.2;II. AN EMPIRICAL METHOD FOR ASSIGNING COORDINATES TO FACIAL FEATURES;474
12.2.3;III. CONCLUSIONS;481
12.2.4;ACKNOWLEDGMENTS;481
12.2.5;REFERENCES;482
12.3;CHAPTER 21. ON MULTIVARIATE DISPLAY;484
12.3.1;Tables;489
12.3.2;Inside-Out Plots;494
12.3.3;Function Plots;502
12.3.4;Polygons;504
12.3.5;Trees;506
12.3.6;Faces;510
12.3.7;Extensions;514
12.3.8;Conclusions;517
12.3.9;Seeing vs. Reading a Display;520
12.3.10;An Appreciation;522
12.3.11;Acknowledgments;522
12.3.12;References;522
13;PART V: OTHER TOPICS;524
13.1;CHAPTER 22. MINIMAX ESTIMATION OF THE MEAN OF A NORMAL DISTRIBUTION SUBJECT TO DOING WELL AT A POINT;526
13.1.1;SUMMARY;526
13.1.2;I. THE PROBLEM;527
13.1.3;II. OPTIMAL AND SUBOPTIMAL PROCEDURES AND THE CONNECTION TO ROBUST ESTIMATION OF LOCATION;528
13.1.4;III. THE BEHAVIOUR OF µ (t) FOR SMALL t;535
13.1.5;IV. THE BEHAVIOUR OF µ (t) FOR t CLOSE TO 1;540
13.1.6;V. ACKNOWLEDGMENT;542
13.1.7;VI. REFERENCES;542
13.2;CHAPTER 23. SOME NEW DICHOTOMOUS REGRESSION METHODS;544
13.2.1;I. INTRODUCTION;544
13.2.2;II. MODELS FOR PREDICTING THE OUTCOME OF INTENSIVE CARE;545
13.2.3;III. ANALOGIES TO NORMAL REGRESSION AND R2;551
13.2.4;IV. A SCREENING PROCEDURE FOR GOODNESS OF FIT;555
13.2.5;V. EXAMINATION OF THE LIKELIHOOD SURFACE;561
13.2.6;VI. DISCUSSION AND CONCLUSION;565
13.2.7;ACKNOWLEDGMENT;568
13.2.8;REFERENCES;570
13.3;CHAPTER 24. THE APPLICATION OF SPLINE FUNCTIONS TO THE PHARMACOKINETIC ANALYSIS OF METHOTREXATE INFUSED INTO MALIGNANT EFFUSIONS;572
13.3.1;I. INTRODUCTION;572
13.3.2;II. ANALYSIS AND RESULTS;575
13.3.3;III. DISCUSSION;591
13.3.4;IV. ACKNOWLEDGMENTS;591
13.3.5;V. REFERENCES;592
13.4;CHAPTER 25.
SELECTING REPRESENTATIVE POINTS IN NORMAL POPULATIONS;594
13.4.1;REFERENCES;606
13.5;CHAPTER 26. LEAST INFORMATIVE DISTRIBUTIONS;608
13.5.1;I. INTRODUCTION;608
13.5.2;II. THE NORMAL DISTRIBUTION;609
13.5.3;III. THE CASE n = 2;610
13.5.4;IV. EXPONENTIAL DISTRIBUTION;612
13.5.5;V. A MINIMAX FORMULATION;613
13.5.6;VI. REFERENCES;614
13.6;CHAPTER 27. SIGNIFICANCE LEVELS, CONFIDENCE LEVELS, CLOSED REGIONS, CLOSED MODELS, AND DISCRETE DISTRIBUTIONS;616
13.6.1;I. INTRODUCTION;616
13.6.2;II. TESTS;618
13.6.3;III. CONFIDENCE PROCEDURES;619
13.6.4;IV. REMARKS;622
13.6.5;REFERENCES;626