A Permutation and Combination-Based Solution on the Ranking of Multivariate Populations in the Case of Ordered Categorical Responses with Application to the Evaluation of the Indoor Environment

This article deals with the problem of classifying and ranking several multivariate populations of interests using the permutation and combination approach providing also an inferential validity of the procedure. The need to define an appropriate classification of populations, i.e., products, services, teaching courses, degree programs, and so on, is very common within many areas of applied research. Many times the populations of interest are multivariate in nature meaning that many aspects of that populations can be simultaneously observed on the same unit/subject. From a statistical point of view, when the response variable of interest is multivariate in nature, the problem may become quite difficult to cope with especially in case of ordered categorical responses, due to the large dimensionality of the parametric space. Nonparametric inference based on the NPC methodology however, allows us to overcome these limitations, without the need of referring to assume any specified random distribution.     

Multivariate normal estimation: the case (n < p)

Estimation in the multivariate context when the number of observations available is less than the number of variables is a classical theoretical problem. In order to ensure estimability, one has to assume certain constraints on the parameters. A method for maximum likelihood estimation under constraints is proposed to solve this problem. Even in the extreme case where only a single multivariate observation is available, this may provide a feasible solution. It simultaneously provides a simple, straightforward methodology to allow for specific structures within and between covariance matrices of several populations. This methodology yields exact maximum likelihood estimates.     

The Multivariate Randomized Complete Block Design: A Novel Permutation Solution in Case of Ordered Categorical Variables

In this article, multivariate extensions of the combination-based test statistics for the comparison of several treatments in the multivariate Randomized Complete Block designs are introduced in case of categorical response variables. Several tests for the multivariate Randomized Complete Block designs, including MANOVA procedure, are compared with the method proposed via a Monte Carlo simulation study. The method has also been applied to a real case study in the field of sensorial testing studies. Results suggest that in each experimental situation where normality of the supposed underlying continuous model is hard to justify and especially when errors have heavy-tailed distributions, the proposed nonparametric procedure can be considered as a valid solution.     

Inferences on a linear combination of K multivariate normal mean vectors

In this paper, the hypothesis testing and confidence region construction for a linear combination of mean vectors for K independent multivariate normal populations are considered. A new generalized pivotal quantity and a new generalized test variable are derived based on the concepts of generalized p-values and generalized confidence regions. When only two populations are considered, our results are equivalent to those proposed by Gamage et al. [Generalized p-values and confidence regions for the multivariate Behrens–Fisher problem and MANOVA, J. Multivariate Aanal. 88 (2004), pp. 117–189] in the bivariate case, which is also known as the bivariate Behrens–Fisher problem. However, in some higher dimension cases, these two results are quite different. The generalized confidence region is illustrated with two numerical examples and the merits of the proposed method are numerically compared with those of the existing methods with respect to their expected areas, coverage probabilities under different scenarios.     

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.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Parameterizations and Fitting of Bi-directed Graph Models to Categorical Data

Abstract.  We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as thenation multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation-independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method.     

Empirical likelihood tests for two‐sample problems via nonparametric density estimation

The authors study the problem of testing whether two populations have the same law by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. They obtain the asymptotic distribution of the test statistic and propose a bootstrap approximation to calibrate the test. A simulation study is carried out in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions.     

Testing equality of generalized variances of k multivariate normal populations

Generalized variance is a measure of dispersion of multivariate data. Comparison of dispersion of multivariate data is one of the favorite issues for multivariate quality control, generalized homogeneity of multidimensional scatter, etc. In this article, the problem of testing equality of generalized variances of k multivariate normal populations by using the Bartlett's modified likelihood ratio test (BMLRT) is proposed. Simulations to compare the Type I error rate and power of the BMLRT and the likelihood ratio test (LRT) methods are performed. These simulations show that the BMLRT method has a better chi-square approximation under the null hypothesis. Finally, a practical example is given.     

Multivariate Approach for Comparative Evaluations of Customer Satisfaction with Application to Transport Services

In problems related to evaluations of products or services (e.g. in customer satisfaction analysis) the main difficulties concern the synthesis of the information, which is necessary for the presence of several evaluators and many response variables (aspects under evaluation). In this article, the problem of determining and comparing the satisfaction of different groups of customers, in the presence of multivariate response variables and using the results of pairwise comparisons is addressed. Within the framework of group ranking methods and multi criteria decision making theory, a new approach, based on nonparametric techniques, for evaluating group satisfaction in a multivariate framework is proposed and the concept of Multivariate Relative Satisfaction is defined. An application to the evaluation of public transport services, like the railway service and the urban bus service, by students of the University of Ferrara (Italy) is also discussed.     

Multiple comparisons with ‘best’ for multivariate normal populations

Comparisons of multivariate normal populations are made using a mul-tivariate approach (instead of reducing the problem to a univariate one). A rather negative finding is that, for comparisons with the ‘best’ of each variate, repeated univariate comparisons appear to be almost as efficient as multivariate comparisons, at least for the bivariate case and, under certain circumstances, for higher dimensional cases. Investigations are done on comparisons with the ‘MAX-best’ population (that one having the largest maximum of the marginal means), the ‘MIN-best’ (having the largest minimum) and the ‘O-best’ (being closest to largest in all marginal means). Detailed results are given for the bivariate normal with extensions indicated for the multivariate.     

Complete ranking procedures with appropriate loss functions

This paper treats the problem of comparing different evaluations of procedures which rank the variances of k normal populations. Procedures are evaluated on the basis of appropriate loss functions for a particular goal. The goal considered involves ranking the variances of k independent normal populations when the corresponding population means are unknown. The variances are ranked by selecting samples of size n from each population and using the sample variances to obtain the ranking. Our results extend those of various authors who looked at the narrower problem of evaluating the standard proceduv 3 associated with selecting the smallest of the population variances (see e.g.,P. Somerville (1975)).

Different loss functions (both parametric and non-parametric) appropriate to the particular goal under consideration are proposed. Procedures are evaluated by the performance of their risk over a particular preference zone. The sample size n, the least favorable parametric configuration, and the maximum value of the risk are three quantities studied for each procedure. When k is small these quantities, calculated by numerical simulation, show which loss functions respond better and which respond worse to increases in sample size. Loss functions are compared with one another according to the extent of this response. Theoretical results are given for the case of asymptotically large k. It is shown that for certain cases the error incurred by using these asymptotic results is small when k is only moderately large.

This work is an outgrowth of and extends that of J. Reeves and M.J. Sobel (1987) by comparing procedures on the basis of the sample size (perpopulation) required to obtain various bounds on the associated risk functions. New methodologies are developed to evaluate complete ranking procedures in different settings.     

On multiple comparisons among K samples subject to unequal patterns of censorship

Two multiple comparisons procedures for determining which of K arbitrarily censored populations differ from each other are proposed. The procedures are based on multiple comparisons using the generalized Wilcoxon and log-rank statistics. The procedures incorporate a pairwise ranking scheme, rather than the joint ranking scheme proposed by Breslow (1970) and Crowley and Thomas (1975). A conservative testing method suggested by an inequality due to ?idák (1967) is given; a numerical example is presented.     

A generalized pivotal quantity approach to portfolio selection

The major problem of mean–variance portfolio optimization is parameter uncertainty. Many methods have been proposed to tackle this problem, including shrinkage methods, resampling techniques, and imposing constraints on the portfolio weights, etc. This paper suggests a new estimation method for mean–variance portfolio weights based on the concept of generalized pivotal quantity (GPQ) in the case when asset returns are multivariate normally distributed and serially independent. Both point and interval estimations of the portfolio weights are considered. Comparing with Markowitz's mean–variance model, resampling and shrinkage methods, we find that the proposed GPQ method typically yields the smallest mean-squared error for the point estimate of the portfolio weights and obtains a satisfactory coverage rate for their simultaneous confidence intervals. Finally, we apply the proposed methodology to address a portfolio rebalancing problem.     

The use of Andrews curves for detecting the out-of-control variables when a multivariate control chart signals

The aim of this paper is to present a new method for solving the problem of detecting the out-of-control variables when a multivariate control chart signals. The main idea is based on Andrews curves. The proposed method is investigated thoroughly and is proved to have interesting results in comparison to a competing method.     

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.     

Parameter Estimation in Large Dynamic Paired Comparison Experiments

Paired comparison data in which the abilities or merits of the objects being compared may be changing over time can be modelled as a non-linear state space model. When the population of objects being compared is large, likelihood-based analyses can be too computationally cumbersome to carry out regularly. This presents a problem for rating populations of chess players and other large groups which often consist of tens of thousands of competitors. This problem is overcome through a computationally simple non-iterative algorithm for fitting a particular dynamic paired comparison model. The algorithm, which improves over the commonly used algorithm of Elo by incorporating the variability in parameter estimates, can be performed regularly even for large populations of competitors. The method is evaluated on simulated data and is applied to ranking the best chess players of all time, and to ranking the top current tennis-players.     

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

Statistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.     

Neurocognitive Effects in Welders Exposed to Aluminium: An Application of the NPC Test and NPC Ranking Methods

The aim of this study is to investigate the effects of exposure to aluminium on the cognitive sphere in a group of welders. Given the intrinsic complexity of the neurocognitive sphere, three different psychomotor variables were used for its investigation allowing the resulting problem to be naturally modelized into a multivariate framework and solved by a nonparametric combination of permutation tests (NPC test). In order to make the treated and control groups comparable, we also stratified the samples on the basis of a risk index that estimates the combined action of biological age and exposure time. Despite the fact that studies in the literature have reached conflicting results, our study highlights a significant drop in attention and memory performances in individuals exposed to aluminium. Finally, we identify a global criterion that summarises the information on the neurocognitive state by applying the nonparametric combination of dependent rankings method (NPC ranking).     

Pairwise likelihood estimation for multivariate mixed Poisson models generated by Gamma intensities

Estimating the parameters of multivariate mixed Poisson models is an important problem in image processing applications, especially for active imaging or astronomy. The classical maximum likelihood approach cannot be used for these models since the corresponding masses cannot be expressed in a simple closed form. This paper studies a maximum pairwise likelihood approach to estimate the parameters of multivariate mixed Poisson models when the mixing distribution is a multivariate Gamma distribution. The consistency and asymptotic normality of this estimator are derived. Simulations conducted on synthetic data illustrate these results and show that the proposed estimator outperforms classical estimators based on the method of moments. An application to change detection in low-flux images is also investigated.