Mathematics

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16 pages, 309 KiB

Open AccessArticle

Variable Selection for Spatial Logistic Autoregressive Models

by Jiaxuan Liang, Yi Cheng, Yuqi Su, Shuyue Xiao and Yunquan Song

Mathematics 2022, 10(17), 3095; https://doi.org/10.3390/math10173095 - 29 Aug 2022

Viewed by 1454

When the spatial response variables are discrete, the spatial logistic autoregressive model adds an additional network structure to the ordinary logistic regression model to improve the classification accuracy. With the emergence of high-dimensional data in various fields, sparse spatial logistic regression models have [...] Read more.

When the spatial response variables are discrete, the spatial logistic autoregressive model adds an additional network structure to the ordinary logistic regression model to improve the classification accuracy. With the emergence of high-dimensional data in various fields, sparse spatial logistic regression models have attracted a great deal of interest from researchers. For the high-dimensional spatial logistic autoregressive model, in this paper, we propose a variable selection method with for the spatial logistic model. To identify important variables and make predictions, one efficient algorithm is employed to solve the penalized likelihood function. Simulations and a real example show that our methods perform well in a limited sample. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

18 pages, 1892 KiB

Open AccessEditor’s ChoiceArticle

A Robust Variable Selection Method for Sparse Online Regression via the Elastic Net Penalty

by Wentao Wang, Jiaxuan Liang, Rong Liu, Yunquan Song and Min Zhang

Mathematics 2022, 10(16), 2985; https://doi.org/10.3390/math10162985 - 18 Aug 2022

Cited by 6 | Viewed by 1748

Abstract

Variable selection has been a hot topic, with various popular methods including lasso, SCAD, and elastic net. These penalized regression algorithms remain sensitive to noisy data. Furthermore, “concept drift” fundamentally distinguishes streaming data learning from batch learning. This article presents a method for [...] Read more.

Variable selection has been a hot topic, with various popular methods including lasso, SCAD, and elastic net. These penalized regression algorithms remain sensitive to noisy data. Furthermore, “concept drift” fundamentally distinguishes streaming data learning from batch learning. This article presents a method for noise-resistant regularization and variable selection in noisy data streams with multicollinearity, dubbed canal-adaptive elastic net, which is similar to elastic net and encourages grouping effects. In comparison to lasso, the canal adaptive elastic net is especially advantageous when the number of predictions (p) is significantly larger than the number of observations (n), and the data are multi-collinear. Numerous simulation experiments have confirmed that canal-adaptive elastic net has higher prediction accuracy than lasso, ridge regression, and elastic net in data with multicollinearity and noise. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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17 pages, 359 KiB

Open AccessArticle

An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem

by Serim Hong, Carlos A. Coelho and Junyong Park

Mathematics 2022, 10(16), 2953; https://doi.org/10.3390/math10162953 - 16 Aug 2022

Cited by 2 | Viewed by 1389

Abstract

The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending [...] Read more.

The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch’s t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens–Fisher problem, exhibiting precise sizes and better powers than Welch’s approach for those cases. Numerical studies show that the Welch’s t-test tends to be a bit more conservative than the test statistics based on the exact or near-exact distribution, in particular when sample sizes are small and unbalanced, situations in which the proposed exact or near-exact distributions obtain higher powers than Welch’s t-test. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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21 pages, 2942 KiB

Open AccessArticle

Inference and Local Influence Assessment in a Multifactor Skew-Normal Linear Mixed Model

by Zeinolabedin Najafi, Karim Zare, Mohammad Reza Mahmoudi, Soheil Shokri and Amir Mosavi

Mathematics 2022, 10(15), 2820; https://doi.org/10.3390/math10152820 - 08 Aug 2022

Cited by 3 | Viewed by 1268

Abstract

This work considers a multifactor linear mixed model under heteroscedasticity in random-effect factors and the skew-normal errors for modeling the correlated datasets. We implement an expectation–maximization (EM) algorithm to achieve the maximum likelihood estimates using conditional distributions of the skew-normal distribution. The EM [...] Read more.

This work considers a multifactor linear mixed model under heteroscedasticity in random-effect factors and the skew-normal errors for modeling the correlated datasets. We implement an expectation–maximization (EM) algorithm to achieve the maximum likelihood estimates using conditional distributions of the skew-normal distribution. The EM algorithm is also implemented to extend the local influence approach under three model perturbation schemes in this model. Furthermore, a Monte Carlo simulation is conducted to evaluate the efficiency of the estimators. Finally, a real data set is used to make an illustrative comparison among the following four scenarios: normal/skew-normal errors and heteroscedasticity/homoscedasticity in random-effect factors. The empirical studies show our methodology can improve the estimates when the model errors follow from a skew-normal distribution. In addition, the local influence analysis indicates that our model can decrease the effects of anomalous observations in comparison to normal ones. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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14 pages, 627 KiB

Open AccessArticle

Cumulative Residual Tsallis Entropy-Based Test of Uniformity and Some New Findings

by Mohamed S. Mohamed, Haroon M. Barakat, Salem A. Alyami and Mohamed A. Abd Elgawad

Mathematics 2022, 10(5), 771; https://doi.org/10.3390/math10050771 - 28 Feb 2022

Cited by 7 | Viewed by 1692

Abstract

The Tsallis entropy is an extension of the Shannon entropy and is used extensively in physics. The cumulative residual Tsallis entropy, which is a generalization of the Tsallis entropy, plays an important role in the measurement uncertainty of random variables and has simple [...] Read more.

The Tsallis entropy is an extension of the Shannon entropy and is used extensively in physics. The cumulative residual Tsallis entropy, which is a generalization of the Tsallis entropy, plays an important role in the measurement uncertainty of random variables and has simple relationships with other important information and reliability measures. In this paper, some novel properties of the cumulative residual Tsallis entropy are disclosed. Moreover, this entropy measure is applied to testing the uniformity, where the limit distribution and an approximation of the distribution of the test statistic are derived. In addition, the property of stability is discussed. Furthermore, the percentage points and power against seven alternative distributions of this test statistic are presented. Finally, to compare the power of the suggested test with that of other tests of uniformity, a simulation study is conducted. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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9 pages, 451 KiB

Open AccessArticle

Bayesian Inference for Stochastic Cusp Catastrophe Model with Partially Observed Data

by Ding-Geng Chen, Haipeng Gao and Chuanshu Ji

Mathematics 2021, 9(24), 3245; https://doi.org/10.3390/math9243245 - 15 Dec 2021

Viewed by 1738

Abstract

The purpose of this paper is to develop a data augmentation technique for statistical inference concerning stochastic cusp catastrophe model subject to missing data and partially observed observations. We propose a Bayesian inference solution that naturally treats missing observations as parameters and we [...] Read more.

The purpose of this paper is to develop a data augmentation technique for statistical inference concerning stochastic cusp catastrophe model subject to missing data and partially observed observations. We propose a Bayesian inference solution that naturally treats missing observations as parameters and we validate this novel approach by conducting a series of Monte Carlo simulation studies assuming the cusp catastrophe model as the underlying model. We demonstrate that this Bayesian data augmentation technique can recover and estimate the underlying parameters from the stochastic cusp catastrophe model. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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12 pages, 745 KiB

Open AccessArticle

Stability of Dependencies of Contingent Subgroups with Merged Groups: Vaccination Case Study

by Tomas Macak

Mathematics 2021, 9(22), 2917; https://doi.org/10.3390/math9222917 - 16 Nov 2021

Viewed by 1174

Abstract

The answers to extreme phenomena both in nature and in business sectors are the constructions of the distribution of random variables with extreme values. Another area in which appropriate theoretical research is conducted regarding the influence of suppressor (third) variables in categorical data. [...] Read more.

The answers to extreme phenomena both in nature and in business sectors are the constructions of the distribution of random variables with extreme values. Another area in which appropriate theoretical research is conducted regarding the influence of suppressor (third) variables in categorical data. When examining dependencies in PivotTables, we often find it necessary to merge data into larger sets (e.g., due to a greater number of theoretical frequencies lower than their critical value). A phenomenon many exist wherein the partial relation is stronger than the zero relation. For example, in such a combination, instability may occur, which indicates contingent subgroups with the merged group. The dependence of dependencies is practically manifested because the data of contingent subgroups indicate inconsistent (inverted) conclusions compared to the associated group. For this reason, this paper aimed to find the critical ratios of partial probabilities in the contingency table of subgroups of the original variables, and to determine the conditions of result consistency and contingency stability, including the proof. For practical use and for the ease of repeating the proposed procedure, the solution is based on a case study that compares the effectiveness of vaccination. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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20 pages, 1545 KiB

Open AccessArticle

Marginalized Two-Part Joint Modeling of Longitudinal Semi-Continuous Responses and Survival Data: With Application to Medical Costs

by Mohadeseh Shojaei Shahrokhabadi, (Din) Ding-Geng Chen, Sayed Jamal Mirkamali, Anoshirvan Kazemnejad and Farid Zayeri

Mathematics 2021, 9(20), 2603; https://doi.org/10.3390/math9202603 - 15 Oct 2021

Cited by 1 | Viewed by 1628

Abstract

Non-negative continuous outcomes with a substantial number of zero values and incomplete longitudinal follow-up are quite common in medical costs data. It is thus critical to incorporate the potential dependence of survival status and longitudinal medical costs in joint modeling, where censorship is [...] Read more.

Non-negative continuous outcomes with a substantial number of zero values and incomplete longitudinal follow-up are quite common in medical costs data. It is thus critical to incorporate the potential dependence of survival status and longitudinal medical costs in joint modeling, where censorship is death-related. Despite the wide use of conventional two-part joint models (CTJMs) to capture zero-inflation, they are limited to conditional interpretations of the regression coefficients in the model’s continuous part. In this paper, we propose a marginalized two-part joint model (MTJM) to jointly analyze semi-continuous longitudinal costs data and survival data. We compare it to the conventional two-part joint model (CTJM) for handling marginal inferences about covariate effects on average costs. We conducted a series of simulation studies to evaluate the superior performance of the proposed MTJM over the CTJM. To illustrate the applicability of the MTJM, we applied the model to a set of real electronic health record (EHR) data recently collected in Iran. We found that the MTJM yielded a smaller standard error, root-mean-square error of estimates, and AIC value, with unbiased parameter estimates. With this MTJM, we identified a significant positive correlation between costs and survival, which was consistent with the simulation results. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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18 pages, 1248 KiB

Open AccessArticle

Compositional Data Modeling through Dirichlet Innovations

by Seitebaleng Makgai, Andriette Bekker and Mohammad Arashi

Mathematics 2021, 9(19), 2477; https://doi.org/10.3390/math9192477 - 03 Oct 2021

Viewed by 1556

Abstract

The Dirichlet distribution is a well-known candidate in modeling compositional data sets. However, in the presence of outliers, the Dirichlet distribution fails to model such data sets, making other model extensions necessary. In this paper, the Kummer–Dirichlet distribution and the gamma distribution are [...] Read more.

The Dirichlet distribution is a well-known candidate in modeling compositional data sets. However, in the presence of outliers, the Dirichlet distribution fails to model such data sets, making other model extensions necessary. In this paper, the Kummer–Dirichlet distribution and the gamma distribution are coupled, using the beta-generating technique. This development results in the proposal of the Kummer–Dirichlet gamma distribution, which presents greater flexibility in modeling compositional data sets. Some general properties, such as the probability density functions and the moments are presented for this new candidate. The method of maximum likelihood is applied in the estimation of the parameters. The usefulness of this model is demonstrated through the application of synthetic and real data sets, where outliers are present. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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12 pages, 328 KiB

Open AccessArticle

High-Dimensional Mahalanobis Distances of Complex Random Vectors

by Deliang Dai and Yuli Liang

Mathematics 2021, 9(16), 1877; https://doi.org/10.3390/math9161877 - 07 Aug 2021

Cited by 1 | Viewed by 2212

Abstract

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed [...] Read more.

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio

c = p / n \to \infty

. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix

F = S_{2}^{- 1} S_{1}

—the product of a sample covariance matrix

S_{1}

(from the independent variable array

(b e {(Z_{i})}_{1 \times n}

) with the inverse of another covariance matrix

S_{2}

(from the independent variable array

{(Z_{j \neq i})}_{p \times n}

)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components

S_{1}

and

S_{2}

of the F-matrix is not required. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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18 pages, 1140 KiB

Open AccessArticle

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

by Wen Su and Yunyun Wang

Mathematics 2021, 9(12), 1402; https://doi.org/10.3390/math9121402 - 17 Jun 2021

Cited by 5 | Viewed by 1895

Abstract

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly [...] Read more.

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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18 pages, 1132 KiB

Open AccessArticle

An α-Monotone Generalized Log-Moyal Distribution with Applications to Environmental Data

by Talha Arslan

Mathematics 2021, 9(12), 1400; https://doi.org/10.3390/math9121400 - 17 Jun 2021

Cited by 2 | Viewed by 1586

Abstract

Modeling environmental data plays a crucial role in explaining environmental phenomena. In some cases, well-known distributions, e.g., Weibull, inverse Weibull, and Gumbel distributions, cannot model environmental events adequately. Therefore, many authors tried to find new statistical distributions to represent environmental phenomena more accurately. [...] Read more.

Modeling environmental data plays a crucial role in explaining environmental phenomena. In some cases, well-known distributions, e.g., Weibull, inverse Weibull, and Gumbel distributions, cannot model environmental events adequately. Therefore, many authors tried to find new statistical distributions to represent environmental phenomena more accurately. In this paper, an

α

-monotone generalized log-Moyal (

α

-GlogM) distribution is introduced and some statistical properties such as cumulative distribution function, hazard rate function (hrf), scale-mixture representation, and moments are derived. The hrf of the

α

-GlogM distribution can form a variety of shapes including the bathtub shape. The

α

-GlogM distribution converges to generalized half-normal (GHN) and inverse GHN distributions. It reduces to slash GHN and

α

-monotone inverse GHN distributions for certain parameter settings. Environmental data sets are used to show implementations of the

α

-GlogM distribution and also to compare its modeling performance with its rivals. The comparisons are carried out using well-known information criteria and goodness-of-fit statistics. The comparison results show that the

α

-GlogM distribution is preferable over its rivals in terms of the modeling capability. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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23 pages, 677 KiB

Open AccessArticle

Flexible Log-Linear Birnbaum–Saunders Model

by Guillermo Martínez-Flórez, Inmaculada Barranco-Chamorro and Héctor W. Gómez

Mathematics 2021, 9(11), 1188; https://doi.org/10.3390/math9111188 - 24 May 2021

Cited by 2 | Viewed by 1502

Abstract

Rieck and Nedelman (1991) introduced the sinh-normal distribution. This model was built as a transformation of a N(0,1) distribution. In this paper, a generalization based on a flexible skew normal distribution is introduced. In this way, a more general model is obtained that [...] Read more.

Rieck and Nedelman (1991) introduced the sinh-normal distribution. This model was built as a transformation of a N(0,1) distribution. In this paper, a generalization based on a flexible skew normal distribution is introduced. In this way, a more general model is obtained that can describe a range of asymmetric, unimodal and bimodal situations. The paper is divided into two parts. First, the properties of this new model, called flexible sinh-normal distribution, are obtained. In the second part, the flexible sinh-normal distribution is related to flexible Birnbaum–Saunders, introduced by Martínez-Flórez et al. (2019), to propose a log-linear model for lifetime data. Applications to real datasets are included to illustrate our findings. Full article

(This article belongs to the Special Issue Mathematical and Computational Statistics and Their Applications)

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