Moments Of Random Vectors Research Articles

Moments of random vectors with skew t distribution and their quadratic forms

Moments of skew t random vectors and their quadratic forms are derived. It is shown that the moments of the sample autocovariance function and of the sample variogram estimator depend on a measure of multivariate kurtosis, but not on a skewness parameter.

Moments of Random Vectors Which Belong to Some Domain of Normal Attraction

Let $X$ be a random vector on $\mathbb{R}^k$ whose distribution $\mu$ belongs to the domain of normal attraction of some operator stable law $\nu$. For a given $\nu$ it has been shown elsewhere that for certain ranges of $\alpha$ depending on $\nu$, either $E|\langle X, \theta\rangle|^\alpha$ is finite for every $\theta \neq 0$ or is infinite for every $\theta \neq 0$. In this paper we show that the set of $\alpha$ for which $E|\langle X, \theta\rangle|^\alpha$ exists depends, in general, on both $\theta$ and $\nu$, and we obtain a complete description of the cases in which $E|\langle X, \theta\rangle|^\alpha$ can be guaranteed either to exist or to diverge, just on the basis of $\theta$ and $\nu$.

Mathematical Statistics: Basic Ideas and Selected Topics.

(NOTE: Each chapter concludes with Problems and Complements, Notes, and References.) 1. Statistical Models, Goals, and Performance Criteria. Data, Models, Parameters, and Statistics. Bayesian Models. The Decision Theoretic Framework. Prediction. Sufficiency. Exponential Families. 2. Methods of Estimation. Basic Heuristics of Estimation. Minimum Contrast Estimates and Estimating Equations. Maximum Likelihood in Multiparameter Exponential Families. Algorithmic Issues. 3. Measures of Performance. Introduction. Bayes Procedures. Minimax Procedures. Unbiased Estimation and Risk Inequalities. Nondecision Theoretic Criteria. 4. Testing and Confidence Regions. Introduction. Choosing a Test Statistic: The Neyman-Pearson Lemma. Uniformly Most Powerful Tests and Monotone Likelihood Ratio Models. Confidence Bounds, Intervals and Regions. The Duality between Confidence Regions and Tests. Uniformly Most Accurate Confidence Bounds. Frequentist and Bayesian Formulations. Prediction Intervals. Likelihood Ratio Procedures. 5. Asymptotic Approximations. Introduction: The Meaning and Uses of Asymptotics. Consistency. First- and Higher-Order Asymptotics: The Delta Method with Applications. Asymptotic Theory in One Dimension. Asymptotic Behavior and Optimality of the Posterior Distribution. 6. Inference in the Multiparameter Case. Inference for Gaussian Linear Models. Asymptotic Estimation Theory in p Dimensions. Large Sample Tests and Confidence Regions. Large Sample Methods for Discrete Data. Generalized Linear Models. Robustness Properties and Semiparametric Models. Appendix A: A Review of Basic Probability Theory. The Basic Model. Elementary Properties of Probability Models. Discrete Probability Models. Conditional Probability and Independence. Compound Experiments. Bernoulli and Multinomial Trials, Sampling with and without Replacement. Probabilities on Euclidean Space. Random Variables and Vectors: Transformations. Independence of Random Variables and Vectors. The Expectation of a Random Variable. Moments. Moment and Cumulant Generating Functions. Some Classical Discrete and Continuous Distributions. Modes of Convergence of Random Variables and Limit Theorems. Further Limit Theorems and Inequalities. Poisson Process. Appendix B: Additional Topics in Probability and Analysis. Conditioning by a Random Variable or Vector. Distribution Theory for Transformations of Random Vectors. Distribution Theory for Samples from a Normal Population. The Bivariate Normal Distribution. Moments of Random Vectors and Matrices. The Multivariate Normal Distribution. Convergence for Random Vectors: Op and Op Notation. Multivariate Calculus. Convexity and Inequalities. Topics in Matrix Theory and Elementary Hilbert Space Theory. Appendix C: Tables. The Standard Normal Distribution. Auxiliary Table of the Standard Normal Distribution. t Distribution Critical Values. X 2 Distribution Critical Values. F Distribution Critical Values. Index.