The Lilliefors normality test is a classical extension of the Kolmogorov–Smirnov goodness-of-fit test tailored to assessing normality. A recent modification improves its robustness to outliers by introducing a subsetting function. In this paper, we propose an alternative approach that replaces subsetting functions with Ordered Weighted Averaging (OWA) functions and further generalizes the test to any location-scale family, not only the normal distribution. We conduct extensive experiments on the power of the resulting test for three representative location-scale families—normal, uniform and shifted-exponential—using different weight vectors to define the OWA functions. The results indicate that the best trade-off between robustness and statistical power is achieved by a well-known special class of OWA functions: the order statistics.

Ordered Weighted Averaging (OWA) functions are a popular tool for the aggregation of real values and have been used successfully in several fields of application. The extension of these OWA functions to the multivariate setting is not unique and has been addressed separately by different disciplines. In this paper, we introduce a unifying perspective by presenting under a common framework different classes of multivariate OWA functions and discuss the main fulfilled properties by each of these classes.

Testing for symmetry about an unknown median is a problem that has attracted the attention of statisticians for decades. Many of the existing tests of symmetry are based on the asymptotic distribution of a skewness coefficient, which typically is asymptotically normal and centered around zero for symmetric distributions. Unfortunately, the asymptotic variance depends on the underlying distribution and differs from one symmetric distribution to another one. A possible way out is to estimate this asymptotic variance from the sample by means of the bootstrap. In this paper, we explore this approach for six different skewness coefficients existing in the literature. Extensive experiments are performed for comparing the performance of the six associated tests of symmetry to that of two state-of-the-art symmetry tests in terms of preservation of the significance level under several symmetric distributions, power under asymmetric distributions and robustness in the presence of outliers. Even though the results show no clear best test, we conclude by providing some guidelines for choosing a test of symmetry based on the needs of the user.

The notion of an orness measure for aggregation functions has been a relevant study subject whose history can be traced back to the early works of Dujmović in 1973. Intuitively, an orness measure quantifies the similarity of an aggregation function to the “or” function and results in an essential tool for decision engineering, field in which the choice of aggregation function is sometimes restricted to a desired value of orness (orness-directed aggregation). In 1988, Yager presented a particular example of orness measure for ordered weighted averaging (OWA) functions and initiated a series of contributions aiming at proposing an axiomatic definition of orness measure for OWA functions. In this paper, we go much further and present an axiomatic definition of orness measure for the whole family of aggregation functions. We end by proposing two natural construction methods for an orness measure for aggregation functions. The particular examples of the (discrete) Choquet integral and uninorms are studied in detail.

The field of aggregation theory aims at formalizing in a mathematical way the process of combining several inputs into a single output, typically both the inputs and the output being elements of a poset. Although the field in itself only dates from the second half of the last century, one could easily trace back further in time prominent examples of aggregation functions. For instance, means were already studied by Cauchy in the early 1820s. Although the most popular aggregation processes have historically been those dealing with real numbers, the interest of practitioners in different types of data is not to be neglected. Some prominent examples are the aggregation of strings, which is nowadays a popular topic for computer scientists and bioinformaticians, and the aggregation of rankings, which has been studied in social choice theory since the eighteenth century. In this article, we propose to abandon the current order-based understanding of an aggregation process and embrace a new geometrically oriented sense upon which a new theory of aggregation could be developed.

In this letter, a characterization of the property of orthomonotnoicity for aggregation functions for multidimensional data is provided.

In the field of information fusion, the problem of data aggregation has been formalized as an order-preserving process that builds upon the property of monotonicity. However, fields such as computational statistics, data analysis, and geometry usually emphasize the role of equivariances to various geometrical transformations in aggregation processes. Admittedly, if we consider a unidimensional data fusion task, both requirements are often compatible with each other. Nevertheless, in this paper, we show that, in the multidimensional setting, the only idempotent functions that are monotone and orthogonal equivariant are the over-simplistic weighted centroids. Even more, this result still holds after replacing monotonicity and orthogonal equivariance by the weaker property of orthomonotonicity. This implies that the aforementioned approaches to the aggregation of multidimensional data are irreconcilable, and that, if a weighted centroid is to be avoided, we must choose between monotonicity and a desirable behavior with regard to orthogonal transformations.

This paper describes how distance-based classification methods could be improved by incorporating ranking rules, which are old acquaintances of aggregation and social choice theorists. Specifically, a novel method incorporating two prominent ranking rules (namely, the plurality and the Borda count ranking rules) into the distance-based classification method of nearest neighbours is presented. Some exploratory experiments have been conducted, obtaining encouraging results. Interestingly, the newly proposed method seems to turn the classic method of nearest neighbours independent of the chosen distance metric while not compromising its performance significantly.

The study of conditions, under which the existence of an “absolute” best winner can be assured, is a hot topic in the field of social choice. Unanimity is an evident example of a condition under which the winner is obvious. However, many more properties weaker than unanimity have been analysed in literature: the presence of a Condorcet winner, strong stochastic transitivity, the presence of a candidate that Borda dominates all other candidates, etc. Unfortunately, one could easily find a prominent ranking rule, for which the outcome does not agree with these relaxed conditions. In this study, we aim to identify a condition weaker than unanimity, but under which the social outcome is still obvious. This condition, defined as the conjunction of three properties already studied by the present authors and hereinafter referred to as acclamation, will be proven to be a meeting point for the most prominent ranking rules in social choice theory, and will be used for introducing an intuitively appealing ranking rule.

Penalty functions have been a common tool in data aggregation for decades. Unfortunately, although the definition of a penalty function has evolved over the years, the use of penalty functions has been reduced to the aggregation of real numbers. However, in this `era of aggregation,' the need of generalizing the current definition in order to comply with the characteristics of new types of data arises. In this paper, we bring to the attention the notion of betweenness relation and propose to replace the currently required property of quasiconvexity of a penalty function by the compatibility with a betweenness relation. Several construction methods for a penalty function are provided based on the use of a monometric. Interestingly, several prominent data aggregation methods are proved to fit into this new framework for penalty-based data aggregation.