Chapter Notes

Tests for redundancy of variables in linear two-group discriminant analysis are well known and frequently used. We give a survey of similar tests, including the one-sample T ² as a special case, in the situation in which only the mean vector (but no covariance matrix) is available in one sample. Then we show that a relation between linear regression and discriminant functions found by Fisher (1936) can be generalized to this situation. Relating regression and discriminant analysis to a multivariate linear model sheds more light on the relationship between them. Practical and didactical advantages of the regression approach to T ² tests and discriminant analysis are outlined.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

A small-sample correction factor for S-estimators

S-estimators are frequently used as robust estimators of regression and of location and dispersion. Under certain differentiability conditions, S-estimators of multivariate location and dispersion parameters are consistent [Davies PL. Asymtotic behaviour of S-estimators of multivariate location parameters and dispersion matrices. Ann Stat. 1987;15(3):1269–1292]. However, it has been observed that the S-estimators of dispersion parameters give biased results in the case of small-sample data sets. In this work, we constructed formulas based on simulation studies, which allow us to compute small-sample correction factors for all sample sizes and dimensions for S-estimators of dispersion parameters without having to carry out any new simulations. We considered real data to illustrate the effects of the small-sample correction factor.     

An iterative approach to distance correlation-based sure independence screening

Feature screening and variable selection are fundamental in analysis of ultrahigh-dimensional data, which are being collected in diverse scientific fields at relatively low cost. Distance correlation-based sure independence screening (DC-SIS) has been proposed to perform feature screening for ultrahigh-dimensional data. The DC-SIS possesses sure screening property and filters out unimportant predictors in a model-free manner. Like all independence screening methods, however, it fails to detect the truly important predictors which are marginally independent of the response variable due to correlations among predictors. When there are many irrelevant predictors which are highly correlated with some strongly active predictors, the independence screening may miss other active predictors with relatively weak marginal signals. To improve the performance of DC-SIS, we introduce an effective iterative procedure based on distance correlation to detect all truly important predictors and potentially interactions in both linear and nonlinear models. Thus, the proposed iterative method possesses the favourable model-free and robust properties. We further illustrate its excellent finite-sample performance through comprehensive simulation studies and an empirical analysis of the rat eye expression data set.     

Detection of non-Gaussianity

We develop two tests sensitive to various departures from composite goodness-of-fit hypothesis of normality. The tests are based on the sums of squares of some components naturally arising in decomposition of the Shapiro–Wilk-type statistic. Each component itself has diagnostic properties. The numbers of squared components in sums are determined via some novel selection rules based on the data. The new solutions prove to be effective tools in detecting a broad spectrum of sources of non-Gaussianity. We also discuss two variants of the new tests adjusted to verification of simple goodness-of-fit hypothesis of normality. These variants also compare well to popular competitors.     

Estimating Archimedean Copulas in High Dimensions

Abstract. This article presents a novel estimation procedure for high‐dimensional Archimedean copulas. In contrast to maximum likelihood estimation, the method presented here does not require derivatives of the Archimedean generator. This is computationally advantageous for high‐dimensional Archimedean copulas in which higher‐order derivatives are needed but are often difficult to obtain. Our procedure is based on a parameter‐dependent transformation of the underlying random variables to a one‐dimensional distribution where a minimum‐distance method is applied. We show strong consistency of the resulting minimum‐distance estimators to the case of known margins as well as to the case of unknown margins when pseudo‐observations are used. Moreover, we conduct a simulation comparing the performance of the proposed estimation procedure with the well‐known maximum likelihood approach according to bias and standard deviation.     

Projection array based designs for computer experiments

In this paper we define a new class of designs for computer experiments. A projection array based design defines sets of simulation runs with properties that extend the conceptual properties of orthogonal array based Latin hypercube sampling, particularly to underlying design structures other than orthogonal arrays. Additionally, we illustrate how these designs can be sequentially augmented to improve the overall projection properties of the initial design or focus on interesting regions of the design space that need further exploration to improve the overall fit of the underlying response surface. We also illustrate how an initial Latin hypercube sample can be expressed as a projection array based design and show how one can augment these designs to improve higher dimensional space filling properties.     

环境规制对中国油料作物产出影响的研究——基于距离函数对技术效率的测度

在综述环境规制对技术效率影响的国内外文献基础上,通过距离函数测算了2001-2010年环境规制对中国东部、中部和西部地区四种油料作物的影响,从无规制、宽松规制及严格规制这三个层次揭示了环境规制对不同区域的大豆、油菜、花生及芝麻在产出技术效率和生产力发挥方面的差异化影响。研究表明:对于所研究的绝大多数地区,环境规制水平的提高能够提升该地区油料作物的产出技术效率,并有利于更大程度地发挥潜在生产力;而在极少数地区,较高的环境规制水平对技术效率的发挥已经起到了微弱的阻碍作用。最后,阐释了同一环境规制对不同产业产生迥异影响的普遍规律,论证了较为完善的环境规制应该是做为一个体系而存在的,为本国制定与完善环境规制提供了一定的参考。     

基于主成分和支持向量机模型在人口预测中的应用

人口问题是一个复杂、受诸多因素影响的社会问题,准确地预测人口数量对于制定国民经济计划和社会发展战略有着深远的意义。针对神经网络预测算法易陷入局部极小、输入数据具有较强相关性的问题,建立了一种基于主元分析(PCA)与支持向量机(SVM)相结合的人口预测模型,并对我国人口总量进行预测。结果表明,该模型比单变量GM(1,1)和BP神经网络模型预测精度有了明显提高。