arXiv subjects

Xin Dang

Publications and source records attributed to Xin Dang.

16 recordsLinked to original sources

arXiv2023

Asymptotic Normality of Gini Correlation in High Dimension with Applications to the K-sample Problem

The categorical Gini correlation proposed by Dang et al. is a dependence measure to characterize independence between categorical and numerical variables. The asymptotic distributions of the sample correlation under dependence and independence have been established when the dimension of the numerical variable is fixed. However, its asymptotic behavior for high dimensional data has not been explored. In this paper, we develop the central limit theorem for the Gini correlation in the more realistic setting where the dimensionality of the numerical variable is diverging. We then construct a powerful and consistent test for the $K$-sample problem based on the asymptotic normality. The proposed test not only avoids computation burden but also gains power over the permutation procedure. Simulation studies and real data illustrations show that the proposed test is more competitive to existing methods across a broad range of realistic situations, especially in unbalanced cases.

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arXiv2023

Grouped feature screening for ultrahigh-dimensional classification via Gini distance correlation

Gini distance correlation (GDC) was recently proposed to measure the dependence between a categorical variable, Y, and a numerical random vector, X. It mutually characterizes independence between X and Y. In this article, we utilize the GDC to establish a feature screening for ultrahigh-dimensional discriminant analysis where the response variable is categorical. It can be used for screening individual features as well as grouped features. The proposed procedure possesses several appealing properties. It is model-free. No model specification is needed. It holds the sure independence screening property and the ranking consistency property. The proposed screening method can also deal with the case that the response has divergent number of categories. We conduct several Monte Carlo simulation studies to examine the finite sample performance of the proposed screening procedure. Real data analysis for two real life datasets are illustrated.

stat.ME↗

arXiv2021

Unadjusted Langevin algorithm for non-convex weakly smooth potentials

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler Maruyama discretization of the Langevin diffusion process, referred as Unadjusted Langevin Algorithm (ULA), studied mostly in the context of smooth (gradient Lipschitz) and strongly log-concave densities, is a considerable hindrance for its deployment in many sciences, including statistics and machine learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods for non-convex distributions. Particularly, we introduce a new mixture weakly smooth condition, under which we prove that ULA will converge with additional log-Sobolev inequality. We also show that ULA for smoothing potential will converge in $L_{2}$-Wasserstein distance. Moreover, using convexification of nonconvex domain \citep{ma2019sampling} in combination with regularization, we establish the convergence in Kullback-Leibler (KL) divergence with the number of iterations to reach $ε$-neighborhood of a target distribution in only polynomial dependence on the dimension. We relax the conditions of \citep{vempala2019rapid} and prove convergence guarantees under isoperimetry, and non-strongly convex at infinity.

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arXiv2020

Black-box sampling for weakly smooth Langevin Monte Carlo using p-generalized Gaussian smoothing

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler-Maruyama discretization of the Langevin diffusion process, also named as Langevin Monte Carlo (LMC), studied mostly in the context of smooth (gradient-Lipschitz) and strongly log-concave densities, a significant constraint for its deployment in many sciences, including computational statistics and statistical learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods. Particularly, we generalize the Gaussian smoothing, approximate the gradient using p-generalized Gaussian smoothing and take advantage of it in the context of black-box sampling. We first present a non-strongly concave and weakly smooth black-box LMC algorithm, ideal for practical applicability of sampling challenges in a general setting.

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arXiv2019

Empirical Likelihood Test for Diagonal Symmetry

Energy distance is a statistical distance between the distributions of random variables, which characterizes the equality of the distributions. Utilizing the energy distance, we develop a nonparametric test for the diagonal symmetry, which is consistent against any fixed alternatives. The test statistic developed in this paper is based on the difference of two $U$-statistics. By applying the jackknife empirical likelihood approach, the standard limiting chi-square distribution with degree freedom of one is established and is used to determine critical value and $p$-value of the test. Simulation studies show that our method is competitive in terms of empirical sizes and empirical powers.

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arXiv2019

Jackknife Empirical Likelihood Approach for K-sample Tests

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

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arXiv2019

A new Gini correlation between quantitative and qualitative variables

We propose a new Gini correlation to measure dependence between a categorical and numerical variables. Analogous to Pearson $R^2$ in ANOVA model, the Gini correlation is interpreted as the ratio of the between-group variation and the total variation, but it characterizes independence (zero Gini correlation mutually implies independence). Closely related to the distance correlation, the Gini correlation is of simple formulation by considering the nature of categorical variable. As a result, the proposed Gini correlation has a lower computational cost than the distance correlation and is more straightforward to perform inference. Simulation and real applications are conducted to demonstrate the advantages.

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arXiv2019

Depth-based Weighted Jackknife Empirical Likelihood for Non-smooth U-structure Equations

In many applications, parameters of interest are estimated by solving some non-smooth estimating equations with $U$-statistic structure. Jackknife empirical likelihood (JEL) approach can solve this problem efficiently by reducing the computation complexity of the empirical likelihood (EL) method. However, as EL, JEL suffers the sensitivity problem to outliers. In this paper, we propose a weighted jackknife empirical likelihood (WJEL) to tackle the above limitation of JEL. The proposed WJEL tilts the JEL function by assigning smaller weights to outliers. The asymptotic of the WJEL ratio statistic is derived. It converges in distribution to a multiple of a chi-square random variable. The multiplying constant depends on the weighting scheme. The self-normalized version of WJEL ratio does not require to know the constant and hence yields the standard chi-square distribution in the limit. Robustness of the proposed method is illustrated by simulation studies and one real data application.

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arXiv2019

Estimating Feature-Label Dependence Using Gini Distance Statistics

Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RKHS). Two Gini distance based dependence measures are explored: Gini distance covariance and Gini distance correlation. Unlike Pearson covariance and correlation, which do not characterize independence, the above Gini distance based measures define dependence as well as independence of random variables. The test statistics are simple to calculate and do not require probability density estimation. Uniform convergence bounds and asymptotic bounds are derived for the test statistics. Comparisons with distance covariance statistics are provided. It is shown that Gini distance statistics converge faster than distance covariance statistics in the uniform convergence bounds, hence tighter upper bounds on both Type I and Type II errors. Moreover, the probability of Gini distance covariance statistic under-performing the distance covariance statistic in Type II error decreases to 0 exponentially with the increase of the sample size. Extensive experimental results are presented to demonstrate the performance of the proposed method.

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arXiv2018

On mutual information estimation for mixed-pair random variables

We study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel method to estimate the mutual information between the two random variables. The estimates enjoy a central limit theorem under some regular conditions on the distributions. The theoretical results are demonstrated by simulation study.

math.ST↗

arXiv2018

Simulation-based inference methods for partially observed Markov model via the R package is2

Partially observed Markov process (POMP) models are powerful tools for time series modeling and analysis. Inherited the flexible framework of R package pomp, the is2 package extends some useful Monte Carlo statistical methodologies to improve on convergence rates. A variety of efficient statistical methods for POMP models have been developed including fixed lag smoothing, second-order iterated smoothing, momentum iterated filtering, average iterated filtering, accelerate iterated filtering and particle iterated filtering. In this paper, we show the utility of these methodologies based on two toy problems. We also demonstrate the potential of some methods in a more complex model, employing a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the extension beyond POMP models and the development of additional methods within the framework provided by is2.

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arXiv2018

Robust and Efficient Boosting Method using the Conditional Risk

Well-known for its simplicity and effectiveness in classification, AdaBoost, however, suffers from overfitting when class-conditional distributions have significant overlap. Moreover, it is very sensitive to noise that appears in the labels. This article tackles the above limitations simultaneously via optimizing a modified loss function (i.e., the conditional risk). The proposed approach has the following two advantages. (1) It is able to directly take into account label uncertainty with an associated label confidence. (2) It introduces a "trustworthiness" measure on training samples via the Bayesian risk rule, and hence the resulting classifier tends to have finite sample performance that is superior to that of the original AdaBoost when there is a large overlap between class conditional distributions. Theoretical properties of the proposed method are investigated. Extensive experimental results using synthetic data and real-world data sets from UCI machine learning repository are provided. The empirical study shows the high competitiveness of the proposed method in predication accuracy and robustness when compared with the original AdaBoost and several existing robust AdaBoost algorithms.

stat.ML↗

arXiv2018

Jackknife Empirical Likelihood Methods for Gini Correlations and their Equality Testing

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.

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arXiv2018

A rank-based Cramér-von-Mises-type test for two samples

We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the two-sample Cramér-von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the Hájek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.

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arXiv2016

Gini Covariance Matrix and its Affine Equivariant Version

We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation-retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM's are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.

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arXiv2016

Symmetric Gini Covariance and Correlation

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.

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