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A data depth can be used to measure the \"depth\" or \"outlyingness\" of a given multivariate sample with respect to its underlying distribution. This leads to a natural center-outward ordering of the sample points. Based on this ordering, quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location, scale, bias, skewness and kurtosis, as well as for comparing inference methods. All graphs are one-dimensional curves in the plane and can be easily visualized and interpreted. A \"sunburst plot\" is presented as a bivariate generalization of the box-plot. DD-(depth versus depth) plots are proposed and examined as graphical inference tools. Some new diagnostic tools for checking multivariate normality are introduced. One of them monitors the exact rate of growth of the maximum deviation from the mean, while the others examine the ratio of the overall dispersion to the dispersion of a certain central region. The affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods.

The existing lexicographical ordering approaches respect the total ordering properties, thus making this approach a very robust solution for multivariate ordering. However, different marginal components derived from various representations of a colour image will lead to different results of multivariate ordering. Moreover, the output of lexicographical ordering only depends on the first component leading to the followed components taking no effect. To address these issues, three new marginal components are obtained by means of quaternion decomposition, and they are employed by fuzzy lexicographical ordering, and thus a new fuzzy extremum estimation algorithm (FEEA) based on quaternion decomposition is proposed in this study. The novel multivariate mathematical morphological operators are also defined according to FEEA. Comparing with the existing solutions, experimental results show that the proposed FEEA performs better results on multivariate extremum estimation, and the presented multivariate mathematical operators can be easily handled and can provide better results on multivariate image filtering.

For many structural design problems univariate extreme value theory is applied to quantify the risk of failure due to extreme levels of some environmental process. In practice, many forms of structure fail owing to a combination of various processes at extreme levels. Recent developments in statistical methodology for multivariate extremes enable the modelling of such behaviour. The aim of this paper is to demonstrate how these ideas can be exploited as part of the design process.

The classical treatment of multivariate extreme values is through componentwise ordering, though in practice most interest is in actual extreme events. Here the point process of observations which are extreme in at least one component is considered. Parametric models for the dependence between components must satisfy certain constraints. Two new techniques for generating such models are presented. Aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data.

In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum likelihood estimators and of the parameters θ 1 and θ 2 have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively. Here, we prove that these estimators are stochastically monotone with respect to θ 1 and θ 2 , respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters θ 1 and θ 2 . For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions under truncation, which is also of independent interest.

Traditionally it has been assumed by ecologists that the dispersion of positions with respect to leks of female capercaillies during the mating season is superior to the dispersion of positions of male capercaillies. However some recently published articles suggest the idea that dispersion is not sex-biased, but both sexes show a similar dispersion. This article introduces a mathematical model to approach the study of the above question by means of an indexed dispersion criterion. This model is developed in detail thus deriving its main mathematical properties. On this basis, the model is applied to the analysis of the motivating problem with real data of positions of leks and female and male capercaillies. The results derived by means of the model put into doubt the traditional assumption of a superior dispersion of females supporting the new theories, that is, both sexes show a similar dispersion.

Statistical process control (SPC) charts play a central role in quality control and management. Many conventional SPC charts are designed under the assumption that the related process distribution is normal. In practice, the normality assumption is often invalid. In such cases, some articles show that certain conventional SPC charts are robust and can still be used as long as their parameters are properly chosen. Some other articles argue that results from such conventional SPC charts would not be reliable and that nonparametric SPC charts should be considered instead. In recent years, many nonparametric SPC charts have been proposed. Most of them are based on the ranking information in process observations collected at different time points. Some of them are based on data categorization and categorical data analysis. In this article, we give some perspectives on issues related to the robustness of conventional SPC charts and to the strengths and limitations of various nonparametric SPC charts.

In this paper, we investigate some multivariate integral stochastic orderings of skew-normal random vectors. We derive the results of the sufficient and/or necessary conditions by applying an identity for$ Ef({\\mathbf Y})-Ef({\\mathbf X}) $ , where$ {\\mathbf X} $and$ {\\mathbf Y} $are multivariate skew-normal random vectors,$ f $satisfies some weak regularity condition. The integral orders considered here are the componentwise convex, copositive, completely-positive orderings and their corresponding increasing ones as well as linear forms of stochastic orderings, which play a vital role in transforming the unmanageable multivariate components into an easy-to-handle univariate variable.

The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption of finite second moments. Applications are provided to hypothesis testing on a data set from materials science and to multivariate statistical quality control. The distance standard deviation is compared to classical scale measures for inference on the spread of heavy-tailed distributions. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini’s mean difference. New expressions for the distance standard deviation are obtained in terms of Gini’s mean difference and the moments of spacings of order statistics. It is also shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread.