# 科研成果

2020
Ai, M., Wang, F., Yu, J., & Zhang, H. (2020). Optimal subsampling for large-scale quantile regression. Journal of Complexity. 访问链接Abstract
To deal with massive data sets, subsampling is known as an effective method which can significantly reduce computational costs in estimating model parameters. In this article, an efficient subsampling method is developed for large-scale quantile regression via Poisson sampling framework, which can solve the memory constraint problem imposed by big data. Under some mild conditions, large sample properties for the estimator involving the weak and strong consistencies, and asymptotic normality are established. Furthermore, the optimal subsampling probabilities are derived according to the A-optimality criterion. It is shown that the estimator based on the optimal subsampling asymptotically achieves a smaller variance than that by the uniform random subsampling. The proposed method is illustrated and evaluated through numerical analyses on both simulated and real data sets.
Yu, J., Wang, H., Ai, M., & Zhang, H. (2020). Optimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators with Massive Data. Journal of the American Statistical Association. 访问链接Abstract
Nonuniform subsampling methods are effective to reduce computational burden and maintain estimation efficiency for massive data. Existing methods mostly focus on subsampling with replacement due to its high computational efficiency. If the data volume is so large that nonuniform subsampling probabilities cannot be calculated all at once, then subsampling with replacement is infeasible to implement. This paper solves this problem using Poisson subsampling. We first derive optimal Poisson subsampling probabilities in the context of  quasi-likelihood estimation under the A- and L-optimality criteria. For a practically implementable algorithm with approximated optimal subsampling probabilities, we establish the consistency and asymptotic normality of the resultant estimators. To deal with the situation that the full data are stored in different blocks or at multiple locations, we develop a distributed subsampling framework, in which statistics are computed simultaneously on smaller partitions of the full data. Asymptotic properties of the resultant aggregated estimator are investigated. We illustrate and evaluate the proposed strategies through numerical experiments on simulated and real data sets.
Huang, H., Gao, Y., Zhang, H., & Li, B. (2020). Weighted Lasso Estimates for Sparse Logistic Regressions: Non-asymptotic Properties with Measurement Error. Acta Mathematica Scientia.Abstract
When we are interested in high-dimensional system and focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We proposed two types of weighted Lasso estimates depending on covariates by the McDiarmid inequality. Given sample size $n$ and dimension of covariates $p$, the finite sample behavior of our proposed methods with a diverging number of predictors is illustrated by non-asymptotic oracle inequalities such as $\ell_{1}$-estimation error and squared prediction error of the unknown parameters. We compare the performance of our methods with former weighted estimates on simulated data, then apply these methods to do real data analysis.
Fan, Y., Zhang, H., & Yan, T. (2020). Asymptotic Theory for Differentially Private Generalized β-models with Parameters Increasing. Statistics and Its Interface, 13(3), 385 – 398. 访问链接
Ai, M., Yu, J., Zhang, H., & Wang, H. (2020). Optimal Subsampling for Big Data Regressions. Statistica Sinica. 访问链接
Zhang, H., & Jia, J. (2020). Elastic-net Regularized High-dimensional Negative Binomial Regression: Consistency and Weak Signals Detection. Statistica Sinica. 访问链接
2019
Denworth, L., & 张慧铭(译),. (2019). P值危机:统计学需要一场变革. 环球科学, 21, 76-81. 访问链接Abstract

2018
Zhang, H., Tan, K., & Bo, L. (2018). COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data. Frontiers of Mathematics in China, 13(4), 967–998. 访问链接Abstract
We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (ab, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.
2017
Zhang, H., Li, B., & Jay, K. G. (2017). A characterization of signed discrete infinitely divisible distributions. Studia Scientiarum Mathematicarum Hungarica, 54(4), 446–470. 访问链接
2016
Zhang, H. (2016). New proofs of Chaundy--Bullard identity in "The Problem of Points''. The Mathematical Intelligencer, 38(1), 4-5. 访问链接
Zhang, H. (2016). Infinite Divisibility and Compound Poisson Law:Related Count Data Models and High-Dimensional Variable Selection. 华中师范大学 硕士论文. 访问链接Abstract
In this master thesis, we explore the probability theory, statistical inference and numerical computation of discrete compound Poisson (DCP) distribution. In particular, we do a very comprehensive literature review of DCP distributions and its applications in related statistical models of count data fields, and especially, we discuss penalized generalized linear model of count data regression.The discrete compound Poisson distributions have the probability generating function in the form of the following: The famous Feller’s characterization of the compound Poisson states that a discrete distribution is compound Poisson if and only if its distribution is discrete infinitely divisible. This is a special case of Levy-Khinchine formula. When the{ai}i=1∞, may take negative values and the sum is absolutely convergent, it is called pseudo discrete compound Poisson distribution.In the first chapter, we introduce an important tool (probability generating function and Fourier transform) as preliminaries and improve the flawed proof of Feller’s characterization, and then we give a short introduction of variable selection method about Lasso and generalization. We close this chapter with the infinitely divisibile prior distribution in Bayesian Lasso and we envisages appropriate zero-inflated distribution as prior distribution which obtains the nonzero sparse estimation of coefficients. The chapter Ⅱ discusses characterizations of DCP distribution(process) with ten methods to prove the probability mass function are given in Appendix, and we give over a hundred kinds of special cases or sub-families of DCP distribution which are listed in a table with references. We use Stein-Chen method and operator semigroup method to obtain the upper bound of the total variation between a sum of independent discrete r.v. and a related discrete compound Poisson r.v., and use row sum in random triangular array to approximate discrete compound Poisson distribution. Chapter Ⅳ studys statistics, parameters estimation, FFT of DCP probability mass. Chapter Ⅴ firstly uses cumulants estimation and Fourier transform estimation to actuarial claim data with zero-inflated and overdispersion properties, then compares its Kolmogorov-Smimov test and Chi-squared test. We give a theorem that a set of count data obeys discrete pseudo compound Poisson distribution if its. probability of zero is larger than the probability of nonzero. Further more, we use this zero-inflated property of pseudo discrete compound Poisson with adding virtual frequency techniques; we get an algorithm to fit any discrete distributions. Chapter V also discusses count GLM related to the DCP distribution and use penalized estimation to select important regression variables. In particular, we consider the Elastic net estimates of negative binomial regression, and we give a necessary and sufficient condition(like Karush-Kuhn-Tucker conditions) for non-zero(zero) coefficient estimates. Using a spider count data, we analysis this real example by negative binomial regression with MLE, Lasso, Elastic net penalties. Next, we set forth the survival functions in discrete frailty model and cured rate models (or long term survivor models with competing causes) which are derived from some DCP distributions. In the last section, we look forword to the future study that mixed Poisson distribution to approximate any discrete distribution, and states the problem of variable selection in mixture components. Due to the complexity of the mixture, it results the high-dimentional problem.
2015

2014
Zhang, H., Liu, Y., & Li, B. (2014). Notes on discrete compound Poisson model with applications to risk theory. Insurance: Mathematics and Economics, 56, 325-336. 访问链接