Orthogonal polynomials,repeated measures,and SPSS

We discuss the construction of discrete orthonormal polynomials, using MAPLE procedures. We also study two important applications of these polynomials in statistics: in multiple linear regression and in repeated measures analysis. In particular, it is argued that the tests given by SPSS for linear and other trends in a within-subject factor are inefficient. Examples are given, including two (from psychology and medicine, respectively) which involve repeated measures and SPSS. Extensive tables of discrete orthornormal polynomials are given in the Appendix.     

Discrete random vectors generated by Taylor expansions

We demonstrate how univariate discrete and multivariate discrete distributions can be generated using Taylor expansions. Some of the results involve use of Bell polynomials.     

Book Reviews

Books reviewed:
J.K. Lindsey, Introductory Statistics: A Modelling Approach
D.R. Brillinger, L.T. Fernholz & S. Morgenthaler (eds), The Practice of Data Analysis: Essays in Honor of John W. Tukey
Thomas P. Ryan, Modern Regression Methods
Samuel Kotz & Cynthia R. Lovelace, Process Capability Indices in Theory and Practice
David A. Harville, Matrix Algebra from a Statistician's Perspective
Steven G. Krantz, A Primer of Mathematical Writing
Nicholas J. Higham, Handbook of Writing for the Mathematical Sciences     

Bayesian regression on non-parametric mixed-effect models with shape-restricted Bernstein polynomials

We develop a Bayesian estimation method to non-parametric mixed-effect models under shape-constrains. The approach uses a hierarchical Bayesian framework and characterizations of shape-constrained Bernstein polynomials (BPs). We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of BPs to ensure the desired shape constraints. The small sample properties of the Bayesian shape-constrained estimators across a range of functions are provided via simulation studies. Two real data analysis are given to illustrate the application of the proposed method.     

Distribution approximation and modelling via orthogonal polynomial sequences

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.     

Simulation‐based sample‐sizing and power calculations in logistic regression with partial prior information

There have been many approximations developed for sample sizing of a logistic regression model with a single normally‐distributed stimulus. Despite this, it has been recognised that there is no consensus as to the best method. In pharmaceutical drug development, simulation provides a powerful tool to characterise the operating characteristics of complex adaptive designs and is an ideal method for determining the sample size for such a problem. In this paper, we address some issues associated with applying simulation to determine the sample size for a given power in the context of logistic regression. These include efficient methods for evaluating the convolution of a logistic function and a normal density and an efficient heuristic approach to searching for the appropriate sample size. We illustrate our approach with three case studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Derrick Bell,CRT, and educational leadership 1995–present

Critical Race Theory (CRT) has become a centered conceptual framework to understand American education and reform (Ladson-Billings and Tate 1995; Solorzano and Yosso; 2001; Decuir and Dixon 2004). Indeed, educational leadership scholars have not been far behind in recognizing the explicative and powerful role of CRT studies in their work (Lopez 2003; Parker and Villalpando 2007). As we acknowledge the role of CRT, we cannot do so without reflecting on the life and works of the quintessential Critical Legal Studies (CLS) scholar Derrick Bell (1930–2011). In this article, we use Bell’s collective works to analyze current trends and research in educational leadership. We bring his works into conversation not only with conceptions of instructional and distributed leadership, but with the palpability that CRT has on the current state of educational reform. More specifically, we use Bell’s theories of interest convergence and conversations around ‘racial remedies’ to understand two recent trends in educational leadership: discourses of social justice leadership and the move toward data-driven leadership behaviors. We ask questions like: what has been the impact of research discourses social justice on the education of African American and Latino urban youth? And, how has the current social structures benefited from such discourses? We conclude with recommendations for educational leadership researchers and professors, and encourage them to consider race as an integral part of their works.     

近代西方世界兴起原因的再思考(下)——从斯密、黑格尔、马克思、桑巴特、韦伯、熊彼特到诺思、肯尼迪和华勒斯坦

本文对近代一些思想家对西方世界兴起的复杂社会原因的探索做了多学科的综合文献回顾。从而发现,依据亚当.斯密的分工和市场扩展理论,黑格尔与马克思等古典学者认为西方世界近代兴起的主要原因来源于分工和自由竞争所带来的劳动生产率的提高;桑巴特、韦伯和熊彼特则把近代西方世界的快速经济增长归结为企业家的追求和企业无限扩张的资本主义精神;经济史学家诺思认为近代西方世界兴起的根本原因在于私有产权制度和现代经济组织;政治学家保尔.肯尼迪则把这一历史现象归结为欧洲近代历史上的政治多元化和历史机缘;而华勒斯坦等历史学家则从非决定论的理论视角讨论了近代欧洲兴起的天时地利。综观西方学术诸家对西方世界兴起原因的解说可以发现,尽管在这个复杂的问题上存在诸多歧见,但保护产权的法律制度以及支撑法律制度工作的现代民族国家和宪政民主政治的生成是西方世界近代兴起的基本原因之一,这一点乃是诸家的理论共识。     

二元Kantorovi多项式在L~p空间的Lipschitz性质

研究了二元Kantorovic多项式在L~p空间的Lipschitz性质,证明该多项式与函数属于同一Lipschltz类.