Recent Advances in Statistics - Papers in Honor of Herman Chernoff on His Sixtieth Birthday

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Recent Advances in Statistics: Papers in Honor of Herman Chernoff on His Sixtieth Birthday

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CONTRIBUTORS

PREFACE

HERMAN CHERNOFF: AN APPRECIATION

PUBLICATIONS OF HERMAN CHERNOFF

PART I: SEQUENTIAL ANALYSIS INCLUDING DESIGNS

CHAPTER 1. OPTIMAL STOPPING OF BROWNIAN MOTION: A COMPARISON TECHNIQUE

I. INTRODUCTION AND SUMMARY

II. STOPPING PROBLEMS FOR BROWNIAN MOTION

III. BANDIT PROBLEMS

IV. GITTINS INDICES FOR NORMAL PROCESSES

V. THE BUYER'S PROBLEM

VI. REFERENCES

CHAPTER 2.

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I. INTRODUCTION

II. THE CONTINUOUS-TIME PROBLEM AND A CLASS OF STOPPING RULES

III. ASYMPTOTIC OPTIMALITY OF THE CLASS C OF STOPPING BOUNDARIES

IV. SOME ASYMPTOTICALLY SUBOPTIMAL STOPPING RULES

V. REFERENCES

CHAPTER 3. THE CRAMÉR-RAO INEQUALITY HOLDS ALMOST EVERYWHERE

I. INTRODUCTION

II. THE FIXED SAMPLE SIZE CASE

III. THE SEQUENTIAL CASE

IV. REMARKS

V. ACKNOWLEDGMENTS

VI. APPENDIX A

VII. APPENDIX B

VIII. REFERENCES

CHAPTER 4. A TWO-SAMPLE SEQUENTIAL TEST FOR SHIFT WITH ONE SAMPLE SIZE FIXED IN ADVANCE

I. INTRODUCTION

II. PROBLEM SPECIFICATION AND THE SUGGESTED PROCEDURE

III. CALCULATION OF ERROR PROBABILITIES

IV. AVERAGE NEW DATA SAMPLE-SIZE

V. COMPARISON WITH A FIXED NUMBER OF NEW DATA OBSERVATIONS

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

VII. APPENDIX

VIII. REFERENCES

CHAPTER 5. ON SEQUENTIAL RANK TESTS

I. INTRODUCTION

II. PRELIMINARIES

III. THE MAIN THEOREM

IV. A SEQUENTIAL PROBABILITY RATIO TEST

V. REFERENCES

PART II: OPTIMIZATION INCLUDING CONTROL THEORY

CHAPTER 6. A NON-HOMOGENEOUS MARKOV MODEL OF A CHAIN-LETTER SCHEME

I. INTRODUCTION

II. A MARKOV MODEL OF THE RECRUITMENT PROCESS AND ITS DIFFUSION APPROXIMATION

III. TERMINATION OF THE RECRUITMENT PROCESS

IV. A BOUND FOR THE SIZE OF THE SUB-CHAIN STARTED BY THE kth PARTICIPANT

V. MONTE CARLO SIMULATION OF THE PROCESS

VI. ACKNOWLEDGMENT

REFERENCES

CHAPTER 7. SET-VALUED PARAMETERS AND SET-VALUED STATISTICS

I. INTRODUCTION

II. PRELIMINARIES

III. SUFFICIENCY

IV. MAXIMUM LIKELIHOOD ESTIMATORS

V. BAYESIAN METHODS

VI. EXAMPLES

REFERENCES

CHAPTER 8. OPTIMAL SEQUENTIAL DECISIONS IN PROBLEMS INVOLVING MORE THAN ONE DECISION MAKER

I. INTRODUCTION

II. A SIMPLE GAME

III. THE STOCHASTIC GAME

IV. A MORE GENERAL GAME

V. DISCUSSION

VI. REFERENCES

CHAPTER 9. AN AVERAGING METHOD FOR STOCHASTIC APPROXIMATIONS WITH DISCONTINUOUS DYNAMICS, CONSTRAINTS, AND STATE DEPENDENT NOISE

I. INTRODUCTION

III. THE PROJECTION METHOD

IV. STATE DEPENDENT NOISE

V. EXAMPLES

REFERENCES

CHAPTER 10. ENLIGHTENED APPROXIMATIONS IN OPTIMAL STOPPING

I. CHARACTERISATIONS

II. THE MAXIMISATION OF A SAMPLE MEAN

III. EVALUATION OF THE GITTINS INDEX

IV. REFERENCES

CHAPTER 11. SURVEY OF CLASSICAL AND BAYESIAN APPROACHES TO THE CHANGE-POINT PROBLEM: FIXED SAMPLE AND SEQUENTIAL PROCEDURES OF TESTING AND ESTIMATION

I. INTRODUCTION

II. TESTING HYPOTHESES CONCERNING CHANGE POINTS

III. ESTIMATING THE LOCATION OF THE SHIFT POINT

IV. DYNAMIC CONTROL PROCEDURES

REFERENCES

PART III: NONPARAMETRICS INCLUDING LARGE SAMPLE THEORY

CHAPTER 12. LARGE DEVIATIONS OF THE MAXIMUM LIKELIHOOD ESTIMATE IN THE MARKOV CHAIN CASE

I. INTRODUCTION

II. PROOF

REFERENCES

CHAPTER 13. BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS

I. INTRODUCTION

II. THE PRIOR

III. CONNECTION WITH THE DIRICHLET PROCESS

IV. COMPUTATIONS

V. IMPROVING THE MONTE CARLO

VI. SUMMARY

REFERENCES

CHAPTER 14. AN EXTENSION OF A THEOREM OF H. CHERNOFF AND E. L. LEHMANN

I. INTRODUCTION

II. TEST CRITERIA FOR THE GAUSSIAN CASE

III. GAUSSIAN THEORY UNDER THE NULL HYPOTHESIS

IV. ASYMPTOTICS UNDER THE NULL HYPOTHESIS

V. ASYMPTOTIC BEHAVIOR UNDER ALTERNATIVES

VI. REFERENCES

CHAPTER 15. THE LIMITING BEHAVIOR OF MULTIPLE ROOTS OF THE LIKELIHOOD EQUATION

I. INTRODUCTION

II. NOTATION AND PRELIMINARY LEMMAS

III. GLOBAL LIMITING BEHAVIOR OF Sn

IV. LOCAL LIMITING BEHAVIOR OF Sn

V. SOME EXAMPLES OF K-L ORDERED FAMILIES

VI. TWO SPECIAL FAMILIES

VII. IMPLICATIONS FOR THE ESTIMATION OF .0

VIII. CONCLUDING REMARKS

REFERENCES

CHAPTER 16. ON SOME RECURSIVE RESIDUAL RANK TESTS FOR CHANGE-POINTS

I. INTRODUCTION

II. RECURSIVE RESIDUAL RANK TESTS FOR THE LOCATION MODEL

III. RECURSIVE RESIDUAL RANK TESTS FOR THE REGRESSION MODEL

IV. ASYMPTOTIC PROPERTIES OF Dn+ AND Dn

V. ASYMPTOTIC RELATIVE EFFICIENCY RESULTS

VI. APPENDIX

VII. ACKNOWLEDGMENT

VIII. REFERENCES

CHAPTER 17. OPTIMAL UNIFORM RATE OF CONVERGENCE FOR NONPARAMETRIC ESTIMATORS OF A DENSITY FUNCTION OR ITS DERIVATIVES

I. INTRODUCTION

II. PROOF OF THEOREM 2

REFERENCES

CHAPTER 18. RANKS AND ORDER STATISTICS

I. INTRODUCTION

II. THE RESULT

III. PROOF

IV. APPENDIX

V. REFERENCES

PART IV: STATISTICAL GRAPHICS

CHAPTER 19. M AND N PLOTS

I. INTRODUCTION

II. EXAMPLES OF M AND N PLOTS

III. SOME PRACTICAL DETAILS

IV. ACKNOWLEDGMENT

V. REFERENCES

CHAPTER 20. INVESTIGATING THE SPACE OF CHERNOFF FACES

I. AN EMPIRICAL METHOD FOR EVALUATING REGIONS OF THE SPACE OF FACE PARAMETER VECTORS

II. AN EMPIRICAL METHOD FOR ASSIGNING COORDINATES TO FACIAL FEATURES

III. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

CHAPTER 21. ON MULTIVARIATE DISPLAY

Tables

Inside-Out Plots

Function Plots

Polygons

Trees

Faces

Extensions

Conclusions

Seeing vs. Reading a Display

An Appreciation

Acknowledgments

References

PART V: OTHER TOPICS

CHAPTER 22. MINIMAX ESTIMATION OF THE MEAN OF A NORMAL DISTRIBUTION SUBJECT TO DOING WELL AT A POINT

SUMMARY

I. THE PROBLEM

II. OPTIMAL AND SUBOPTIMAL PROCEDURES AND THE CONNECTION TO ROBUST ESTIMATION OF LOCATION

III. THE BEHAVIOUR OF µ (t) FOR SMALL t

IV. THE BEHAVIOUR OF µ (t) FOR t CLOSE TO 1

V. ACKNOWLEDGMENT

VI. REFERENCES

CHAPTER 23. SOME NEW DICHOTOMOUS REGRESSION METHODS

I. INTRODUCTION

II. MODELS FOR PREDICTING THE OUTCOME OF INTENSIVE CARE

III. ANALOGIES TO NORMAL REGRESSION AND R2

IV. A SCREENING PROCEDURE FOR GOODNESS OF FIT

V. EXAMINATION OF THE LIKELIHOOD SURFACE

VI. DISCUSSION AND CONCLUSION

ACKNOWLEDGMENT

REFERENCES

CHAPTER 24. THE APPLICATION OF SPLINE FUNCTIONS TO THE PHARMACOKINETIC ANALYSIS OF METHOTREXATE INFUSED INTO MALIGNANT EFFUSIONS

I. INTRODUCTION

II. ANALYSIS AND RESULTS

III. DISCUSSION

IV. ACKNOWLEDGMENTS

V. REFERENCES

CHAPTER 25.

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REFERENCES

CHAPTER 26. LEAST INFORMATIVE DISTRIBUTIONS

I. INTRODUCTION

II. THE NORMAL DISTRIBUTION

III. THE CASE n = 2

IV. EXPONENTIAL DISTRIBUTION

V. A MINIMAX FORMULATION

VI. REFERENCES

CHAPTER 27. SIGNIFICANCE LEVELS, CONFIDENCE LEVELS, CLOSED REGIONS, CLOSED MODELS, AND DISCRETE DISTRIBUTIONS

I. INTRODUCTION

II. TESTS

III. CONFIDENCE PROCEDURES

IV. REMARKS

REFERENCES