Derivatives of the Incomplete Beta Function

The incomplete beta function is defined as where Beta(p, q) is the beta function. Dutka (1981) gave a history of the development and numerical evaluation of this function. In this article, an algorithm for computing first and second derivatives of Ix,p,q with respect to p and q is described. The algorithm is useful, for example, when fitting parameters to a censored beta, truncated beta, or a truncated beta-binomial model.

Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations

This article develops confidence interval procedures for functions of simple, partial, and squared multiple correlation coefficients. It is assumed that the observed multivariate data represent a random sample from a distribution that possesses infinite moments, but there is no requirement that the distribution be normal. The coverage error of conventional one-sided large sample intervals decreases at rate 1√n as n increases, where n is an index of sample size. The coverage error of the proposed intervals decreases at rate 1/n as n increases. The results of a simulation study that evaluates the performance of the proposed intervals is reported and the intervals are illustrated on a real data set

Derivatives of the Incomplete Beta Function

The incomplete beta function is defined as where Beta(p, q) is the beta function. Dutka (1981) gave a history of the development and numerical evaluation of this function. In this article, an algorithm for computing first and second derivatives of Ix,p,q with respect to p and q is described. The algorithm is useful, for example, when fitting parameters to a censored beta, truncated beta, or a truncated beta-binomial model

Taking Ownership: Our Pledge to Educate All of Detroit's Children

Excellent Schools Detroit represents a broad and diverse cross section of Detroit's education, government, civic and community, parent, organized labor, and philanthropic leaders who are committed to ensuring that all Detroit children receive the great education they deserve. This citywide education plan reflects months of discussions and deliberations by coalition members, as well as a series of six community meetings in November and December, youth focus groups, small group discussions with multiple stakeholders, and other outreach efforts. We appreciate the thoughtful recommendations from the many Detroiters who are as passionate as we are about the need to prepare all students for college, careers, and life in the 21st century

A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications

AbstractThis article describes a local parameterization of orthogonal and semi-orthogonal matrices. The parameterization leads to a unified approach for obtaining the asymptotic joint distributions of estimators of singular-values and -vectors, and of eigen-values and -vectors. The singular- or eigen-values can have arbitrary multiplicities. The approach is illustrated on principal components analyzes, canonical correlation analysis, inter-battery factory analysis, and reduced-rank regression

The mixed model for multivariate repeated measures: validity conditions and an approximate test

likelihood ratio test, matrix quadratic form, multivariate sphericity, Wishart distribution,