MULTIPARAMETRIC STATISTICS

By JOSEPH RAMIREZ

CONTENTS CHAPTER 1 INTRODUCTION: THE DEVELOPMENT OF MULTIPARAMETRIC STATISTICS THE STEIN E?ECT THE KOLMOGOROV ASYMPTOTICS SPECTRAL THEORY OF INCREASING RANDOM MATRICES CONSTRUCTING MULTIPARAMETRIC PROCEDURES OPTIMAL SOLUTION TO EMPIRICAL LINEAR EQUATIONS CHAPTER 2 FUNDAMENTAL PROBLEM OF STATISTICS 21 SHRINKAGE OF SAMPLE MEAN VECTOR SHRINKAGE FOR NORMAL DISTRIBUTIONS SHRINKAGE FOR A WIDE CLASS OF DISTRIBUTIONS CONCLUSIONS 22 SHRINKAGE OF UNBIASED ESTIMATORS SPECIAL SHRINKAGE OF NORMAL ESTIMATORS SHRINKAGE OF ARBITRARY UNBIASED ESTIMATORS LIMIT QUADRATIC RISK OF SHRINKAGE ESTIMATORS 23 SHRINKAGE OF IN?NITE-DIMENSIONAL VECTORS NORMAL DISTRIBUTIONS WIDE CLASS OF DISTRIBUTIONS 24 UNIMPROVABLE COMPONENT-WISE ESTIMATION ESTIMATOR FOR THE DENSITY OF PARAMETERS ESTIMATOR FOR THE BEST ESTIMATING FUNCTION CHAPTER 3 SPECTRAL THEORY OF LARGE SAMPLE COVARIANCE MATRICES 31 SPECTRAL FUNCTIONS OF LARGE SAMPLE COVARIANCE MATRICES GRAM MATRICES SAMPLE COVARIANCE MATRICES LIMIT SPECTRA 32 SPECTRAL FUNCTIONS OF IN?NITE SAMPLE COVARIANCE MATRICES DISPERSION EQUATIONS FOR IN?NITE GRAM MATRICES DISPERSION EQUATIONS FOR SAMPLE COVARIANCE MATRICES LIMIT SPECTRAL EQUATIONS 33 NORMALIZATION OF QUALITY FUNCTIONS SPECTRAL FUNCTIONS OF SAMPLE COVARIANCE MATRICES NORMAL EVALUATION OF SAMPLE-DEPENDENT FUNCTIONALS CHAPTER 4 ASYMPTOTICALLY UNIMPROVABLE SOLUTION OF MULTIVARIATE PROBLEMS 41 ESTIMATORS OF LARGE INVERSE COVARIANCE MATRICES PROBLEM SETTING SHRINKAGE FOR INVERSE COVARIANCE MATRICES GENERALIZED RIDGE ESTIMATORS ASYMPTOTICALLY UNIMPROVABLE ESTIMATOR PROOFS FOR SECTION 41 42 MATRIX SHRINKAGE ESTIMATORS OF EXPECTATION VECTORS LIMIT QUADRATIC RISK FOR ESTIMATORS OF VECTORS MINIMIZATION OF THE LIMIT QUADRATIC RISK STATISTICS TO APPROXIMATE LIMIT RISK STATISTICS TO APPROXIMATE THE EXTREMUM SOLUTION 43 MULTIPARAMETRIC SAMPLE LINEAR REGRESSION FUNCTIONALS OF RANDOM GRAM MATRICES FUNCTIONALS IN THE REGRESSION PROBLEM MINIMIZATION OF QUADRATIC RISK SPECIAL CASES CHAPTER 5 MULTIPARAMETRIC DISCRIMINANT ANALYSIS 51 DISCRIMINANT ANALYSIS OF INDEPENDENT VARIABLES A PRIORI WEIGHTING OF VARIABLES EMPIRICAL WEIGHTING OF VARIABLES MINIMUM ERROR PROBABILITY FOR EMPIRICAL WEIGHTING STATISTICS TO ESTIMATE PROBABILITIES OF ERRORS CONTRIBUTION OF A SMALL NUMBER OF VARIABLES SELECTION OF VARIABLES BY THRESHOLD 52 DISCRIMINANT ANALYSIS OF DEPENDENT VARIABLES ASYMPTOTICAL SETTING MOMENTS OF GENERALIZED DISCRIMINANT FUNCTION LIMIT PROBABILITIES OF ERRORS BEST-IN-THE-LIMIT DISCRIMINANT PROCEDURE THE EXTENSION TO A WIDE CLASS OF DISTRIBUTIONS ESTIMATING THE ERROR PROBABILITY CHAPTER 6 THEORY OF SOLUTION TO HIGH-ORDER SYSTEMS OF EMPIRICAL LINEAR ALGEBRAIC EQUATIONS 61 THE BEST BAYES SOLUTION 62 ASYMPTOTICALLY UNIMPROVABLE SOLUTION SPECTRAL FUNCTIONS OF LARGE GRAM MATRICES LIMIT SPECTRAL FUNCTIONS OF GRAM MATRICES QUADRATIC RISK OF PSEUDOSOLUTIONS MINIMIZATION OF THE LIMIT RISK SHRINKAGE-RIDGE PSEUDOSOLUTION PROOFS FOR SECTION 62