PROBABILITY AND STATISTICAL MODELS WITH APPLICATIONS
CONTENTS PART I APPROXIMATION, BOUNDS, AND INEQUALITIES 1 NONUNIFORM BOUNDS IN PROBABILITY APPROXIMATIONS USING STEIN’S METHOD 1 1 INTRODUCTION 1 2 POISSON APPROXIMATION 1 3 BINOMIAL APPROXIMATION: BINARY EXPANSION OF A RANDOM INTEGER 1 4 NORMAL APPROXIMATION 1 5 CONCLUSION REFERENCES 2 PROBABILITY INEQUALITIES FOR MULTIVARIATE DISTRIBUTIONS WITH APPLICATIONS TO STATISTICS 2 1 INTRODUCTION AND SUMMARY 2 2 POSITIVE DEPENDENCE AND PRODUCT-TYPE INEQUALITIES 2 3 NEGATIVE DEPENDENCE AND PRODUCT-TYPE INEQUALITIES 2 4 BONFERRONI-TYPE INEQUALITIES 2 5 APPLICATIONS 2 5 1 SEQUENTIAL ANALYSIS 2 5 2 A DISCRETE SCAN STATISTIC 2 5 3 AN APPROXIMATION FOR A MULTINOMIAL DISTRIBUTION 2 5 4 A CONDITIONAL DISCRETE SCAN STATISTIC 2 5 5 SIMULTANEOUS PREDICTION IN TIME SERIES MODELS REFERENCES 3 APPLICATIONS OF COMPOUND POISSON APPROXIMATION 3 1 INTRODUCTION 3 2 FIRST APPLICATIONS 3 2 1 RUNS 3 2 2 SEQUENCE MATCHING 3 3 WORD COUNTS 3 4 DISCUSSION AND NUMERICAL EXAMPLES REFERENCES 4 COMPOUND POISSON APPROXIMATION FOR SUMS OF DEPENDENT RANDOM VARIABLES 4 1 INTRODUCTION 4 2 PRELIMINARIES AND NOTATIONS 4 3 MAIN RESULTS 4 4 EXAMPLES OF APPLICATIONS 4 4 1 A COMPOUND POISSON APPROXIMATION FOR TRUNCATED MOVING SUM OF I I D R V S 4 4 2 THE NUMBER OF OVERLAPPING SUCCESS RUNS IN A STATIONARY TWO-STATE MARKOV CHAIN REFERENCES 5 UNI?ED VARIANCE BOUNDS AND A STEIN-TYPE IDENTITY 5 1 INTRODUCTION 5 2 PROPERTIES OF THE TRANSFORMATION 5 3 APPLICATION TO VARIANCE BOUNDS REFERENCES 6 PROBABILITY INEQUALITIES FOR U-STATISTICS 6 1 INTRODUCTION 6 2 PRELIMINARIES 6 3 PROBABILITY INEQUALITIES REFERENCES II PROBABILITY AND STOCHASTIC PROCESSES 7 THEORY AND APPLICATIONS OF DECOUPLING ˜ 7 1 COMPLETE DECOUPLING OF MARGINAL LAWS AND ONE-SIDED BOUNDS 7 2 TANGENT SEQUENCES AND CONDITIONALLY INDEPENDENT VARIABLES 7 3 BASIC DECOUPLING INEQUALITIES FOR TANGENT SEQUENCES 7 4 APPLICATIONS TO MARTINGALE INEQUALITIES AND EXPONENTIAL TAIL PROBABILITY BOUNDS 7 5 DECOUPLING OF MULTILINEAR FORMS, U-STATISTICS AND U-PROCESSES 7 6 TOTAL DECOUPLING OF STOPPING TIMES 7 7 PRINCIPLE OF CONDITIONING IN WEAK CONVERGENCE 7 8 CONCLUSION REFERENCES 8 A NOTE ON THE PROBABILITY OF RAPID EXTINCTION OF ALLELES IN A WRIGHT-FISHER PROCESS 8 1 INTRODUCTION 8 2 THE LOWER BOUND FOR BOUNDARY SETS REFERENCES 9 STOCHASTIC INTEGRAL FUNCTIONALS IN AN ASYMPTOTIC SPLIT STATE SPACE 9 1 INTRODUCTION 9 2 PRELIMINARIES 9 3 PHASE MERGING SCHEME FOR MARKOV JUMP PROCESSES 9 4 AVERAGE OF STOCHASTIC INTEGRAL FUNCTIONAL 9 5 DI?USION APPROXIMATION OF STOCHASTIC INTEGRAL FUNCTIONAL 9 5 1 SINGLE SPLITTING STATE SPACE 9 5 2 DOUBLE SPLIT STATE SPACE 9 6 INTEGRAL FUNCTIONAL WITH PERTURBED KERNEL REFERENCES 10 BUSY PERIODS FOR SOME QUEUES WITH DETERMINISTIC INTERARRIVAL OR SERVICE TIMES ` 10 1 INTRODUCTION 10 2 PRELIMINARIES: A BASIC CLASS OF POLYNOMIALS 10 2 1 CONSTRUCTION OF THE BASIC POLYNOMIALS 10 2 2 A GENERALIZED APPELL STRUCTURE 10 3 THE DG/M (Q)/1 QUEUE 10 3 1 MODEL AND NOTATION 10 3 2 EXACT DISTRIBUTION OF NR 10 4 THE M (Q)/DG/1 QUEUE 10 4 1 MODEL AND NOTATION 10 4 2 EXACT DISTRIBUTION OF NR REFERENCES 11 THE EVOLUTION OF POPULATION STRUCTURE OF THE PERTURBED NON-HOMOGENEOUS SEMI-MARKOV SYSTEM 11 1 INTRODUCTION 11 2 THE PERTURBED NON-HOMOGENEOUS SEMI-MARKOV SYSTEM 11 3 THE EXPECTED POPULATION STRUCTURE WITH RESPECT TO THE FIRST PASSAGE TIME PROBABILITIES 11 4 THE EXPECTED POPULATION STRUCTURE WITH RESPECT TO THE DURATION OF A MEMBERSHIP IN A STATE 11 5 THE EXPECTED POPULATION STRUCTURE WITH RESPECT TO THE STATE OCCUPANCY OF A MEMBERSHIP 11 6 THE EXPECTED POPULATION STRUCTURE WITH RESPECT TO THE COUNTING TRANSITION PROBABILITIES REFERENCES III DISTRIBUTIONS, CHARACTERIZATIONS, AND APPLICATIONS 12 CHARACTERIZATIONS OF SOME EXPONENTIAL FAMILIES BASED ON SURVIVAL DISTRIBUTIONS AND MOMENTS 12 1 INTRODUCTION 12 2 AN AUXILIARY LEMMA 12 3 CHARACTERIZATIONS BASED ON SURVIVAL DISTRIBUTIONS 12 4 CHARACTERIZATIONS BASED ON MOMENTS REFERENCES 13 BIVARIATE DISTRIBUTIONS COMPATIBLE OR NEARLY COMPATIBLE WITH GIVEN CONDITIONAL INFORMATION 13 1 INTRODUCTION 13 2 IMPRECISE SPECI?CATION 13 3 PRECISE SPECI?CATION 13 4 AN EXAMPLE REFERENCES 14 A CHARACTERIZATION OF A DISTRIBUTION ARISING FROM ABSORPTION SAMPLING 14 1 INTRODUCTION 14 2 THE CHARACTERIZATION THEOREM 14 3 AN APPLICATION REFERENCES 15 RE?NEMENTS OF INEQUALITIES FOR SYMMETRIC FUNCTIONS REFERENCES 16 GENERAL OCCUPANCY DISTRIBUTIONS 16 1 INTRODUCTION 16 2 A GENERAL RANDOM OCCUPANCY MODEL 16 3 SPECIAL OCCUPANCY DISTRIBUTIONS 16 3 1 GEOMETRIC PROBABILITIES 16 3 2 BERNOULLI PROBABILITIES REFERENCES 17 A SKEW T DISTRIBUTION 17 1 INTRODUCTION 17 2 DERIVATION OF SKEW T DENSITY 17 3 PROPERTIES OF SKEW T DISTRIBUTION 17 4 A FIRST BIVARIATE SKEW T DISTRIBUTION 17 5 A SECOND BIVARIATE SKEW T DISTRIBUTION REFERENCES 18 ON THE POSTERIOR MOMENTS FOR TRUNCATION PARAMETER DISTRIBUTIONS AND IDENTI?ABILITY BY POSTERIOR MEAN FOR EXPONENTIAL DISTRIBUTION WITH LOCATION PARAMETERS 18 1 INTRODUCTION 18 2 POSTERIOR MOMENTS 18 3 EXAMPLES 18 4 IDENTI?ABILITY BY POSTERIOR MEAN 18 5 AN ILLUSTRATIVE EXAMPLE REFERENCES 19 DISTRIBUTIONS OF RANDOM VOLUMES WITHOUT USING INTEGRAL GEOMETRY TECHNIQUES 19 1 INTRODUCTION 19 2 EVALUATION OF ARBITRARY MOMENTS OF THE RANDOM VOLUMES 19 2 1 MATRIX-VARIATE DISTRIBUTIONS FOR X 19 2 2 TYPE-1 BETA DISTRIBUTION FOR X 19 2 3 THE CASE WHEN THE ROWS OF X ARE INDEPENDENTLY DISTRIBUTED 19 2 4 TYPE-1 BETA DISTRIBUTED INDEPENDENT ROWS OF X 19 2 5 TYPE-2 BETA DISTRIBUTED INDEPENDENT ROWS OF X 19 2 6 INDEPENDENTLY GAUSSIAN DISTRIBUTED POINTS 19 2 7 DISTRIBUTIONS OF THE R-CONTENTS REFERENCES IV TIME SERIES, LINEAR, AND NON-LINEAR MODELS 20 COINTEGRATION OF ECONOMIC TIME SERIES 20 1 INTRODUCTION 20 2 REGRESSION MODELS 20 3 SIMULTANEOUS EQUATION MODELS 20 4 CANONICAL ANALYSIS AND THE REDUCED RANK REGRESSION ESTIMATOR 20 5 AUTOREGRESSIVE PROCESSES 20 6 NONSTATIONARY MODELS 20 7 COINTEGRATED MODELS 20 8 ASYMPTOTIC DISTRIBUTION OF ESTIMATORS AND TEST CRITERION REFERENCES 21 ON SOME POWER PROPERTIES OF GOODNESS-OF-FIT TESTS IN TIME SERIES ANALYSIS 21 1 TESTING SPECTRAL DENSITY FITS 21 2 LOCAL POWER CONSIDERATIONS 21 3 COMPARISON REFERENCES 22 LINEAR CONSTRAINTS ON A LINEAR MODEL 22 1 INTRODUCTION 22 2 GEOMETRIC INTERPRETATION OF THE ROLE OF THE LINEAR CONSTRAINTS REFERENCES 23 M-METHODS IN GENERALIZED NONLINEAR MODELS 23 1 INTRODUCTION 23 2 DE?NITIONS AND ASSUMPTIONS 23 3 ASYMPTOTIC RESULTS 23 4 TEST OF SIGNI?CANCE AND COMPUTATIONAL ALGORITHM 23 4 1 SUBHYPOTHESIS TESTING 23 4 2 NONLINEAR HYPOTHESIS TESTING 23 4 3 COMPUTATIONAL ALGORITHM REFERENCES V INFERENCE AND APPLICATIONS 24 EXTENSIONS OF A VARIATION OF THE ISOPERIMETRIC PROBLEM 24 1 INTRODUCTION 24 2 INFORMATION RETRIEVAL PROBLEM 24 3 INFORMATION RETRIEVAL WITHOUT MEASUREMENT ERROR 24 4 USEFUL INFORMATION IN A VARIABLE 24 5 ALLOCATION OF STORAGE SPACE 24 6 THE ISOPERIMETRIC PROBLEM 24 7 EXTENSIONS REFERENCES 25 ON FINDING A SINGLE POSITIVE UNIT IN GROUP TESTING 25 1 INTRODUCTION 25 2 DESCRIPTION OF PROPERTIES, NUMERICAL RESULTS 25 3 SOME FORMULAS FOR PROCEDURE RDH 25 4 THE GREEDY PROCEDURE RG 25 5 CONCLUSIONS 25 6 CHANGING THE PRIOR WITH PROCEDURE RDH 25 7 ROBUSTNESS OF PROCEDURE RDH FOR Q KNOWN REFERENCES 26 TESTING HYPOTHESES ON VARIANCES IN THE PRESENCE OF CORRELATIONS 26 1 BIVARIATE NORMAL POPULATION 26 2 MODIFYING THE HYPOTHESIS 26 3 NONNULL MOMENTS 26 4 NULL CASE 26 5 THE CONDITIONAL HYPOTHESIS REFERENCES 27 ESTIMATING THE SMALLEST SCALE PARAMETER: UNIVERSAL DOMINATION RESULTS 27 1 INTRODUCTION 27 2 MAIN RESULTS REFERENCES 28 ON SENSITIVITY OF EXPONENTIAL RATE OF CONVERGENCE FOR THE MAXIMUM LIKELIHOOD ESTIMATOR 28 1 INTRODUCTION 28 2 MAIN RESULTS 28 3 SOME APPLICATIONS 28 3 1 EXPONENTIAL MODEL 28 3 2 FAMILIES OF T-DISTRIBUTIONS WITH LOCATION PARAMETER 28 4 DISCUSSION REFERENCES 29 A CLOSER LOOK AT WEIGHTED LIKELIHOOD IN THE CONTEXT OF MIXTURES 29 1 INTRODUCTION 29 2 BACKGROUND 29 3 SIMULATION EXPERIMENTS AND RESULTS 29 3 1 NORMAL MIXTURE WITH EQUAL COMPONENT VARIANCE 29 3 2 NORMAL MIXTURES WITH UNEQUAL COMPONENT VARIANCE 29 3 3 OTHER MODELS 29 4 MODEL SELECTION 29 5 CONCLUSIONS REFERENCES 30 ON NONPARAMETRIC FUNCTION ESTIMATION WITH IN?NITE-ORDER FLAT-TOP KERNELS 30 1 INTRODUCTION: A GENERAL FAMILY OF FLAT-TOP KERNELS OF IN?NITE ORDER 30 2 MULTIVARIATE DENSITY ESTIMATION: A REVIEW 30 3 FURTHER ISSUES ON DENSITY ESTIMATION 30 3 1 CASE OF SMOOTH DENSITY OVER A FINITE DOMAIN 30 3 2 CASE OF IN?NITE DOMAIN WITH SOME DISCONTINUITIES REFERENCES 31 MULTIPOLISHING LARGE TWO-WAY TABLES 31 1 INTRODUCTION 31 2 BILINEAR MULTIPOLISHERS 31 2 1 CHOOSING THE AUXILIARY VARIABLES EQUAL TO THE E?ECTS OF THE ADDITIVE FIT 31 2 2 ROBUST ALTERNATIVES 31 3 MATRIX APPROXIMATIONS 31 3 1 APPROXIMATIONS OF TWO-WAY TABLES 31 4 DISPLAYS 31 5 EXAMPLE 31 6 CONCLUDING REMARKS REFERENCES 32 ON DISTANCES AND MEASURES OF INFORMATION: A CASE OF DIVERSITY 32 1 INTRODUCTION 32 2 MEASURING INFORMATION – MEASURES OF INFORMATION 32 3 PROPERTIES OF MEASURES OF INFORMATION 32 4 MEASURES OF INFORMATION AND INFERENCE 32 5 APPLICATIONS 32 6 CONCLUSIONS REFERENCES REPRESENTATION FORMULAE FOR PROBABILITIES OF CORRECT CLASSI?CATION 33 1 INTRODUCTION 33 2 VECTOR ALGEBRAIC PRELIMINARIES 33 3 DISTRIBUTIONAL RESULTS 33 3 1 REPRESENTATION FORMULAE BASED UPON THE TWO-DIMENSIONAL GAUSSIAN LAW 33 3 2 REPRESENTATION FORMULAE BASED UPON THE DOUBLY NONCENTRAL F-DISTRIBUTION REFERENCES 34 ESTIMATION OF CYCLING E?ECT ON RELIABILITY ? 34 1 MODELS 34 2 SEMIPARAMETRIC ESTIMATION 34 2 1 THE FIRST MODEL 34 2 2 THE SECOND MODEL REFERENCES VI APPLICATIONS TO BIOLOGY AND MEDICINE 35 A NEW TEST FOR TREATMENT VS CONTROL IN AN ORDERED 2 × 3 CONTINGENCY TABLE 35 1 INTRODUCTION 35 2 NEW TEST, IMPLEMENTATION AND EXAMPLE 35 3 SIMULATION STUDY 35 4 THEORETICAL PROPERTIES APPENDIX REFERENCES LIMITING RESULTS FOR THE PROPORTION OF TRIALS ON TREATMENT I