Standard Gini covariance and Gini correlation play important roles in measuring the dependence of random variables with heavy tails. However, the asymmetry brings a substantial difficulty in interpretation. In this paper, we propose a symmetric Gini-type covariance and a symmetric Gini correlation (ρ g ) based on the joint rank function. The proposed correlation ρ g is more robust than the Pearson correlation but less robust than the Kendall's τ correlation. We establish the relationship between ρ g and the linear correlation ρ for a class of random vectors in the family of elliptical distributions, which allows us to estimate ρ based on estimation of ρ g . The asymptotic normality of the resulting estimators of ρ are studied through two approaches: one from influence function and the other from U-statistics and the delta method. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, regular Gini, Pearson and Kendall's τ under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure ρ g demonstrates superior finite sample performance, which makes it attractive in applications.

The Gini correlation plays an important role in measuring dependence of random variables with heavy tailed distributions, whose properties are a mixture of Pearson's and Spearman's correlations. Due to the structure of this dependence measure, there are two Gini correlations between each pair of random variables, which are not equal in general. Both the Gini correlation and the equality of the two Gini correlations play important roles in Economics. In the literature, there are limited papers focusing on the inference of the Gini correlations and their equality testing. In this paper, we develop the jackknife empirical likelihood (JEL) approach for the single Gini correlation, for testing the equality of the two Gini correlations, and for the Gini correlations' differences of two independent samples. The standard limiting chi-square distributions of those jackknife empirical likelihood ratio statistics are established and used to construct confidence intervals, rejection regions, and to calculate p-values of the tests. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals, as well as in terms of power of the tests. The proposed methods are illustrated in an application on a real data set from UCI Machine Learning Repository. noindent

The categorical Gini correlation is an alternative measure of dependence between categorical and numerical variables, which characterizes the independence of the variables. A non‐parametric test based on the categorical Gini correlation for the equality of K distributions is developed. By applying the jackknife empirical likelihood approach, the standard limiting chi‐squared distribution with degrees of freedom of K − 1 is established and is used to determine the critical value and p‐value of the test. Simulation studies show that the proposed method is competitive with existing methods in terms of power of the tests in most cases. The proposed method is illustrated in an application on a real dataset.

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (2002) studied the polynomial transformations of Gaussian FARIMA(0,d,0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (1996, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p,d,q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.Comment: accepted by Statistical Inference for Stochastic Processes, 28 page

Let {X n :n∈N}be a linear process with bounded probability density function f(x). We study the estimation of the quadratic functional ∫ R f 2(x)dx. With a Fourier transform on the kernel function and the projection method, it is shown that, under certain mild conditions, the estimator 2 false/ false( n false( n − 1 false) h n false) ∑ 1 ≤ i < j ≤ n K false( false( X i − X j false) false/ h n false) has similar asymptotical properties as the i.i.d. case studied in Giné and Nickl if the linear process {X n :n∈N}has the defined short range dependence. We also provide an application to L 2 2 divergence and the extension to multi‐variate linear processes. The simulation study for linear processes with Gaussian and α‐stable innovations confirms our theoretical results. As an illustration, we estimate the L 2 2 divergences among the density functions of average annual river flows for four rivers and obtain promising results.