Comparison of combinatoric and likelihood ratio procedures for classifying samples

A random sample is to be classified as coming from one of two normally distributed populations with known parameters. Combinatoric procedures which classify the sample based upon the sample mean(s) and variance(s) are described for the univariate and multivariate problems. Comparisons of misclassification probabilities are made between the combinatoric and the likelihood ratio procedure in the univariate case and between two alternative combinatoric procedures in the bivariate case.     

Finite population corrections for multivariate Bayes sampling

We consider the adjustment, based upon a sample of size n, of collections of vectors drawn from either an infinite or finite population. The vectors may be judged to be either normally distributed or, more generally, second-order exchangeable. We develop the work of Goldstein and Wooff (1998) to show how the familiar univariate finite population corrections (FPCs) naturally generalise to individual quantities in the multivariate population. The types of information we gain by sampling are identified with the orthogonal canonical variable directions derived from a generalised eigenvalue problem. These canonical directions share the same co-ordinate representation for all sample sizes and, for equally defined individuals, all population sizes enabling simple comparisons between both the effects of different sample sizes and of different population sizes. We conclude by considering how the FPC is modified for multivariate cluster sampling with exchangeable clusters. In univariate two-stage cluster sampling, we may decompose the variance of the population mean into the sum of the variance of cluster means and the variance of the cluster members within clusters. The first term has a FPC relating to the sampling fraction of clusters, the second term has a FPC relating to the sampling fraction of cluster size. We illustrate how this generalises in the multivariate case. We decompose the variance into two terms: the first relating to multivariate finite population sampling of clusters and the second to multivariate finite population sampling within clusters. We solve two generalised eigenvalue problems to show how to generalise the univariate to the multivariate: each of the two FPCs attaches to one, and only one, of the two eigenbases.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

Combined multiple testing of multivariate survival times by censored empirical likelihood

In each study testing the survival experience of one or more populations, one must not only choose an appropriate class of tests, but further an appropriate weight function. As the optimal choice depends on the true shape of the hazard ratio, one is often not capable of getting the best results with respect to a specific dataset. For the univariate case several methods were proposed to conquer this problem. However, most of the interesting datasets contain multivariate observations nowadays. In this work we propose a multivariate version of a method based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. By considering the conditional hazards, we take the correlation between the components into account with the goal of obtaining a test that exhibits a high power irrespective of the shape of the hazard ratio under the alternative hypothesis.     

A new functional statistic for multivariate normality

The test statistics of assessing multivariate normality based on Roy’s union-intersection principle (Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1953) are generalizations of univariate normality, and are formed as the optimal value of a nonlinear multivariate function. Due to the difficulty of solving multivariate optimization problems, researchers have proposed various approximations. However, this paper shows that the (nearly) global solution contrarily results in unsatisfactory power performance in Monte Carlo simulations. Thus, instead of searching for a true optimal solution, this study proposes a functional statistic constructed by the q% quantile of the objective function values. A comparative Monte Carlo analysis shows that the proposed method is superior to two highly recommended tests when detecting widely-selected alternatives that characterize the various properties of multivariate normality.     

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.     

On algorithms for ordinary least squares regression spline fitting: A comparative study

Regression spline smoothing is a popular approach for conducting nonparametric regression. An important issue associated with it is the choice of a "theoretically best" set of knots. Different statistical model selection methods, such as Akaike's information criterion and generalized cross-validation, have been applied to derive different "theoretically best" sets of knots. Typically these best knot sets are defined implicitly as the optimizers of some objective functions. Hence another equally important issue concerning regression spline smoothing is how to optimize such objective functions. In this article different numerical algorithms that are designed for carrying out such optimization problems are compared by means of a simulation study. Both the univariate and bivariate smoothing settings will be considered. Based on the simulation results, recommendations for choosing a suitable optimization algorithm under various settings will be provided.     

Some new results on univariate and multivariate permutation tests for ordinal categorical variables under restricted alternatives

In several sciences, especially when dealing with performance evaluation, complex testing problems may arise due in particular to the presence of multidimensional categorical data. In such cases the application of nonparametric methods can represent a reasonable approach. In this paper, we consider the problem of testing whether a “treatment” is stochastically larger than a “control” when univariate and multivariate ordinal categorical data are present. We propose a solution based on the nonparametric combination of dependent permutation tests (Pesarin in Multivariate permutation test with application to biostatistics. Wiley, Chichester, 2001), on variable transformation, and on tests on moments. The solution requires the transformation of categorical response variables into numeric variables and the breaking up of the original problem’s hypotheses into partial sub-hypotheses regarding the moments of the transformed variables. This type of problem is considered to be almost impossible to analyze within likelihood ratio tests, especially in the multivariate case (Wang in J Am Stat Assoc 91:1676–1683, 1996). A comparative simulation study is also presented along with an application example.     

Tests of multinormality based on location vectors and scatter matrices

Classical univariate measures of asymmetry such as Pearson’s (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example.     

Multivariate trimmed means based on the Tukey depth

In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey's depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10.     

On multivariate quantile regression

To detect the dependence on the covariates in the lower and upper tails of the response distribution, regression quantiles are very useful tools in linear model problems with univariate response. We consider here a notion of regression quantiles for problems with multivariate responses. The approach is based on minimizing a loss function equivalent to that in the case of univariate response. To construct an affine equivariant notion of multivariate regression quantiles, we have considered a transformation retransformation procedure based on ‘data-driven coordinate systems’. We indicate some algorithm to compute the proposed estimates and establish asymptotic normality for them. We also, suggest an adaptive procedure to select the optimal data-driven coordinate system. We discuss the performance of our estimates with the help of a finite sample simulation study and to illustrate our methodology, we analyzed an interesting data-set on blood pressures of a group of women and another one on the dependence of sales performances on creative test scores.     

Pairwise testing of group mean vectors in MANOVA with small samples

Several, multivariate, pairwise, multiple comparison procedures are proposed as follow-ups for a significant multivariate analysis of variance. The Peritz procedure is generalized from univariate to several multivariate applications. Procedures are evaluated using overall power, any-pair power and all-pairs power applied to mean vectors with common sample sizes of 4, 5, and 9. Monte Carlo simulation demonstrated greater power than previously proposed univariate procedures in many conditions especially for all-pairs power. The multivariate Peritz procedure based on the Lawley–Hotelling trace was found to be most powerful in many conditions.     

Block unimodality for multivariate Bayesian robustness

Summary The development of Bayesian robustness has been growing in the last decade. The theory has extensively dealt with the univariate parameter case. Among the vast amount of proposals in the literature, only a few of them have a straightforward extension to the multivariate case. In this paper we consider the multidimensional version of the class of ε-contaminated prior distributions, with unimodal contaminations. In the multivariate case there is not a unique definition of unimodality and one's choice must be based on statistical ground. Here we propose the use of the block unimodal distributions, which proved to be very suitable for modelling situations where the coordinates of the parameter ϑ are deemed, a priori, weakly correlated.     

New multivariate aging notions based on the corrected orthant and the standard construction

ABSTRACT

Recently, some well-known univariate aging classes of lifetime distributions have been characterized by means of properties of their quantile functions and excess-wealth functions. The generalization of the univariate aging notions to the multivariate case involve, among other factors, appropriate definitions of multivariate quantiles or regression representation and related notions, which are able to correctly describe the intrinsic characteristic of the concepts of aging that should be generalized. The multivariate versions of these notions, which are characterized by using the multivariate u-quantiles and the multivariate excess-wealth function, are considered in this paper. Relationships between such multivariate aging classes are studied, and examples are provided.     

An R package and a study of methods for computing empirical likelihood

Empirical likelihood (EL) is an important nonparametric statistical methodology. We develop a package in R called el.convex to implement EL for inference about a multivariate mean. This package contains five functions which use different optimization algorithms but meanwhile seek the same goal. These functions are based on the theory of convex optimization; they are Newton, Davidon–Fletcher–Powell, Broyden–Fletcher–Goldfarb–Shanno, conjugate gradient method, and damped Newton, respectively. We also compare them with the function el.test in the existing R package emplik, and discuss their relative advantages and disadvantages.     

Characterizing and Approximating Infinite Scale Mixtures of Normals

ABSTRACT

The purpose of this study is to approximate and identify infinite scale mixtures of normals, SMN. A new method for approximating any infinite SMN with a known mixing measure by a finite SMN is presented. In the new method, the modulus of continuity of the normal family as a function of the scale is used to discretize the mixing measure. This method will be used to approximate univariate and multivariate SMN with mean 0. In the multivariate case, two different methods are used to approximate the infinite SMN. Several results related to SMN are proved and other known ones are presented. For example, SMN are characterized by their corresponding Laplace transforms.     

Diagnostics for identifying influential cases in manova

Measures of influence of multivariate cases on the estimated parameter matrix in MANOVA are developed. The development is based on the confidence region resulting from the likelihood ratio criterion, and is patterned after the development of Cook's univariate D_Imeasure. Influence measures corresponding to the other 3 common multivariate test criteria are presented. Relationships to the measures of Caroni (1987), and Barrett and Ling (1988) are noted. A numerical example is included.     

Testing for multivariate normality via univariate tests: A case study using lead isotope ratio data

Samples from ore bodies, mined for copper in antiquity, can be characterized by measurements on three lead isotope ratios. Given sufficient samples, it is possible to estimate the lead isotope field-a three-dimensional construct-that characterizes the ore body. For the purposes of estimating the extent of a field, or assessing whether bronze artefacts could have been made using copper from a particular field, it is often assumed that fields have a trivariate normal distribution. Using recently published data, for which the sample sizes are larger than usual, this paper casts doubt on this assumption. A variety of tests of univariate normality are applied, both to the original lead isotope ratios and to transformations of them based on principal component analysis; the paper can be read as a case study in the use of tests of univariate normality for assessing multivariate normality. This is not an optimal approach, but is sufficient in the cases considered to suggest that fields are, in fact, 'non-normal'. A direct test of multivariate normality confirms this. Some implications for the use of lead isotope ratio data in archaeology are discussed.     

Applications of empirical characteristic functions in some multivariate problems

In this note we extend univariate tests for normality and symmetry based on empirical characteristic functions to the multivariate case.     

Statistical inference for a general class of asymmetric distributions

We consider a general class of asymmetric univariate distributions depending on a real-valued parameter α, which includes the entire family of univariate symmetric distributions as a special case. We discuss the connections between our proposal and other families of skew distributions that have been studied in the statistical literature. A key element in the construction of such families of distributions is that they can be stochastically represented as the product of two independent random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes as well as extensions to the multivariate case. We also study statistical inference for this class based on the method of moments and maximum likelihood. We give special attention to the skew-power exponential distribution, but other cases like the skew-t distribution are also considered. Finally, the statistical methods are illustrated with 3 examples based on real datasets.