Research

19 pages, 392 KiB

Open AccessArticle

Optimal Non-Asymptotic Bounds for the Sparse β Model

by Xiaowei Yang, Lu Pan, Kun Cheng and Chao Liu

Mathematics 2023, 11(22), 4685; https://doi.org/10.3390/math11224685 - 17 Nov 2023

Viewed by 530

This paper investigates the sparse

β

model with

𝓁_{1}

penalty in the field of network data models, which is a hot topic in both statistical and social network research. We present a refined algorithm designed for parameter estimation in the proposed model. [...] Read more.

This paper investigates the sparse

β

model with

𝓁_{1}

penalty in the field of network data models, which is a hot topic in both statistical and social network research. We present a refined algorithm designed for parameter estimation in the proposed model. Its effectiveness is highlighted through its alignment with the proximal gradient descent method, stemming from the convexity of the loss function. We study the estimation consistency and establish an optimal bound for the proposed estimator. Empirical validations facilitated through meticulously designed simulation studies corroborate the efficacy of our methodology. These assessments highlight the prospective contributions of our methodology to the advanced field of network data analysis. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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12 pages, 298 KiB

Open AccessArticle

Sharper Concentration Inequalities for Median-of-Mean Processes

by Guangqiang Teng, Yanpeng Li, Boping Tian and Jie Li

Mathematics 2023, 11(17), 3730; https://doi.org/10.3390/math11173730 - 30 Aug 2023

Viewed by 787

Abstract

The Median-of-Mean (MoM) estimation is an efficient statistical method for handling data with contamination. In this paper, we propose a variance-dependent MoM estimation method using the tail probability of a binomial distribution. The bound of this method is better than the classical Hoeffding [...] Read more.

The Median-of-Mean (MoM) estimation is an efficient statistical method for handling data with contamination. In this paper, we propose a variance-dependent MoM estimation method using the tail probability of a binomial distribution. The bound of this method is better than the classical Hoeffding method under mild conditions. This method is then used to study the concentration of variance-dependent MoM empirical processes and sub-Gaussian intrinsic moment norm. Finally, we give the bound of the variance-dependent MoM estimator with distribution-free contaminated data. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

16 pages, 315 KiB

Open AccessArticle

On a Low-Rank Matrix Single-Index Model

by The Tien Mai

Mathematics 2023, 11(9), 2065; https://doi.org/10.3390/math11092065 - 26 Apr 2023

Cited by 1 | Viewed by 739

Abstract

In this paper, we conduct a theoretical examination of a low-rank matrix single-index model. This model has recently been introduced in the field of biostatistics, but its theoretical properties for jointly estimating the link function and the coefficient matrix have not yet been [...] Read more.

In this paper, we conduct a theoretical examination of a low-rank matrix single-index model. This model has recently been introduced in the field of biostatistics, but its theoretical properties for jointly estimating the link function and the coefficient matrix have not yet been fully explored. In this paper, we make use of the PAC-Bayesian bounds technique to provide a thorough theoretical understanding of the joint estimation of the link function and the coefficient matrix. This allows us to gain a deeper insight into the properties of this model and its potential applications in different fields. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

14 pages, 295 KiB

Open AccessArticle

Non-Asymptotic Bounds of AIPW Estimators for Means with Missingness at Random

by Fei Wang and Yuhao Deng

Mathematics 2023, 11(4), 818; https://doi.org/10.3390/math11040818 - 06 Feb 2023

Viewed by 1086

Abstract

The augmented inverse probability weighting is well known for its double robustness in missing data and causal inference. If either the propensity score model or the outcome regression model is correctly specified, the estimator is guaranteed to be consistent. Another important property of [...] Read more.

The augmented inverse probability weighting is well known for its double robustness in missing data and causal inference. If either the propensity score model or the outcome regression model is correctly specified, the estimator is guaranteed to be consistent. Another important property of the augmented inverse probability weighting is that it can achieve first-order equivalence to the oracle estimator in which all nuisance parameters are known, even if the fitted models do not converge at the parametric root-n rate. We explore the non-asymptotic properties of the augmented inverse probability weighting estimator to infer the population mean with missingness at random. We also consider inferences of the mean outcomes on the observed group and on the unobserved group. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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23 pages, 586 KiB

Open AccessArticle

Representation Theorem and Functional CLT for RKHS-Based Function-on-Function Regressions

by Hengzhen Huang, Guangni Mo, Haiou Li and Hong-Bin Fang

Mathematics 2022, 10(14), 2507; https://doi.org/10.3390/math10142507 - 19 Jul 2022

Viewed by 1144

Abstract

We investigate a nonparametric, varying coefficient regression approach for modeling and estimating the regression effects caused by two functionally correlated datasets. Due to modern biomedical technology to measure multiple patient features during a time interval or intermittently at several discrete time points to [...] Read more.

We investigate a nonparametric, varying coefficient regression approach for modeling and estimating the regression effects caused by two functionally correlated datasets. Due to modern biomedical technology to measure multiple patient features during a time interval or intermittently at several discrete time points to review underlying biological mechanisms, statistical models that do not properly incorporate interventions and their dynamic responses may lead to biased estimates of the intervention effects. We propose a shared parameter change point function-on-function regression model to evaluate the pre- and post-intervention time trends and develop a likelihood-based method for estimating the intervention effects and other parameters. We also propose new methods for estimating and hypothesis testing regression parameters for functional data via reproducing kernel Hilbert space. The estimators of regression parameters are closed-form without computation of the inverse of a large matrix, and hence are less computationally demanding and more applicable. By establishing a representation theorem and a functional central limit theorem, the asymptotic properties of the proposed estimators are obtained, and the corresponding hypothesis tests are proposed. Application and the statistical properties of our method are demonstrated through an immunotherapy clinical trial of advanced myeloma and simulation studies. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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29 pages, 551 KiB

Open AccessArticle

Sharper Sub-Weibull Concentrations

by Huiming Zhang and Haoyu Wei

Mathematics 2022, 10(13), 2252; https://doi.org/10.3390/math10132252 - 27 Jun 2022

Cited by 9 | Viewed by 1584

Abstract

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: [...] Read more.

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an

ℓ_{2}

-error of estimated coefficients in negative binomial regressions when the heavy-tailed covariates are sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions. In applying random matrices, we derive non-asymptotic versions of Bai-Yin’s theorem for sub-Weibull entries with exponential tail bounds. Finally, by demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator without the second-moment condition, we discuss and define the sub-Weibull type robust estimator for independent observations

{X_{i}}_{i = 1}^{n}

without exponential-moment conditions. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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15 pages, 409 KiB

Open AccessArticle

Group Logistic Regression Models with l_p,q Regularization

by Yanfang Zhang, Chuanhua Wei and Xiaolin Liu

Mathematics 2022, 10(13), 2227; https://doi.org/10.3390/math10132227 - 25 Jun 2022

Cited by 6 | Viewed by 1478

Abstract

In this paper, we proposed a logistic regression model with

l_{p, q}

regularization that could give a group sparse solution. The model could be applied to variable-selection problems with sparse group structures. In the context of big data, the solutions for [...] Read more.

In this paper, we proposed a logistic regression model with

l_{p, q}

regularization that could give a group sparse solution. The model could be applied to variable-selection problems with sparse group structures. In the context of big data, the solutions for practical problems are often group sparse, so it is necessary to study this kind of model. We defined the model from three perspectives: theoretical, algorithmic and numeric. From the theoretical perspective, by introducing the notion of the group restricted eigenvalue condition, we gave the oracle inequality, which was an important property for the variable-selection problems. The global recovery bound was also established for the logistic regression model with

l_{p, q}

regularization. From the algorithmic perspective, we applied the well-known alternating direction method of multipliers (ADMM) algorithm to solve the model. The subproblems for the ADMM algorithm were solved effectively. From the numerical perspective, we performed experiments for simulated data and real data in the factor stock selection. We employed the ADMM algorithm that we presented in the paper to solve the model. The numerical results were also presented. We found that the model was effective in terms of variable selection and prediction. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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25 pages, 414 KiB

Open AccessEditor’s ChoiceArticle

Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations

by Shaomin Li, Haoyu Wei and Xiaoyu Lei

Mathematics 2022, 10(10), 1700; https://doi.org/10.3390/math10101700 - 16 May 2022

Cited by 2 | Viewed by 1721

Abstract

Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion [...] Read more.

Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion estimation for the heterogeneous NBR models, which model the dispersion parameter as a function. Specifically, we proposed a double regression and applied a double

ℓ_{1}

-penalty to both regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for the lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demonstrate that the new methods are effective. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

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