José M. González-Barrios

On the Construction of Families of Absolutely Continuous Copulas with Given Restrictions

The problem of constructing copulas with a given diagonal section has been studied by Sungur and Yang (1996) and Fredricks and Nelsen (1997a); (b); (2002). The results of Sungur and Yang are especially relevant because, among other results, they have proven that an Archimedean copula is characterized by its diagonal section. The results obtained by Fredricks and Nelsen allow one to build a singular copula with a given a diagonal section. In all cases, the resulting copulas are symmetric. In this article, we provide a family of absolutely continuous copulas with a fixed diagonal, which can differ from another absolutely continuous copula almost everywhere with respect to Lebesgue measure. It is important to mention that the asymmetry in the proposed methodology is not an issue.

A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree

We present a procedure for the identification of clusters in multivariate data sets, based on the comparison between the k nearest neighbors graph, Gk, and the minimal spanning tree, MST. Our key statistic is the random quantity the smallest k such that Gk contains the MST. Under regularity assumptions, we show that for i.i.d. data from a density on with compact support having one connected component, , where n denotes sample size, a bound that seems to be sharp, according to simulations. This leads to a consistent test for the identification of crisp clusters. We illustrate the use of our procedure with an example.

ON CONVERGENCE THEOREMS FOR QUANTILES

Given a real random variable Xits quantiles are always well defined, unlike its expectation. Based on a general definition of population quantiles we generalize known results for the case in which the quantiles are not unique. We propose estimators for the non unique case that also work in the unique case, giving a general approach for real random variables. Some simulations and applications of these results are also included.

A Statistical Method for the Determination of the Appropriate Order in a General Class of Time Series Models

Abstract We propose a method to determine the order q of a model in a general class of time series models. For the subset of linear moving average models (MA(q)), our method is compared with that of the sample autocorrelations. Since the sample autocorrelation is meant to detect a linear structure of dependence between random variables, it turns out to be more suitable for the linear case. However, our method presents a competitive option in that case, and for nonlinear models (NLMA(q)) it is shown to work better. The main advantages of our approach are that it does not make assumptions on the existence of moments and on the distribution of the noise involved in the moving average models. We also include an example with real data corresponding to the daily returns of the exchange rate process of mexican pesos and american dollars.

The Application of a New Dependency Measure to Principal Component Analysis

Abstract In this article we study the relationship between principal component analysis and a multivariate dependency measure. It is shown, via simulated examples and real data, that the information provided by principal components is compatible with that obtained via the dependency measure δ. Furthermore, we show that in some instances in which principal component analysis fails to give reasonable results due to nonlinearity among the random variables, the dependency statistic δ still provides good results. Finally, we give some ideas about using the statistic δ in order to reduce the dimensionality of a given data set.

A nonparametric symmetry test for absolutely continuous bivariate copulas

Based on the works by Klement and Mesiar (Comment Math Univ Carolinae 47:141–148, 2006) and Nelsen (Stat Pap 48:329–336, 2007) on maximal asymmetry of copulas, we define and study the concept of tri-symmetry and we propose a simple statistic to test symmetry of a bivariate copula, given a random sample of an absolutely continuous bivariate random vector. We also make a power comparison against some other well known nonparametric symmetry tests.

A Small-Sample Nonparametric Independence Test for the Archimedean Family of Bivariate Copulas

In this paper we study the problem of independence of two continuous random variables using the fact that there exists a unique copula that characterizes independence, and that such copula is of Archimedean type. We use properties of the empirical diagonal to build nonparametric independence tests for small samples, under the assumption that the underlying copula belongs to the Archimedean family, giving solution to an open problem proposed by Alsina et al.[2].

A Nonparametric Test for Symmetry Based on Freeman and Halton's Ideas on Contingency Tables

In this article, we propose a nonparametric method to test for symmetry in bivariate data. By using the extension of Fishers exact treatment for 2 × 2 contingency tables proposed by Freeman and Halton (1951), we can test the hypothesis of equal distribution for two samples of integer valued variables. Then, by counting the number of observations belonging to each cell of a symmetric, appropriately built grid, we can produce the two samples of integers required to use this test for equal distribution. The resulting test for symmetry is potentially extendible to higher dimensions. A simulation study is performed to compare with some known tests (Bowker, 1948; Hollander, 1971; and its improvement given in Krampe and Kuhnt, 2007). Our proposal represents a competitive option as a test for symmetry.

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model. We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.