Symmetrical Independence Tests for Two Random Vectors with Arbitrary Dimensional Graphs

Test of independence between random vectors X and Y is an essential task in statistical inference.One type of testing methods is based on the minimal spanning tree of variables X and Y.The main idea is to generate the minimal spanning tree for one random vector X,and for each edges in minimal spanning tree,the corresponding rank number can be calculated based on another random vector Y.The resulting test statistics are constructed by these rank numbers.However,the existed statistics are not symmetrical tests about the random vectors X and Y such that the power performance from minimal spanning tree of X is not the same as that from minimal spanning tree of Y.In addition,the conclusion from minimal spanning tree of X might conflict with that from minimal spanning tree of Y.In order to solve these problems,we propose several symmetrical independence tests for X and Y.The exact distributions of test statistics are investigated when the sample size is small.Also,we study the asymptotic properties of the statistics.A permutation method is introduced for getting critical values of the statistics.Compared with the existing methods,our proposed methods are more efficient demonstrated by numerical analysis.

A distribution-free test of independence based on a modified mean variance index

Cui and Zhong(2019),(Computational Statistics&Data Analysis,139,117–133)proposed a test based on the mean variance(MV)index to test independence between a categorical random variable Y with R categories and a continuous random variable X.They ingeniously proved the asymptotic normality of the MV test statistic when R diverges to infinity,which brings many merits to the MV test,including making it more convenient for independence testing when R is large.This paper considers a new test called the integral Pearson chi-square(IPC)test,whose test statistic can be viewed as a modified MV test statistic.A central limit theorem of the martin-gale difference is used to show that the asymptotic null distribution of the standardized IPC test statistic when R is diverging is also a normal distribution,rendering the IPC test sharing many merits with the MV test.As an application of such a theoretical finding,the IPC test is extended to test independence between continuous random variables.The finite sample performance of the proposed test is assessed by Monte Carlo simulations,and a real data example is presented for illustration.

A New Approach to Learn the Equivalence Class of Bayesian Network

It's a well-known fact that constraint-based algorithms for learning Bayesian network(BN) structure reckon on a large number of conditional independence(C1) tests.Therefore,it is difficult to learn a BN for indicating the original causal relations in the true graph.In this paper,a two-phase method for learning equivalence class of BN is introduced.The first phase of the method learns a skeleton of the BN by CI tests.In this way,it reduces the number of tests compared with other existing algorithms and decreases the running time drastically.The second phase of the method orients edges that exist in all BN equivalence classes.Our method is tested on the ALARM network and experimental results show that our approach outperforms the other algorithms.

The abstract of doctoral dissertation‘Some research on hypothesis testing and nonparametric variable screening problems for high dimensional data’

In this thesis,we construct test statistic for association test and independence test in high dimension,respectively,and study the corresponding theoretical properties under some regularity conditions.Meanwhile,we propose a nonparametric variable screening procedure for sparse additive model with multivariate response in untra-high dimension and established some screening properties.

Distance-covariance-based tests for heteroscedasticity in nonlinear regressions

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

Bi-objective evolutionary Bayesian network structure learning via skeleton constraint

Bayesian network is a popular approach to uncertainty knowledge representation and reasoning. Structure learning is the first step to learn a Bayesian network. Score-based methods are one of the most popular ways of learning the structure. In most cases, the score of Bayesian network is defined as adding the log-likelihood score and complexity score by using the penalty function. If the penalty function is set unreasonably, it may hurt the performance of structure search. Thus, Bayesian network structure learning is essentially a bi-objective optimization problem. However, the existing bi-objective structure learning algorithms can only be applied to small-scale networks. To this end, this paper proposes a bi-objective evolutionary Bayesian network structure learning algorithm via skeleton constraint (BBS) for the medium-scale networks. To boost the performance of searching, BBS introduces the random order prior (ROP) initial operator. ROP generates a skeleton to constrain the searching space, which is the key to expanding the scale of structure learning problems. Then, the acyclic structures are guaranteed by adding the orders of variables in the initial skeleton. After that, BBS designs the Pareto rank based crossover and skeleton guided mutation operators. The operators operate on the skeleton obtained in ROP to make the search more targeted. Finally, BBS provides a strategy to choose the final solution. The experimental results show that BBS can always find the structure which is closer to the ground truth compared with the single-objective structure learning methods. Furthermore, compared with the existing bi-objective structure learning methods, BBS is scalable and can be applied to medium-scale Bayesian network datasets. On the educational problem of discovering the influencing factors of students’ academic performance, BBS provides higher quality solutions and is featured with the flexibility of solution selection compared with the widely-used Bayesian network structure learning methods.