Statistical Inference in Linear Models

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (15 January 2023) | Viewed by 17826

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Center of Mathematics and Applications and Department of Mathematics, University of Beira Interior, 6201-001 Covilhã, Portugal
Interests: applied statistics; computational mathematical methods; distribution theory; linear models; prediction; statistical inference
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Special Issue Information

Dear Colleagues,

Linear models are very important statistical models with a role in several fields of science, and are of practical importance in statistics. The most typical is the linear regression model. Many phenomena, such as those in biology, medicine, economics, management, geology, meteorology, agriculture and industry, can be approximately described by linear models. The further research and development of linear models is still a very active research subject.

In this Special Issue, we invite front-line researchers and authors to submit novel and original research. Potential topics include, but are not limited to:

  • Prediction and testing in linear models;
  • Regression and linear models;
  • Econometrics;
  • Robustness of relevant statistical methods;
  • Modelling and simulation;
  • Estimation of variance components;
  • Parameter estimation in linear models
  • Sampling techniques;
  • Applications of linear models;
  • Design and analysis of experiments;
  • Statistical applications; 
  • Generalized linear models.

The collection of papers will be complete in the sense that it will reflect the state of the art in the area and contain all recent important developments, keeping in mind that the Issue is for the benefit of both young researchers coming into the field as well as seasoned researchers.

For further information, please send an email to: sandraf@ubi.pt.

Dr. Sandra Ferreira
Guest Editor

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Published Papers (11 papers)

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Research

22 pages, 3067 KiB  
Article
A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data
by Daniel Maposa, Amon Masache and Precious Mdlongwa
Math. Comput. Appl. 2023, 28(4), 86; https://doi.org/10.3390/mca28040086 - 24 Jul 2023
Viewed by 790
Abstract
Exploration of solar irradiance can greatly assist in understanding how renewable energy can be better harnessed. It helps in establishing the solar irradiance climate in a particular region for effective and efficient harvesting of solar energy. Understanding the climate provides planners, designers and [...] Read more.
Exploration of solar irradiance can greatly assist in understanding how renewable energy can be better harnessed. It helps in establishing the solar irradiance climate in a particular region for effective and efficient harvesting of solar energy. Understanding the climate provides planners, designers and investors in the solar power generation sector with critical information. However, a detailed exploration of these climatic characteristics has not yet been studied for the Southern African data. Very little exploration is being done through the use of measures of centrality only. These descriptive statistics may be misleading. As a result, we overcome limitations in the currently used deterministic models through the application of distributional modelling through quantile functions. Deterministic and stochastic elements in the data were combined and analysed simultaneously when fitting quantile distributional function models. The fitted models were then used to find population means as explorative parameters that consist of both deterministic and stochastic properties of the data. The application of QFs has been shown to be a practical tool and gives more information than approaches that focus separately on either measures of central tendency or empirical distributions. Seasonal effects were detected in the data from the whole region and can be attributed to the cyclical behaviour exhibited. Daily maximum solar irradiation is taking place within two hours of midday and monthly accumulates in summer months. Windhoek is receiving the best daily total mean, while the maximum monthly accumulated total mean is taking place in Durban. Developing separate solar irradiation models for summer and winter is highly recommended. Though robust and rigorous, quantile distributional function modelling enables exploration and understanding of all components of the behaviour of the data being studied. Therefore, a starting base for understanding Southern Africa’s solar climate was developed in this study. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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19 pages, 510 KiB  
Article
A New Sine Family of Generalized Distributions: Statistical Inference with Applications
by SidAhmed Benchiha, Laxmi Prasad Sapkota, Aned Al Mutairi, Vijay Kumar, Rana H. Khashab, Ahmed M. Gemeay, Mohammed Elgarhy and Said G. Nassr
Math. Comput. Appl. 2023, 28(4), 83; https://doi.org/10.3390/mca28040083 - 16 Jul 2023
Cited by 4 | Viewed by 1691
Abstract
In this article, we extensively study a family of distributions using the trigonometric function. We add an extra parameter to the sine transformation family and name it the alpha-sine-G family of distributions. Some important functional forms and properties of the family are provided [...] Read more.
In this article, we extensively study a family of distributions using the trigonometric function. We add an extra parameter to the sine transformation family and name it the alpha-sine-G family of distributions. Some important functional forms and properties of the family are provided in a general form. A specific sub-model alpha-sine Weibull of this family is also introduced using the Weibull distribution as a parent distribution and studied deeply. The statistical properties of this new distribution are investigated and intended parameters are estimated using the maximum likelihood, maximum product of spacings, least square, weighted least square, and minimum distance methods. For further justification of these estimates, a simulation experiment is carried out. Two real data sets are analyzed to show the suggested model’s application. The suggested model performed well compares to some existing models considered in the study. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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9 pages, 1002 KiB  
Article
Computation of the Distribution of the Sum of Independent Negative Binomial Random Variables
by Marc Girondot and Jon Barry
Math. Comput. Appl. 2023, 28(3), 63; https://doi.org/10.3390/mca28030063 - 28 Apr 2023
Cited by 1 | Viewed by 1968
Abstract
The distribution of the sum of negative binomial random variables has a special role in insurance mathematics, actuarial sciences, and ecology. Two methods to estimate this distribution have been published: a finite-sum exact expression and a series expression by convolution. We compare both [...] Read more.
The distribution of the sum of negative binomial random variables has a special role in insurance mathematics, actuarial sciences, and ecology. Two methods to estimate this distribution have been published: a finite-sum exact expression and a series expression by convolution. We compare both methods, as well as a new normalized saddlepoint approximation, and normal and single distribution negative binomial approximations. We show that the exact series expression used lots of memory when the number of random variables was high (>7). The normalized saddlepoint approximation gives an output with a high relative error (around 3–5%), which can be a problem in some situations. The convolution method is a good compromise for applied practitioners, considering the amount of memory used, the computing time, and the precision of the estimates. However, a simplistic implementation of the algorithm could produce incorrect results due to the non-monotony of the convergence rate. The tolerance limit must be chosen depending on the expected magnitude order of the estimate, for which we used the answer generated by the saddlepoint approximation. Finally, the normal and negative binomial approximations should not be used, as they produced outputs with a very low accuracy. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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14 pages, 502 KiB  
Article
Prediction Interval for Compound Conway–Maxwell–Poisson Regression Model with Application to Vehicle Insurance Claim Data
by Jahnavi Merupula, V. S. Vaidyanathan and Christophe Chesneau
Math. Comput. Appl. 2023, 28(2), 39; https://doi.org/10.3390/mca28020039 - 09 Mar 2023
Viewed by 1352
Abstract
Regression models in which the response variable has a compound distribution have applications in actuarial science. For example, the aggregate claim amount in a vehicle insurance portfolio can be modeled using a compound Poisson distribution. In this paper, we propose a regression model, [...] Read more.
Regression models in which the response variable has a compound distribution have applications in actuarial science. For example, the aggregate claim amount in a vehicle insurance portfolio can be modeled using a compound Poisson distribution. In this paper, we propose a regression model, wherein the response variable is assumed to have a compound Conway–Maxwell–Poisson (CMP) distribution. This distribution is a parsimonious two-parameter Poisson distribution that accounts for both over- and under-dispersed count data, making it more suitable for application in various fields. A two-part methodology in the framework of a generalized linear model is proposed to estimate the parameters. Additionally, a method to obtain the prediction interval of the response variable is developed. The workings of the proposed methodology are illustrated through simulated data. An application of the compound CMP regression model to real-life vehicle insurance claims data is presented. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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25 pages, 3248 KiB  
Article
The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data
by Suleman Nasiru, Abdul Ghaniyyu Abubakari and Christophe Chesneau
Math. Comput. Appl. 2023, 28(1), 25; https://doi.org/10.3390/mca28010025 - 14 Feb 2023
Cited by 2 | Viewed by 1649
Abstract
The usefulness of (probability) distributions in the field of biomedical science cannot be underestimated. Hence, several distributions have been used in this field to perform statistical analyses and make inferences. In this study, we develop the arctan power (AP) distribution and illustrate its [...] Read more.
The usefulness of (probability) distributions in the field of biomedical science cannot be underestimated. Hence, several distributions have been used in this field to perform statistical analyses and make inferences. In this study, we develop the arctan power (AP) distribution and illustrate its application using biomedical data. The distribution is flexible in the sense that its probability density function exhibits characteristics such as left-skewedness, right-skewedness, and J and reversed-J shapes. The characteristic of the corresponding hazard rate function also suggests that the distribution is capable of modeling data with monotonic and non-monotonic failure rates. A bivariate extension of the AP distribution is also created to model the interdependence of two random variables or pairs of data. The application reveals that the AP distribution provides a better fit to the biomedical data than other existing distributions. The parameters of the distribution can also be fairly accurately estimated using a Bayesian approach, which is also elaborated. To end the study, the quantile and modal regression models based on the AP distribution provided better fits to the biomedical data than other existing regression models. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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15 pages, 1203 KiB  
Article
Forecasting Financial and Macroeconomic Variables Using an Adaptive Parameter VAR-KF Model
by Nat Promma and Nawinda Chutsagulprom
Math. Comput. Appl. 2023, 28(1), 19; https://doi.org/10.3390/mca28010019 - 02 Feb 2023
Cited by 2 | Viewed by 1463
Abstract
The primary objective of this article is to present an adaptive parameter VAR-KF technique (APVAR-KF) to forecast stock market performance and macroeconomic factors. The method exploits a vector autoregressive model as a system identification technique, and the Kalman filter is served as a [...] Read more.
The primary objective of this article is to present an adaptive parameter VAR-KF technique (APVAR-KF) to forecast stock market performance and macroeconomic factors. The method exploits a vector autoregressive model as a system identification technique, and the Kalman filter is served as a recursive state parameter estimation tool. A further development was designed by incorporating the GARCH model to quantify an automatic observation covariance matrix in the Kalman filter step. To verify the efficiency of our proposed method, we conducted an experimental simulation applied to the main stock exchange index, real effective exchange rate and consumer price index of Thailand and Indonesia from January 1997 to May 2021. The APVAR-KF method is generally shown to have a superior performance relative to the conventional VAR(1) model and the VAR-KF model with constant parameters. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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14 pages, 1410 KiB  
Article
Pseudo-Poisson Distributions with Concomitant Variables
by Barry C. Arnold and Bangalore G. Manjunath
Math. Comput. Appl. 2023, 28(1), 11; https://doi.org/10.3390/mca28010011 - 12 Jan 2023
Viewed by 1398
Abstract
It has been argued in Arnold and Manjunath (2021) that the bivariate pseudo-Poisson distribution will be the model of choice for bivariate data with one equidispersed marginal and the other marginal over-dispersed. This is due to its simple structure, straightforward parameter estimation and [...] Read more.
It has been argued in Arnold and Manjunath (2021) that the bivariate pseudo-Poisson distribution will be the model of choice for bivariate data with one equidispersed marginal and the other marginal over-dispersed. This is due to its simple structure, straightforward parameter estimation and fast computation. In the current note, we introduce the effects of concomitant variables on the bivariate pseudo-Poisson parameters and explore the distributional and inferential aspects of the augmented models. We also include a small simulation study and an example of application to real-life data. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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27 pages, 4189 KiB  
Article
New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression
by Suleman Nasiru, Abdul Ghaniyyu Abubakari and Christophe Chesneau
Math. Comput. Appl. 2022, 27(6), 105; https://doi.org/10.3390/mca27060105 - 03 Dec 2022
Cited by 3 | Viewed by 1626
Abstract
Probability distributions are very useful in modeling lifetime datasets. However, no specific distribution is suitable for all kinds of datasets. In this study, the bounded truncated Cauchy power exponential distribution is proposed for modeling datasets on the unit interval. The probability density function [...] Read more.
Probability distributions are very useful in modeling lifetime datasets. However, no specific distribution is suitable for all kinds of datasets. In this study, the bounded truncated Cauchy power exponential distribution is proposed for modeling datasets on the unit interval. The probability density function exhibits desirable shapes, such as left-skewed, right-skewed, reversed J, and bathtub shapes, whereas the hazard rate function displays J and bathtub shapes. For the purpose of modeling dependence between measures in a dataset, a bivariate extension of the proposed distribution is developed. The bivariate probability density function displays monotonic and non-monotonic shapes, making it suitable for modeling complex bivariate relations. Subsequently, the applications of the distribution are illustrated using COVID-19 data. The results revealed that the new distribution provides a better fit to the datasets compared to other existing distributions. Finally, a new quantile regression model is developed and its application demonstrated. The generated quantile regression model offers a decent fit to the data, according to the residual analysis. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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23 pages, 4958 KiB  
Article
Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves
by Abdisalam Hassan Muse, Christophe Chesneau, Oscar Ngesa and Samuel Mwalili
Math. Comput. Appl. 2022, 27(6), 104; https://doi.org/10.3390/mca27060104 - 30 Nov 2022
Cited by 3 | Viewed by 1800
Abstract
This study aims to propose a flexible, fully parametric hazard-based regression model for censored time-to-event data with crossing survival curves. We call it the accelerated hazard (AH) model. The AH model can be written with or without a baseline distribution for lifetimes. The [...] Read more.
This study aims to propose a flexible, fully parametric hazard-based regression model for censored time-to-event data with crossing survival curves. We call it the accelerated hazard (AH) model. The AH model can be written with or without a baseline distribution for lifetimes. The former assumption results in parametric regression models, whereas the latter results in semi-parametric regression models, which are by far the most commonly used in time-to-event analysis. However, under certain conditions, a parametric hazard-based regression model may produce more efficient estimates than a semi-parametric model. The parametric AH model, on the other hand, is inappropriate when the baseline distribution is exponential because it is constant over time; similarly, when the baseline distribution is the Weibull distribution, the AH model coincides with the accelerated failure time (AFT) and proportional hazard (PH) models. The use of a versatile parametric baseline distribution (generalized log-logistic distribution) for modeling the baseline hazard rate function is investigated. For the parameters of the proposed AH model, the classical (via maximum likelihood estimation) and Bayesian approaches using noninformative priors are discussed. A comprehensive simulation study was conducted to assess the performance of the proposed model’s estimators. A real-life right-censored gastric cancer dataset with crossover survival curves is used to demonstrate the tractability and utility of the proposed fully parametric AH model. The study concluded that the parametric AH model is effective and could be useful for assessing a variety of survival data types with crossover survival curves. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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15 pages, 811 KiB  
Article
Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA
by Marwa K. H. Hassan and Christophe Chesneau
Math. Comput. Appl. 2022, 27(4), 72; https://doi.org/10.3390/mca27040072 - 19 Aug 2022
Cited by 6 | Viewed by 1373
Abstract
The generalized half-logistic distribution is ideal to fit the lifetime of some products, such as ball bearings and electrical insulation. In this paper, we aim to extend this scope by creating a motivated bivariate version. We thus introduce the bivariate generalized half-logistic distribution [...] Read more.
The generalized half-logistic distribution is ideal to fit the lifetime of some products, such as ball bearings and electrical insulation. In this paper, we aim to extend this scope by creating a motivated bivariate version. We thus introduce the bivariate generalized half-logistic distribution using the Farlie Gumbel Morgenstern (FGM) copula, which is called the FGM bivariate generalized half-logistic distribution (FGMBGHLD for short). In particular, the FGMBGHLD finds application in describing bivariate lifetime datasets that have weak correlations between variables. Some statistical properties and functions of our new distribution, such as the product moments, moment generating function, reliability function, and hazard rate function, are derived. We discuss the maximum likelihood estimation method of the FGMBGHLD parameters. As an application of the FGMBGHLD in reliability, we consider the stress–strength model when the stress and strength random variables are dependent. We also derive the point and interval estimates of the stress–strength coefficient. Finally, we use the data from the household income and expenditure survey of KSA 2018 for Saudi households by administrative region to demonstrate the practicability of the proposed model. A comparison with a modern bivariate Weibull distribution is performed. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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24 pages, 515 KiB  
Article
Odd Exponential-Logarithmic Family of Distributions: Features and Modeling
by Christophe Chesneau, Lishamol Tomy, Meenu Jose and Kuttappan Vallikkattil Jayamol
Math. Comput. Appl. 2022, 27(4), 68; https://doi.org/10.3390/mca27040068 - 08 Aug 2022
Cited by 3 | Viewed by 1543
Abstract
This paper introduces a general family of continuous distributions, based on the exponential-logarithmic distribution and the odd transformation. It is called the “odd exponential logarithmic family”. We intend to create novel distributions with desired qualities for practical applications, using the unique properties of [...] Read more.
This paper introduces a general family of continuous distributions, based on the exponential-logarithmic distribution and the odd transformation. It is called the “odd exponential logarithmic family”. We intend to create novel distributions with desired qualities for practical applications, using the unique properties of the exponential-logarithmic distribution as an initial inspiration. Thus, we present some special members of this family that stand out for the versatile shape properties of their corresponding functions. Then, a comprehensive mathematical treatment of the family is provided, including some asymptotic properties, the determination of the quantile function, a useful sum expression of the probability density function, tractable series expressions for the moments, moment generating function, Rényi entropy and Shannon entropy, as well as results on order statistics and stochastic ordering. We estimate the model parameters quite efficiently by the method of maximum likelihood, with discussions on the observed information matrix and a complete simulation study. As a major interest, the odd exponential logarithmic models reveal how to successfully accommodate various kinds of data. This aspect is demonstrated by using three practical data sets, showing that an odd exponential logarithmic model outperforms two strong competitors in terms of data fitting. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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