An efficient algorithm for Harrison-Stevens forecasting using the multi-process multivariate dynamic linear model

This paper develops a computationally efficient algorithm for Harrison-Stevens forecasting in a multivariate time series which has correlated errors. The algorithm uses the observation vector one component at a time on the multiprocess multivariate dynamic linear model. This gives a computationally efficient, robust, quick adapting forecasting method for non stationary multivariate time series.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

METHODS FOR ESTIMATING PRINCIPAL POINTS

ABSTRACT

Principal points of a distribution have been introduced by Flury [1] Flury, D.B. 1990. Principal Points. Biometrika, 77: 33–41. [Crossref], [Web of Science ®] , [Google Scholar] who tackled the problem of optimal grouping in multivariate data. In essence, principal points are the theoretical counterparts of cluster means obtained by a k-means clustering algorithm. There has been considerable effort to find efficient estimation procedures for principal points. It is well known that under certain conditions the k-means estimator is a consistent and asymptotically normal estimator of the population principal points. In this paper some material on principal points is reviewed and new algorithms for the estimation of principal points in univariate distributions (univariate principal points) are proposed. Additionally, the Bootstrap approach is applied to assess the variability of the suggested estimators.     

ASYMPTOTIC DISTRIBUTION OF THE LARGEST EIGENVALUE

ABSTRACT

This paper studies the asymptotic distribution of the largest eigenvalue of the sample covariance matrix. The multivariate distribution for the population is assumed to be elliptical with finite kurtosis 3κ. An expression as an expectation is obtained for the distribution function of the largest eigenvalue regardless of the multiplicity, m, of the population's largest eigenvalue. The asymptotic distribution function and density function are evaluated numerically for m = 2,3,4,5. The bootstrap of the average of the m largest eigenvalues is shown to be consistent for any underlying distribution with finite fourth-order cumulants.     

The extended generalized lambda distribution system for fitting distributions to data: history,completion of theory,tables, applications,the “final word” on moment fits

The generalized lambda distribution, GLD(λ₁, λ₂ λ₃, λ₄), is a four-parameter family that has been used for fitting distributions to a wide variety of data sets. The analysis of the λ₃ and λ₄ values that actually yield valid distributions has (until now) been incomplete. Moreover, because of computational problems and theoretical shortcomings, the moment space over which the GLD can be applied has been limited. This paper completes the analysis of the λ₃ and λ₄ values that are associated with valid distributions, improves previous computational methods to reduce errors associated with fitting data, expands the parameter space over which the GLD can be used, and uses a four-parameter generalized beta distribution to cover the portion of the parameter space where the GLD is not applicable. In short, the paper extends the GLD to an EGLD system that can be used for fitting distributions to data sets that that are cited in the literature as actually occurring in practice. Examples of use of the proposed system are included     

General multivariate linear models for longitudional studies

Longitudinal studies occcur frequently in many different disciplines. To fully utilize the potential value of the information contained in a longitudinal data, various multivariate linear models have been proposed. The methodology and analysis are somewhat unique in their own ways and their relationships are not well understood and presented. This article describes a general multivaritate linear model for longitudinal data and attempts to provide a constructive formulation of the components in the mean response profile. The objective is to point out the extension and connections of some well-known models that have been obscured by different areas of application. More imporiantly, the model is expressed in a unified regression form from the subject matter considerations. Such an approach is simpler and more intuitive than other ways to modeling and parameter estimation. As a cmsequeace the analyses of the general class cf models for longitudional data can be casily implemented with standard software.     

A family of bivariate distributions with non-elliptical contours

In this paper, a family of copulas with two parameters is proposed and its dependence analysis is performed. The corresponding family of bivariate distributions with specified marginals is constructed. For normal marginals, the new distributions are non-elliptical and can be applied in data analysis. They provide various alternative hypotheses for testing normality. Finally, an example is given.     

Bayes Predictor of One-Parameter Exponential Family Type Population Mean Under Balanced Loss Function

We obtain a Bayes predictor and a Bayes prediction risk of the mean of a finite population relative to the balanced loss function. The predictive expected losses associated with classical and standard Bayes predictors are derived and compared with that of a Bayes predictor under a balanced loss function. Specific expressions for a regular exponential family distributed superpopulation are presented and illustrated for some well-known superpopulations.     

Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.