An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)_s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)₁₂ process, is approximated by the moment approximation technique.     

On tests in variance components models 2

A procedure for finding exact tests for some hypotheses on variance components in unbalanced models is proposed. It is based on F-distributed statistics got by an orthogonal decomposition of the sample space.     

Independence distribution preserving joint covariance structures for the multivariate two-group case

We characterize the general nonnegative-definite and positive-definite joint observation covariance structures for the two-group case such that the two sample mean vectors are independent of the two corresponding sample covariance matrices. Also, the sample covariance matrices are distributed as independent noncentral or central Wishart random matrices. We derive and utilize a representation of the general common non-negative-definite solution to a particular system of matrix equations with idempotent coefficient matrices.     

On the independence between two generalized second degree polynomial statistics and their covariance

The condition of independence between two generalized second degree polynomial statistics is worked out in this paper under the assumption that the covariance matrix of the underlying normal distribution may be singular. The covariance between two such forms is also worked out without the assumption of normality. The results are indicated in compact matrix form.     

A History of the Development of Craig's Theorem

Craig's theorem on the independence of quadratic forms in normal variates is traced from its first form, for iid standard normal variates, to the form for variates following an arbitrary nonsingular joint normal distribution. This article gives the main thrust of the development and makes recommendations on coverage of the theorem in courses and textbooks. The history of Craig's theorem is not a happy one. The authors of the earlier articles in the literature tended to make errors of a linear-algebraic nature. Authors of more recently published textbooks have given incorrect or misleadingly incomplete coverage of Craig's theorem and its proof.     

A note on general quadratic forms of nonstationary stochastic processes

In this paper, general quadratic forms of nonstationary, α-mixing time series are considered. Under mixing and moment assumptions, asymptotically normality of these forms are derived. These results do not assume that the variance of the generalized quadratic form has a limit, thus allowing for general types of nonstationarity. However, without well-defined limits, it is not possible to understand the differences in sampling properties of quadratic forms of nonstationary and stationary processes. To understand these differences, the nonstationary process is placed within the locally stationary framework. Under the assumption that the nonstationary process is locally stationary the asymptotic expectation and variance of the weighted sample covariance of the discrete Fourier transforms (an important class of quadratic forms) is derived and shown to be very different to its stationary counterpart.     

Dirichlet forms: Some infinite-dimensional examples

The author presents three examples of a Markov process taking values in an infinite-dimensional state space and analyzes the sample path behaviour using the theory of Dirichlet forms.     

Statistical applications of the multivariate skew normal distribution

Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.     

A Scale-space Approach for Detecting Non-stationarities in Time Series

Abstract.  The presented method called Significant Non-stationarities, represents an exploratory tool for identifying significant changes in the mean, the variance, and the first-lag autocorrelation coefficient of a time series. The changes are detected on different time scales. The statistical inference for each scale is based on accurate approximation of the probability distribution, using test statistics being ratios of quadratic forms. No assumptions concerning the autocovariance function of the time series are made as the dependence structure is estimated non-parametrically. The results of the analyses are summarized in significance maps showing at which time points and on which time scales significant changes in the parameters occur. The performance of the given method is thoroughly studied by simulations in terms of observed significance level and power. Several examples, including a real temperature data set, are studied. The examples illustrate that it is important to carry out the analysis on several time horizons.     

MRPII中能力需求计划的分析与实现

能力需求计划部分是MRPII系统中联系主生产计划、物料需求计划和车间任务管理的重要中间环节。文章探讨了利用“倒排”和“正排”方法相结合实现能力需求计划的原理和思路 ,并且提供了用VisualBasic和SQL实现报表及图表的部分算法。