Examples and problems in mathematical statistics

Examples and problems in mathematical statistics

Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplic...

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Detalles Bibliográficos
Autor principal: Zacks, Shelemyahu, 1932- (-)
Formato: Libro electrónico
Idioma:Inglés
Publicado: Hoboken, New Jersey : Wiley 2014.
Edición:1st ed
Colección:Wiley series in probability and statistics
Materias:
Mathematical statistics > Problems, exercises, etc.
Statistics > Problems, exercises, etc.
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009684436406719
Tabla de Contenidos:
  • Intro
  • Examples and Problems in Mathematical Statistics
  • Contents
  • Preface
  • List of Random Variables
  • List of Abbreviations
  • 1 Basic Probability Theory
  • PART I: THEORY
  • 1.1 OPERATIONS ON SETS
  • 1.2 ALGEBRA AND σ-FIELDS
  • 1.3 PROBABILITY SPACES
  • 1.4 CONDITIONAL PROBABILITIES AND INDEPENDENCE
  • 1.5 RANDOM VARIABLES AND THEIR DISTRIBUTIONS
  • 1.6 THE LEBESGUE AND STIELTJES INTEGRALS
  • 1.6.1 General Definition of Expected Value: The Lebesgue Integral
  • 1.6.2 The Stieltjes-Riemann Integral
  • 1.6.3 Mixtures of Discrete and Absolutely Continuous Distributions
  • 1.6.4 Quantiles of Distributions
  • 1.6.5 Transformations
  • 1.7 JOINT DISTRIBUTIONS, CONDITIONAL DISTRIBUTIONS AND INDEPENDENCE
  • 1.7.1 Joint Distributions
  • 1.7.2 Conditional Expectations: General Definition
  • 1.7.3 Independence
  • 1.8 MOMENTS AND RELATED FUNCTIONALS
  • 1.9 MODES OF CONVERGENCE
  • 1.10 WEAK CONVERGENCE
  • 1.11 LAWS OF LARGE NUMBERS
  • 1.11.1 The Weak Law of Large Numbers (WLLN)
  • 1.11.2 The Strong Law of Large Numbers (SLLN)
  • 1.12 CENTRAL LIMIT THEOREM
  • 1.13 MISCELLANEOUS RESULTS
  • 1.13.1 Law of the Iterated Logarithm
  • 1.13.2 Uniform Integrability
  • 1.13.3 Inequalities
  • 1.13.4 The Delta Method
  • 1.13.5 The Symbols op and Op
  • 1.13.6 The Empirical Distribution and Sample Quantiles
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS TO SELECTED PROBLEMS
  • 2 Statistical Distributions
  • PART I: THEORY
  • 2.1 INTRODUCTORY REMARKS
  • 2.2 FAMILIES OF DISCRETE DISTRIBUTIONS
  • 2.2.1 Binomial Distributions
  • 2.2.2 Hypergeometric Distributions
  • 2.2.3 Poisson Distributions
  • 2.2.4 Geometric, Pascal, and Negative Binomial Distributions
  • 2.3 SOME FAMILIES OF CONTINUOUS DISTRIBUTIONS
  • 2.3.1 Rectangular Distributions
  • 2.3.2 Beta Distributions
  • 2.3.3 Gamma Distributions
  • 2.3.4 Weibull and Extreme Value Distributions.
  • 2.3.5 Normal Distributions
  • 2.3.6 Normal Approximations
  • 2.4 TRANSFORMATIONS
  • 2.4.1 One-to-One Transformations of Several Variables
  • 2.4.2 Distribution of Sums
  • 2.4.3 Distribution of Ratios
  • 2.5 VARIANCES AND COVARIANCES OF SAMPLE MOMENTS
  • 2.6 DISCRETE MULTIVARIATE DISTRIBUTIONS
  • 2.6.1 The Multinomial Distribution
  • 2.6.2 Multivariate Negative Binomial
  • 2.6.3 Multivariate Hypergeometric Distributions
  • 2.7 MULTINORMAL DISTRIBUTIONS
  • 2.7.1 Basic Theory
  • 2.7.2 Distribution of Subvectors and Distributions of Linear Forms
  • 2.7.3 Independence of Linear Forms
  • 2.8 DISTRIBUTIONS OF SYMMETRIC QUADRATIC FORMS OF NORMAL VARIABLES
  • 2.9 INDEPENDENCE OF LINEAR AND QUADRATIC FORMS OF NORMAL VARIABLES
  • 2.10 THE ORDER STATISTICS
  • 2.11 t-DISTRIBUTIONS
  • 2.12 F-DISTRIBUTIONS
  • 2.13 THE DISTRIBUTION OF THE SAMPLE CORRELATION
  • 2.14 EXPONENTIAL TYPE FAMILIES
  • 2.15 APPROXIMATING THE DISTRIBUTION OF THE SAMPLE MEAN: EDGEWORTH AND SADDLEPOINT APPROXIMATIONS
  • 2.15.1 Edgeworth Expansion
  • 2.15.2 Saddlepoint Approximation
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS TO SELECTED PROBLEMS
  • 3 Sufficient Statistics and the Information in Samples
  • PART I: THEORY
  • 3.1 INTRODUCTION
  • 3.2 DEFINITION AND CHARACTERIZATION OF SUFFICIENT STATISTICS
  • 3.2.1 Introductory Discussion
  • 3.2.2 Theoretical Formulation
  • 3.3 LIKELIHOOD FUNCTIONS AND MINIMAL SUFFICIENT STATISTICS
  • 3.4 SUFFICIENT STATISTICS AND EXPONENTIAL TYPE FAMILIES
  • 3.5 SUFFICIENCY AND COMPLETENESS
  • 3.6 SUFFICIENCY AND ANCILLARITY
  • 3.7 INFORMATION FUNCTIONS AND SUFFICIENCY
  • 3.7.1 The Fisher Information
  • 3.7.2 The Kullback-Leibler Information
  • 3.8 THE FISHER INFORMATION MATRIX
  • 3.9 SENSITIVITY TO CHANGES IN PARAMETERS
  • 3.9.1 The Hellinger Distance
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS TO SELECTED PROBLEMS.
  • 4 Testing Statistical Hypotheses
  • PART I: THEORY
  • 4.1 THE GENERAL FRAMEWORK
  • 4.2 THE NEYMAN-PEARSON FUNDAMENTAL LEMMA
  • 4.3 TESTING ONE-SIDED COMPOSITE HYPOTHESES IN MLR MODELS
  • 4.4 TESTING TWO-SIDED HYPOTHESES IN ONE-PARAMETER EXPONENTIAL FAMILIES
  • 4.5 TESTING COMPOSITE HYPOTHESES WITH NUISANCE PARAMETERS-UNBIASED TESTS
  • 4.6 LIKELIHOOD RATIO TESTS
  • 4.6.1 Testing in Normal Regression Theory
  • 4.6.2 Comparison of Normal Means: The Analysis of Variance
  • 4.7 THE ANALYSIS OF CONTINGENCY TABLES
  • 4.7.1 The Structure of Multi-Way Contingency Tables and the Statistical Model
  • 4.7.2 Testing the Significance of Association
  • 4.7.3 The Analysis of Tables
  • 4.7.4 Likelihood Ratio Tests for Categorical Data
  • 4.8 SEQUENTIAL TESTING OF HYPOTHESES
  • 4.8.1 The Wald Sequential Probability Ratio Test
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS TO SELECTED PROBLEMS
  • 5 Statistical Estimation
  • PART I: THEORY
  • 5.1 GENERAL DISCUSSION
  • 5.2 UNBIASED ESTIMATORS
  • 5.2.1 General Definition and Example
  • 5.2.2 Minimum Variance Unbiased Estimators
  • 5.2.3 The Cramér-Rao Lower Bound for the One-Parameter Case
  • 5.2.4 Extension of the Cramér-Rao Inequality to Multiparameter Cases
  • 5.2.5 General Inequalities of the Cramér-Rao Type
  • 5.3 THE EFFICIENCY OF UNBIASED ESTIMATORS IN REGULAR CASES
  • 5.4 BEST LINEAR UNBIASED AND LEAST-SQUARES ESTIMATORS
  • 5.4.1 BLUEs of the Mean
  • 5.4.2 Least-Squares and BLUEs in Linear Models
  • 5.4.3 Best Linear Combinations of Order Statistics
  • 5.5 STABILIZING THE LSE: RIDGE REGRESSIONS
  • 5.6 MAXIMUM LIKELIHOOD ESTIMATORS
  • 5.6.1 Definition and Examples
  • 5.6.2 MLEs in Exponential Type Families
  • 5.6.3 The Invariance Principle
  • 5.6.4 MLE of the Parameters of Tolerance Distributions
  • 5.7 EQUIVARIANT ESTIMATORS
  • 5.7.1 The Structure of Equivariant Estimators.
  • 5.7.2 Minimum MSE Equivariant Estimators
  • 5.7.3 Minimum Risk Equivariant Estimators
  • 5.7.4 The Pitman Estimators
  • 5.8 ESTIMATING EQUATIONS
  • 5.8.1 Moment-Equations Estimators
  • 5.8.2 General Theory of Estimating Functions
  • 5.9 PRETEST ESTIMATORS
  • 5.10 ROBUST ESTIMATION OF THE LOCATION AND SCALE PARAMETERS OF SYMMETRIC DISTRIBUTIONS
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS OF SELECTED PROBLEMS
  • 6 Confidence and Tolerance Intervals
  • PART I: THEORY
  • 6.1 GENERAL INTRODUCTION
  • 6.2 THE CONSTRUCTION OF CONFIDENCE INTERVALS
  • 6.3 OPTIMAL CONFIDENCE INTERVALS
  • 6.4 TOLERANCE INTERVALS
  • 6.5 DISTRIBUTION FREE CONFIDENCE AND TOLERANCE INTERVALS
  • 6.6 SIMULTANEOUS CONFIDENCE INTERVALS
  • 6.7 TWO-STAGE AND SEQUENTIAL SAMPLING FOR FIXED WIDTH CONFIDENCE INTERVALS
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTION TO SELECTED PROBLEMS
  • 7 Large Sample Theory for Estimation and Testing
  • PART I: THEORY
  • 7.1 CONSISTENCY OF ESTIMATORS AND TESTS
  • 7.2 CONSISTENCY OF THE MLE
  • 7.3 ASYMPTOTIC NORMALITY AND EFFICIENCY OF CONSISTENT ESTIMATORS
  • 7.4 SECOND-ORDER EFFICIENCY OF BAN ESTIMATORS
  • 7.5 LARGE SAMPLE CONFIDENCE INTERVALS
  • 7.6 EDGEWORTH AND SADDLEPOINT APPROXIMATIONS TO THE DISTRIBUTION OF THE MLE: ONE-PARAMETER CANONICAL EXPONENTIAL FAMILIES
  • 7.7 LARGE SAMPLE TESTS
  • 7.8 PITMAN'S ASYMPTOTIC EFFICIENCY OF TESTS
  • 7.9 ASYMPTOTIC PROPERTIES OF SAMPLE QUANTILES
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTION OF SELECTED PROBLEMS
  • 8 Bayesian Analysis in Testing and Estimation
  • PART I: THEORY
  • 8.1 THE BAYESIAN FRAMEWORK
  • 8.1.1 Prior, Posterior, and Predictive Distributions
  • 8.1.2 Noninformative and Improper Prior Distributions
  • 8.1.3 Risk Functions and Bayes Procedures
  • 8.2 BAYESIAN TESTING OF HYPOTHESIS
  • 8.2.1 Testing Simple Hypothesis.
  • 8.2.2 Testing Composite Hypotheses
  • 8.2.3 Bayes Sequential Testing of Hypotheses
  • 8.3 BAYESIAN CREDIBILITY AND PREDICTION INTERVALS
  • 8.3.1 Credibility Intervals
  • 8.3.2 Prediction Intervals
  • 8.4 BAYESIAN ESTIMATION
  • 8.4.1 General Discussion and Examples
  • 8.4.2 Hierarchical Models
  • 8.4.3 The Normal Dynamic Linear Model
  • 8.5 APPROXIMATION METHODS
  • 8.5.1 Analytical Approximations
  • 8.5.2 Numerical Approximations
  • 8.6 EMPIRICAL BAYES ESTIMATORS
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS OF SELECTED PROBLEMS
  • 9 Advanced Topics in Estimation Theory
  • PART I: THEORY
  • 9.1 MINIMAX ESTIMATORS
  • 9.2 MINIMUM RISK EQUIVARIANT, BAYES EQUIVARIANT, AND STRUCTURAL ESTIMATORS
  • 9.2.1 Formal Bayes Estimators for Invariant Priors
  • 9.2.2 Equivariant Estimators Based on Structural Distributions
  • 9.3 THE ADMISSIBILITY OF ESTIMATORS
  • 9.3.1 Some Basic Results
  • 9.3.2 The Inadmissibility of Some Commonly Used Estimators
  • 9.3.3 Minimax and Admissible Estimators of the Location Parameter
  • 9.3.4 The Relationship of Empirical Bayes and Stein-Type Estimators of the Location Parameter in the Normal Case
  • PART II: EXAMPLES
  • PART III: PROBLEMS
  • PART IV: SOLUTIONS OF SELECTED PROBLEMS
  • References
  • Author Index
  • Subject Index
  • WILEY SERIES IN PROBABILITY AND STATISTICS.

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