紹介
Mathematical statistics typically represents one of the most difficult challenges in statistics, particularly for those with more applied, rather than mathematical, interests and backgrounds. Most textbooks on the subject provide little or no review of the advanced calculus topics upon which much of mathematical statistics relies and furthermore contain material that is wholly theoretical, thus presenting even greater challenges to those interested in applying advanced statistics to a specific area. Mathematical Statistics with Applications presents the background concepts and builds the technical sophistication needed to move on to more advanced studies in multivariate analysis, decision theory, stochastic processes, or computational statistics. Applications embedded within theoretical discussions clearly demonstrate the utility of the theory in a useful and relevant field of application and allow readers to avoid sudden exposure to purely theoretical materials.
With its clear explanations and more than usual emphasis on applications and computation, this text reaches out to the many students and professionals more interested in the practical use of statistics to enrich their work in areas such as communications, computer science, economics, astronomy, and public health.
目次
INTRODUCTION REVIEW OF MATHEMATICS Introduction Combinatorics Pascal's Triangle Newton's Binomial Formula Exponential Function Stirling's Formula Multinomial Theorem Monotonic Functions Convergence and Divergence Taylor's Theorem Differentiation and Summation Some Properties of Integration Integration by Parts Region of Feasibility Multiple Integration Jacobian Maxima and Minima Lagrange Multiplier L'Hopital's Rule Partial Fraction Expansion Cauchy-Schwarz Inequality Generating Functions Difference Equations Vectors, Matrices and Determinants Real Numbers PROBABILITY THEORY Introduction Subjective Probability, Relative Frequency and Empirical Probability Sample Space Decomposition of a Union of Events: Disjoint Events Sigma Algebra and Probability Space Rules and Axioms of Probability Theory Conditional Probability Law of Total Probability Bayes Rule Sampling With and Without Replacement Probability and SIMULATION Borel Sets Measure Theory in Probability Application of Probability Theory: Decision Analysis RANDOM VARIABLES Introduction Discrete Random Variables Cumulative Distribution Function Continuous Random Variables Joint Distributions Independent Random Variables Distribution of the Sum of Two Independent Random Variables Moments, Expected Values and Variance Covariance and Correlation Distribution of a Function of a Random Variable Multivariate Distributions and Marginal Densities Conditional Expectations Conditional Variance and Covariance Moment Generating Functions Characteristic Functions Probability Generating Functions DISCRETE DISTRIBUTIONS Introduction Bernoulli Distribution Binomial Distribution Multinomial Distribution Hypergeometric Distribution k-Variate Hypergeometric Distribution Geometric Distribution Negative Binomial Distribution Negative Multinomial Distribution Poisson Distribution Discrete Uniform Distribution Lesser Known Distributions Joint Distributions Convolutions Compound Distributions Branching Processes Hierarchical Distributions CONTINUOUS RANDOM VARIABLES Location and Scale Parameters Distribution of Functions of Random Variables Uniform Distribution Normal Distribution Exponential Distribution Poisson Process Gamma Distribution Beta Distribution Chi-square Distribution Student's t-Distribution F-Distribution Cauchy Distribution Exponential Family Hierarchical Models-Mixture Distributions Other Distributions Distributional Relationships Additional Distributional Findings DISTRIBUTIONS OF ORDER STATISTICS Introduction Rank Ordering The Probability Integral Transformation Distributions of Order Statistics in i.i.d. Samples Expectations of Minimum and Maximum Order Statistics Distributions of Single Order Statistics Joint Distributions of Order Statistics ASYMPTOTIC DISTRIBUTION THEORY Introduction Introducing Probability to the Limit Process Introduction to Convergence in Distribution Non-convergence Introduction to Convergence in Probability Convergence Almost Surely (with Probability One) Convergence in rth Mean Relationships Between Convergence Modalities Application of Convergence in Distribution Properties of Convergence in Probability The Law of Large Numbers and Chebyshev's Inequality The Central Limit Theorem Proof of the Central Limit Theorem The Delta Method Convergence Almost Surely (with probability one) POINT ESTIMATION Introduction Method of Moments Estimators Maximum Likelihood Estimators Bayes Estimators Sufficient Statistics Exponential Families Other Estimators* Criteria of a Good Point Estimator HYPOTHESIS TESTING Statistical Reasoning and Hypothesis Testing Discovery, the Scientific Method, and Statistical Hypothesis Testing Simple Hypothesis Testing Statistical Significance The Two Sample Test Two Sided vs. One Sided Testing Likelihood Ratios and the Neyman Pearson Lemma One SampleTesting and the Normal Distribution Two Sample Testing for the Normal Distribution Likelihood Ratio Test and the Binomial Distribution Likelihood Ratio Test and the Poisson Distribution The Multiple Testing Issue Nonparametric Testing Goodness of Fit Testing Fisher's Exact Test Sample Size Computations INTERVAL ESTIMATION Introduction Definition Constructing Confidence Intervals Bayesian Credible Intervals Approximate Confidence Intervals and MLE Pivot The Bootstrap Method* Criteria of a Good Interval Estimator Confidence Intervals and Hypothesis Tests INTRODUCTION TO COMPUTATIONAL METHODS The Newton-Raphson Method The EM Algorithm Simulation Markov Chains Markov Chain Monte Carlo Methods INDEX
