Nenparametric Two-Group Classification: Concepts and a SAS-Based Software Package

This article introduces BestClass, a set of SAS macros, available in the mainframe and workstation environment, designed for solving two-group classification problems using a class of recently developed nonparametric classification methods. The criteria used to estimate the classification function are based on either minimizing a function of the absolute deviations from the surface which separates the groups, or directly minimizing a function of the number of misclassified entities in the training sample. The solution techniques used by BestClass to estimate the classification rule use the mathematical programming routines of the SAS/OR software. Recently, a number of research studies have reported that under certain data conditions this class of classification methods can provide more accurate classification results than existing methods, such as Fisher's linear discriminant function and logistic regression. However, these robust classification methods have not yet been implemented in the major statistical packages, and hence are beyond the reach of those statistical analysts who are unfamiliar with mathematical programming techniques. We use a limited simulation experiment and an example to compare and contrast properties of the methods included in Best-Class with existing parametric and nonparametric methods. We believe that BestClass contributes significantly to the field of nonparametric classification analysis, in that it provides the statistical community with convenient access to this recently developed class of methods. BestClass is available from the authors.     

Functional sliced inverse regression analysis

Most of the usual multivariate methods have been extended to the context of functional data analysis. Our contribution concerns the study of sliced inverse regression (SIR) when the response variable is real but the regressor is a function. In the first part, we show how the relevant properties of SIR remain essentially the same in the functional context under suitable conditions. Unfortunately, the estimation procedure used in the multivariate case cannot be directly transposed to the functional one. Then, we propose a solution that overcomes this difficulty and we show the consistency of the estimates of the parameters of the model.     

On proximity between PCA in the frequency domain and usual PCA

The principal components analysis (PCA) in the frequency domain of a stationary p-dimensional time series (X _n ) _n∈? leads to a summarizing time series written as a linear combination series X′ _n =∑ _m C _m ° X _n?m . Therefore, we observe that, when the coefficients C _m , m≠0, are close to 0, this PCA is close to the usual PCA, that is the PCA in the temporal domain. When the coefficients tend to 0, the corresponding limit is said to satisfy a property noted 𝒫, of which we will study the consequences. Finally, we will examine, for any series, the proximity between the two PCAs.     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

Kullback–Leibler information of a censored variable and its applications

In this paper, we study the Kullback–Leibler (KL) information of a censored variable, which we will simply call it censored KL information. The censored KL information is shown to have the necessary monotonicity property in addition to inherent properties of nonnegativity and characterization. We also present a representation of the censored KL information in terms of the relative risk and study its relation with the Fisher information in censored data. Finally, we evaluate the estimated censored KL information as a goodness-of-fit test statistic.     

R–estimation of normed bivariate density functions

Normed bivariate density funtions were introduced by HOEFFDING (1940/41). In the present paper estimators for normed bivariate ranks and on a FOURIER series expansion in LEGENDRE polynomials. The estimation of normed bivarate density functions under positive dependence is also described     

A correlation-theoretical interpretation of schur analysis

This paper is heavily leaned on the author's recent investigations concerning SCHUR analysis of non-negative Hermitian block matrices. The parameters of the matrix balls and the triangular choice scheme which describe a non-negative Hermitian block matrix will be interpreted in the framework of correlation theory     

Simple Computation of Maximum Likelihood Estimates in Latent Class Model with Equality and Constant Constraints

A simple computational method for estimation of parameters via a type of EM algorithm is proposed in restricted latent class analysis, where equality and constant constraints are considered. These constraints create difficulty in estimation. In order to simply and stably estimate parameters in restricted latent class analysis, a simple computational method using only first-order differentials is proposed, where the step-halving method is adopted. A simulation study shows that in almost all cases the new method gives parameter sequences monotonously increasing the Q-function in the EM algorithm. Analysis of real data is provided.     

A Comparison of Hierarchical Methods for Clustering Functional Data

Functional data analysis (FDA)—the analysis of data that can be considered a set of observed continuous functions—is an increasingly common class of statistical analysis. One of the most widely used FDA methods is the cluster analysis of functional data; however, little work has been done to compare the performance of clustering methods on functional data. In this article, a simulation study compares the performance of four major hierarchical methods for clustering functional data. The simulated data varied in three ways: the nature of the signal functions (periodic, non periodic, or mixed), the amount of noise added to the signal functions, and the pattern of the true cluster sizes. The Rand index was used to compare the performance of each clustering method. As a secondary goal, clustering methods were also compared when the number of clusters has been misspecified. To illustrate the results, a real set of functional data was clustered where the true clustering structure is believed to be known. Comparing the clustering methods for the real data set confirmed the findings of the simulation. This study yields concrete suggestions to future researchers to determine the best method for clustering their functional data.     

Discriminant Analysis for the von Mises-Fisher Distribution

The von Mises-Fisher distribution is widely used for modeling directional data. In this article, we derive the discriminant rules based on this distribution to assign objects into pre-existing classes. We determine a distance between two von Mises-Fisher populations and we calculate estimates of the misclassification probabilities. We also analyze the behavior of the distance between two von Mises-Fisher populations and of the estimates of the misclassification probabilities when we modify the parameters of the populations or the samples size or the dimension of the sphere. Finally, we present an example with real spherical data available in the literature.