OPTIMIZING METHODS IN STATISTICS

By ELIZABETH JOHNSON

CONTENTS PART I SELECTED CONCEPTS OF STATISTICAL DECISION THEORY RANDOM VARIABLES PROBABILITY DISTRIBUTION FUNCTION MOMENTS AND MOMENT-GENERATING FUNCTION CLASSICAL INEQUALITIES WITH MOMENTS MOMENT-GENERATING FUNCTION PARTIAL MOMENTS AND STOCHASTIC DOMINANCE PERFORMANCE FUNCTIONS STOCHASTIC DOMINANCE QUANTILE FUNCTIONS AND FENCHEL DUALITY ENTROPY AND DISTRIBUTION DIVERGENCE MEASURES SHANNON ENTROPY AND KULLBACK–LEIBLER DIVERGENCE RENYI ENTROPY AND RENYI DIVERGENCE DEVIATION, RISK, AND ERROR MEASURES DEVIATION MEASURES RISK ENVELOPES AND RISK IDENTI?ERS AVERSE MEASURES OF RISK ERROR MEASURES PROBABILISTIC INEQUALITIES BASIC PROBABILISTIC INEQUALITIES CHEBYSHEV’S INEQUALITIES WITH DEVIATION MEASURES ONE-SIDED CHEBYSHEV’S INEQUALITIES TWO-SIDED CHEBYSHEV’S INEQUALITIES KOLMOGOROV’S INEQUALITIES WITH DEVIATION MEASURES PART II STATISTICAL DECISION PROBLEMS MAXIMUM LIKELIHOOD METHOD MAXIMUM LIKELIHOOD PRINCIPLE PARAMETRIC ESTIMATION NONPARAMETRIC ESTIMATION ESTIMATION OF PROBABILITY DISTRIBUTIONS OF RANDOM VECTORS ENTROPY MAXIMIZATION SHANNON ENTROPY MAXIMIZATION RELATIVE ENTROPY MINIMIZATION RENYI ENTROPY MAXIMIZATION ENTROPY MAXIMIZATION WITH DEVIATION MEASURES ENTROPY MAXIMIZATION WITH CONSTRAINTS ON AVERAGE ?-QUANTILE FOR DIFFERENT CON?DENCE LEVELS REGRESSION MODELS GENERALIZED LINEAR REGRESSION AND ERROR DECOMPOSITION LEAST SQUARES LINEAR REGRESSION UNCONSTRAINED LEAST SQUARES LINEAR REGRESSION LEAST SQUARES LINEAR REGRESSION WITH CONSTRAINTS MEDIAN REGRESSION UNCONSTRAINED MEDIAN REGRESSION MEDIAN REGRESSION WITH CONSTRAINTS QUANTILE REGRESSION UNCONSTRAINED QUANTILE REGRESSION QUANTILE REGRESSION WITH CONSTRAINTS SPECIAL TYPES OF LINEAR REGRESSION MIXED QUANTILE REGRESSION UNBIASED LINEAR REGRESSION RISK ACCEPTABLE REGRESSION ROBUST REGRESSION CLASSI?CATION LOGISTIC REGRESSION SUPPORT VECTOR MACHINE STANDARD SVM SVM WITH SOFT MARGIN ALTERNATIVE FORMULATIONS OF SVM WITH SOFT MARGIN CHOICE OF MAPPING : KERNEL APPROACH STATISTICAL DECISION MODELS WITH RISK AND DEVIATION MINIMIZATION OF RISK AND DEVIATION DEVIATION MINIMIZATION IN PORTFOLIO SELECTION MARKOWITZ PORTFOLIO SELECTION MARKOWITZ-TYPE PORTFOLIO SELECTION GENERALIZED CAPITAL ASSET PRICING MODEL CAPM REINTERPRETATION: INVERSE PORTFOLIO PROBLEM DEVIATION MINIMIZATION IN OPTIMAL HEDGING CASH MATCHING WITH DEVIATION CONSTRAINTS RISK MINIMIZATION IN PORTFOLIO THEORY APPLICATIONS WITH DOWNSIDE RISK CONSTRAINTS PORTFOLIO OPTIMIZATION WITH DRAWDOWN MEASURE DRAWDOWN MEASURE LINEAR PROGRAMMING REPRESENTATIONS PORTFOLIO PROBLEM FORMULATIONS CAPM WITH CDAR PART III PORTFOLIO SAFEGUARD CASE STUDIES PORTFOLIO SAFEGUARD CASE STUDIES CASE STUDY: OPTIMAL HEDGING OF CDO BOOK (MEANABS_DEV, MEANABS_PEN, POLYNOM_ABS, CARDN) PROBLEM 1: PROBLEM_CDOHEDGE_1 PROBLEM 2: PROBLEM_CDOHEDGE_2 HEDGING PORTFOLIO OF OPTIONS MORTGAGE PIPELINE HEDGING PROBLEM 1: PROBLEM_CVAR_DEV PROBLEM 2: PROBLEM_MEANABS_DEV PROBLEM 3: PROBLEM_STD_DEV PROBLEM 4: PROBLEM_TWO_TAILED_VAR75 PROBLEM 5: PROBLEM_TWO_TAILED_VAR90 CASH MATCHING BOND PORTFOLIO CASH FLOW MATCHING WITH CVAR CONSTRAINTS PORTFOLIO REPLICATION WITH RISK CONSTRAINTS STYLE CLASSI?CATION WITH QUANTILE REGRESSION CVAR ESTIMATION THROUGH EXPLANATORY FACTORS WITH MIXED QUANTILE REGRESSION OPTIMAL TEST SELECTION PROBLEM 1: PROBLEM_DETERMINISTIC_LP_MODEL PROBLEM 2: PROBLEM_ROBUST_MODEL PROBLEM 3: PROBLEM_STOCHASTIC_MODEL SPARSE SIGNAL RECONSTRUCTION: A CARDINALITY APPROACH PROBLEM 1: PROBLEM_CONSTR_CARDINALITY PROBLEM 2: PROBLEM_MINIMIZE_CARDINALITY PROBLEM 3: PROBLEM_CONSTR_POLYNOMABS SPARSE RECONSTRUCTION PROBLEMS FROM SPARCO TOOLBOX PROBLEM 1: L1 RELAXED PROBLEM 2: L1 RELAXED D PROBLEM 3: L2 D OPTIMAL INTENSITY-MODULATED RADIATION THERAPY TREATMENT-PLANNING PROBLEM PORTFOLIO OPTIMIZATION—CVAR VERSUS STANDARD DEVIATION PROBLEM 1: PROBLEM_ST_DEV_COVARIANCES_2P9 PROBLEM 2: PROBLEM_ST_DEV_SCENARIOS_2P9 PROBLEM 3: PROBLEM_MIN_CVAR_DEV_2P9 PORTFOLIO CREDIT-RISK OPTIMIZATION MODELED BY SCENARIOS AND MIXTURES OF NORMAL DISTRIBUTIONS PROBLEM 1: PROBLEM_VAR_LLN PROBLEM 2: PROBLEM_CVAR_LLN PROBLEM 3: PROBLEM_AVG_VAR_CLT PROBLEM 4: PROBLEM_AVG_CVAR_CLT PROBLEM 5: PROBLEM_AVG_VAR_CLT (ALTERNATIVE FORMULATION) PROBLEM 6: PROBLEM_AVG_CVAR_CLT (ALTERNATIVE FORMULATION) SVM WITH VAR AND CVAR PROBLEM 1A: -SVM WITH CVAR PROBLEM 1B: VAR-SVM PROBLEM 2A: EXTENDED -SVM WITH CVAR PROBLEM 2B: EXTENDED VAR-SVM PROBLEM 3A: ROBUST -SVM WITH CVAR PROBLEM 3B: ROBUST VAR-SVM IMPLIED COPULA CDO PRICING MODEL: ENTROPY APPROACH PROBLEM 1A: PROBLEM_HULL_MEANSQUARE PROBLEM 1B: PROBLEM_HULL_VARIANCE PROBLEM 2: PROBLEM_CCC LOGISTIC REGRESSION AND REGULARIZED LOGISTIC REGRESSION FOR ESTIMATING THE PROBABILITY OF CESAREAN SECTION PROBLEM 1: LOGISTIC REGRESSION PROBLEM 2: REGULARIZED LOGISTIC REGRESSION PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS ON A SINGLE SAMPLE PATH PROBLEM 1: CONSTRAINT ON MAXIMUM DRAWDOWN PROBLEM 2: CONSTRAINT ON AVERAGE DRAWDOWN PROBLEM 3: CONSTRAINT ON CDAR PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS ON MULTIPLE SAMPLE PATHS PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS: SINGLE PATH VERSUS MULTIPLE PATHS PROBLEM 1: MULTIPLE SAMPLE PATHS PROBLEM 2: AGGREGATED SAMPLE PATH REFERENCES