MATHEMATICAL STATISTICS

By C. FIMIA

CONTENTS PROBABILITY AND DISTRIBUTIONS INTRODUCTION SETS REVIEW OF SET THEORY SET FUNCTIONS THE PROBABILITY SET FUNCTION COUNTING RULES ADDITIONAL PROPERTIES OF PROBABILITY CONDITIONAL PROBABILITY AND INDEPENDENCE INDEPENDENCE SIMULATIONS RANDOM VARIABLES DISCRETE RANDOM VARIABLES TRANSFORMATIONS CONTINUOUS RANDOM VARIABLES QUANTILES MIXTURES OF DISCRETE AND CONTINUOUS TYPE DISTRIBUTIONS EXPECTATION OF A RANDOM VARIABLE R COMPUTATION FOR AN ESTIMATION OF THE EXPECTED GAIN SOME SPECIAL EXPECTATIONS 1 10 IMPORTANT INEQUALITIES MULTIVARIATE DISTRIBUTIONS DISTRIBUTIONS OF TWO RANDOM VARIABLES MARGINAL DISTRIBUTIONS EXPECTATION TRANSFORMATIONS: BIVARIATE RANDOM VARIABLES CONDITIONAL DISTRIBUTIONS AND EXPECTATIONS INDEPENDENT RANDOM VARIABLES THE CORRELATION COE?CIENT EXTENSION TO SEVERAL RANDOM VARIABLES ?MULTIVARIATE VARIANCE-COVARIANCE MATRIX TRANSFORMATIONS FOR SEVERAL RANDOM VARIABLES LINEAR COMBINATIONS OF RANDOM VARIABLES SOME SPECIAL DISTRIBUTIONS THE BINOMIAL AND RELATED DISTRIBUTIONS NEGATIVE BINOMIAL AND GEOMETRIC DISTRIBUTIONS MULTINOMIAL DISTRIBUTION HYPERGEOMETRIC DISTRIBUTION THE POISSON DISTRIBUTION THE ?, ?2, AND ? DISTRIBUTIONS THE ?2-DISTRIBUTION THE ?-DISTRIBUTION THE NORMAL DISTRIBUTION ?CONTAMINATED NORMALS THE MULTIVARIATE NORMAL DISTRIBUTION BIVARIATE NORMAL DISTRIBUTION ?MULTIVARIATE NORMAL DISTRIBUTION, GENERAL CASE ?APPLICATIONS T- AND F-DISTRIBUTIONS THE T-DISTRIBUTION THE F-DISTRIBUTION STUDENT’S THEOREM ?MIXTURE DISTRIBUTIONS SOME ELEMENTARY STATISTICAL INFERENCES SAMPLING AND STATISTICS POINT ESTIMATORS HISTOGRAM ESTIMATES OF PMFS AND PDFS CON?DENCE INTERVALS CON?DENCE INTERVALS FOR DI?ERENCE IN MEANS CON?DENCE INTERVAL FOR DI?ERENCE IN PROPORTIONS ?CON?DENCE INTERVALS FOR PARAMETERS OF DISCRETE DISTRIBUTIONS ORDER STATISTICS CON?DENCE INTERVALS FOR QUANTILES INTRODUCTION TO HYPOTHESIS TESTING ADDITIONAL COMMENTS ABOUT STATISTICAL TESTS OBSERVED SIGNI?CANCE LEVEL, P-VALUE CHI-SQUARE TESTS THE METHOD OF MONTE CARLO ACCEPT–REJECT GENERATION ALGORITHM BOOTSTRAP PROCEDURES PERCENTILE BOOTSTRAP CON?DENCE INTERVALS BOOTSTRAP TESTING PROCEDURES 4 10 ?TOLERANCE LIMITS FOR DISTRIBUTIONS CONSISTENCY AND LIMITING DISTRIBUTIONS CONVERGENCE IN PROBABILITY CONVERGENCE IN DISTRIBUTION BOUNDED IN PROBABILITY ?-METHOD MOMENT GENERATING FUNCTION TECHNIQUE CENTRAL LIMIT THEOREM ?EXTENSIONS TO MULTIVARIATE DISTRIBUTIONS MAXIMUM LIKELIHOOD METHODS MAXIMUM LIKELIHOOD ESTIMATION RAO–CRAM´ER LOWER BOUND AND E?CIENCY MAXIMUM LIKELIHOOD TESTS MULTIPARAMETER CASE: ESTIMATION MULTIPARAMETER CASE: TESTING THE EM ALGORITHM SU?CIENCY MEASURES OF QUALITY OF ESTIMATORS A SU?CIENT STATISTIC FOR A PARAMETER PROPERTIES OF A SU?CIENT STATISTIC COMPLETENESS AND UNIQUENESS THE EXPONENTIAL CLASS OF DISTRIBUTIONS FUNCTIONS OF A PARAMETER BOOTSTRAP STANDARD ERRORS THE CASE OF SEVERAL PARAMETERS MINIMAL SU?CIENCY AND ANCILLARY STATISTICS SU?CIENCY, COMPLETENESS, AND INDEPENDENCE OPTIMAL TESTS OF HYPOTHESES MOST POWERFUL TESTS UNIFORMLY MOST POWERFUL TESTS LIKELIHOOD RATIO TESTS LIKELIHOOD RATIO TESTS FOR TESTING MEANS OF NORMAL DISTRI- BUTIONS LIKELIHOOD RATIO TESTS FOR TESTING VARIANCES OF NORMAL DIS- TRIBUTIONS ?THE SEQUENTIAL PROBABILITY RATIO TEST ?MINIMAX AND CLASSI?CATION PROCEDURES MINIMAX PROCEDURES CLASSI?CATION INFERENCES ABOUT NORMAL LINEAR MODELS ONE-WAY ANOVA NONCENTRAL ?2 AND F-DISTRIBUTIONS MULTIPLE COMPARISONS TWO-WAY ANOVA INTERACTION BETWEEN FACTORS A REGRESSION PROBLEM MAXIMUM LIKELIHOOD ESTIMATES ?GEOMETRY OF THE LEAST SQUARES FIT A TEST OF INDEPENDENCE THE DISTRIBUTIONS OF CERTAIN QUADRATIC FORMS THE INDEPENDENCE OF CERTAIN QUADRATIC FORMS 10 NONPARAMETRIC AND ROBUST STATISTICS 10 1 LOCATION MODELS 10 2 SAMPLE MEDIAN AND THE SIGN TEST 10 2 1 ASYMPTOTIC RELATIVE E?CIENCY 10 2 2 ESTIMATING EQUATIONS BASED ON THE SIGN TEST 10 2 3 CON?DENCE INTERVAL FOR THE MEDIAN 10 3 SIGNED-RANK WILCOXON 10 3 1 ASYMPTOTIC RELATIVE E?CIENCY 10 3 2 ESTIMATING EQUATIONS BASED ON SIGNED-RANK WILCOXON 10 3 3 CON?DENCE INTERVAL FOR THE MEDIAN 10 3 4 MONTE CARLO INVESTIGATION 10 4 MANN–WHITNEY–WILCOXON PROCEDURE 10 4 1 ASYMPTOTIC RELATIVE E?CIENCY 10 4 2 ESTIMATING EQUATIONS BASED ON THE MANN–WHITNEY–WILCOXON 10 4 3 CON?DENCE INTERVAL FOR THE SHIFT PARAMETER ? 10 4 4 MONTE CARLO INVESTIGATION OF POWER 10 5 ?GENERAL RANK SCORES 10 5 1 E?CACY 10 5 2 ESTIMATING EQUATIONS BASED ON GENERAL SCORES 10 5 3 OPTIMIZATION: BEST ESTIMATES 10 6 ?ADAPTIVE PROCEDURES 10 7 SIMPLE LINEAR MODEL 10 8 MEASURES OF ASSOCIATION 10 8 1 KENDALL’S ? 10 8 2 SPEARMAN’S RHO 10 9 ROBUST CONCEPTS 10 9 1 LOCATION MODEL 10 9 2 LINEAR MODEL 11 BAYESIAN STATISTICS 11 1 BAYESIAN PROCEDURES 11 1 1 PRIOR AND POSTERIOR DISTRIBUTIONS 11 1 2 BAYESIAN POINT ESTIMATION 11 1 3 BAYESIAN INTERVAL ESTIMATION 11 1 4 BAYESIAN TESTING PROCEDURES 11 1 5 BAYESIAN SEQUENTIAL PROCEDURES 11 2 MORE BAYESIAN TERMINOLOGY AND IDEAS 11 3 GIBBS SAMPLER 11 4 MODERN BAYESIAN METHODS 11 4 1 EMPIRICAL BAYES