Components Of Random Vector Research Articles

RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS.

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding's , Blum-Kiefer-Rosenblatt's and Bergsma-Dassios-Yanagimoto's . Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high dimensional settings. We establish explicit relationships between and Pearson's correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative, (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on and are sorted in a descending order.

Scalar measure of the relationship between several random vectors

The problem of estimating the closeness of co-relation (interdependence) between several random vectors of arbitrary dimension is considered. These random vectors can have arbitrary multidimensional continuous distribution laws. Earlier, within the framework of the entropy approach, indicators were obtained for estimating the closeness of the correlation relationship between the components of one random vector and between two random vectors. The goal of the study is to generalize the previously obtained results to the case of several random vectors. The analytical expression for the coefficient of the closeness of co-relation between several random vectors is obtained. This coefficient is expressed through the indices of determination of conditional regressions between the components of random vectors. For the introduced scalar measure of the relationship, a number of particular results were obtained, which turned out to be known correlation coefficients. A rather simple formula expressed through the determinants of each random vector and the determinant of their combination is derived for the case of Gaussian random vectors. The proposed coefficient can be used to study the network structures consisting of many subsystems. In particular, the interpretation of the correlation coefficient between the elements of the network structure and the other elements can be introduced as the correlation coefficient of the system at the vertex. The introduced measure is quite simply interpretable and allows an unambiguous assessing of the closeness of co-relation between several random vectors of arbitrary dimensions and can be used on real data samples. An example of calculating the closeness of the co-relation between three Gaussian random vectors is presented.

Testing independence in high-dimensional multivariate normal data

A simple statistic was proposed for testing the complete independence of components of random vector having a multivariate normal distribution. Using the martingale central limit theorem, it was proved that this test statistic converges in distribution to the normal as long as both the sample size and dimension go to infinity. Simulations were also carried out to performance evaluation of the proposed test for dealing with the high-dimensional data and the results were compared to those of the state-of-the-art tests. The results confirmed that, in terms of the average relative error criterion, the proposed test has adequate type I error rate control and comparable power to those tests with smaller size distortions.

Scalar measure of the interdependence between random vectors

The problem of assessing tightness of the interdependence between random vectors of different dimensionality is considered. These random vectors can obey arbitrary multidimensional continuous distribution laws. An analytical expression is derived for the coefficient of tightness of the interdependence between random vectors. It is expressed in terms of the coefficients of determination of conditional regressions between the components of random vectors. For the case of Gaussian random vectors, a simpler formula is obtained, expressed through the determinants of each of the random vectors and determinant of their association. It is shown that the introduced coefficient meets all the basic requirements imposed on the degree of tightness of the interdependence between random vectors. This approach is more preferable compared to the method of canonical correlations providing determination of the actual tightness of the interdependence between random vectors. Moreover, it can also be used in case of non-linear correlation dependence between the components of random vectors. The measure thus introduced is rather simply interpretable and can be applied in practice to real data samplings. Examples of calculating the tightness of the interdependence between Gaussian random vectors of different dimensionality are given.

Simultaneous Tests for Independence Among Components of Random Vector by Step-Down Multiple Comparison Procedure

In this article, we consider testing independence among components of random vector in multivariate normal population. For testing independence, we use the modified likelihood ratio test statistic which is improved an approximation to χ2 distribution of the likelihood ratio test statistic. In order to perform simultaneous tests for independence among components of random vector, we use the step-down multiple comparison procedure based on the closed testing procedure proposed by Marcus et al. (1976). Finally, we perform Monte Carlo simulations and present numerical results.

Nonparametric estimation of copula functions for dependence modelling

AbstractCopulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness‐of‐fit test for parametric copula models.

Some Generalizations of Prokhorov's Results on Khinchin-Type Inequalities for Polynomials

In this paper, we prove the Khinchin-type inequality $({\bf E}|f(\xi)|^p)^{1/p}\le C(p,d)\,{\bf E}|f(\xi)|$ for polynomials in components of random vectors in a space with logarithmically concave density.

Optimal sample size for dependent components of random vectors in Monte-Carlo methods

Optimal sample size for dependent components of random vectors in Monte-Carlo methods

New Index of Reliability

The index, β\N, is often used as a measure of second-moment reliability has several limitations. The new index, γ\N, avoids these limitations and has the same numerical value as β\N in elementary cases of second-moment reliability. γ\N is a function of the (generalized) Tchebysheff upper-bound to the probability of failure. γ\N is calculated for many different univariate, multivariate, and infinite-dimensional problems and, whenever possible, is compared with β\N. Cases are studied in which distribution characteristics other than the first two moments are known. These include knowledge of absolute moments, of moments of order higher than two, of positivity or unimodality, of the expectation of functions other than powers, of constraints on the maximum value of the probability density, of independence among the components of random vectors. γ\N is also used to make reliability comparisons between cases with the same value of β\N but with different safety regions.