PRINCIPLES OF STATISTICAL INFERENCE

By DAVID WILLIAMS

CONTENTS PRELIMINARIES SUMMARY STARTING POINT ROLE OF FORMAL THEORY OF INFERENCE SOME SIMPLE MODELS FORMULATION OF OBJECTIVES TWO BROAD APPROACHES TO STATISTICAL INFERENCE SOME FURTHER DISCUSSION PARAMETERS NOTES SOME CONCEPTS AND SIMPLE APPLICATIONS LIKELIHOOD SUF?CIENCY EXPONENTIAL FAMILY CHOICE OF PRIORS FOR EXPONENTIAL FAMILY PROBLEMS SIMPLE FREQUENTIST DISCUSSION PIVOTS SIGNI?CANCE TESTS GENERAL REMARKS SIMPLE SIGNI?CANCE TEST ONE- AND TWO-SIDED TESTS RELATION WITH ACCEPTANCE AND REJECTION FORMULATION OF ALTERNATIVES AND TEST STATISTICS RELATION WITH INTERVAL ESTIMATION INTERPRETATION OF SIGNI?CANCE TESTS BAYESIAN TESTING MORE COMPLICATED SITUATIONS GENERAL BAYESIAN FORMULATION FREQUENTIST ANALYSIS SOME MORE GENERAL FREQUENTIST DEVELOPMENTS SOME FURTHER BAYESIAN EXAMPLES INTERPRETATIONS OF UNCERTAINTY BROAD ROLES OF PROBABILITY FREQUENTIST INTERPRETATION OF UPPER LIMITS NEYMAN–PEARSON OPERATIONAL CRITERIA SOME GENERAL ASPECTS OF THE FREQUENTIST APPROACH YET MORE ON THE FREQUENTIST APPROACH PERSONALISTIC PROBABILITY IMPERSONAL DEGREE OF BELIEF REFERENCE PRIORS 5 10 TEMPORAL COHERENCY 5 11 DEGREE OF BELIEF AND FREQUENCY 5 12 STATISTICAL IMPLEMENTATION OF BAYESIAN ANALYSIS 5 13 MODEL UNCERTAINTY 5 14 CONSISTENCY OF DATA AND PRIOR 5 15 RELEVANCE OF FREQUENTIST ASSESSMENT 5 16 SEQUENTIAL STOPPING 5 17 A SIMPLE CLASSI?CATION PROBLEM ASYMPTOTIC THEORY SCALAR PARAMETER MULTIDIMENSIONAL PARAMETER NUISANCE PARAMETERS TESTS AND MODEL REDUCTION COMPARATIVE DISCUSSION PRO?LE LIKELIHOOD AS AN INFORMATION SUMMARIZER CONSTRAINED ESTIMATION SEMI-ASYMPTOTIC ARGUMENTS 6 10 NUMERICAL-ANALYTIC ASPECTS 6 11 HIGHER-ORDER ASYMPTOTICS FURTHER ASPECTS OF MAXIMUM LIKELIHOOD MULTIMODAL LIKELIHOODS IRREGULAR FORM SINGULAR INFORMATION MATRIX FAILURE OF MODEL UNUSUAL PARAMETER SPACE MODI?ED LIKELIHOODS ADDITIONAL OBJECTIVES PREDICTION DECISION ANALYSIS POINT ESTIMATION NON-LIKELIHOOD-BASED METHODS RANDOMIZATION-BASED ANALYSIS SAMPLING A ?NITE POPULATION DESIGN OF EXPERIMENTS