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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.

Since multivariate lifetime data frequently occur in applications, various properties of multivariate distributions have been previously considered to model and describe the main concepts of aging commonly considered in the univariate setting. The generalization of univariate aging notions to the multivariate case involves, among other factors, appropriate definitions of multivariate quantiles and related notions, which are able to correctly describe the intrinsic characteristics of the concepts of aging that should be generalized, and which provide useful tools in the applications. A new multivariate version of the well-known New Better than Used in Expectation univariate aging notion is provided, by means of the concepts of the upper corrected orthant and multivariate excess-wealth function. Some of its properties are described, with particular attention paid to those that can be useful in the analysis of real data sets. Finally, through an example it is illustrated how the new multivariate aging notion influences the final results in the analysis of data on tumor growth from the Comprehensive Cohort Study performed by the German Breast Cancer Study Group.

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernandez-Ponce et al. (1996) , is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernandez-Ponce et al. (1998) . The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.

We study the concept of multivariate dispersion order, defined as the existence of an expansion function that maps a random vector to another one, for multivariate distributions with the same dependence structure. As a particular case, we can order the multivariate t-distribution family in dispersion sense. Finally, we use these results in the problem of detection and characterization of influential observations in regression analysis. This problem can often be used to compare two multivariate t-distributions.

A multivariate dispersion ordering based on quantiles more widely separated is defined. This new multivariate dispersion ordering is a generalization of the classic univariate version. If we vary the ordering of the components in the multivariate random variable then the comparison could not be possible. We provide a characterization using a multivariate expansion function. The relationship among various multivariate orderings is also considered. Finally, several examples illustrate the method of this paper.