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11 pages, 271 KiB

Open AccessArticle

Stability of the Stochastic Ginzburg–Landau–Newell Equations in Two Dimensions

by Jing Wang and Yan Zheng

Axioms 2024, 13(6), 412; https://doi.org/10.3390/axioms13060412 - 19 Jun 2024

Viewed by 313

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of [...] Read more.

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple

H^{2} \subset H^{1} \subset H^{0}

to obtain a bound on the difference between the original solution and the perturbed solution. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

21 pages, 2294 KiB

Open AccessArticle

Statistical Advancement of a Flexible Unitary Distribution and Its Applications

by Hugo S. Salinas, Hassan S. Bakouch, Fatimah E. Almuhayfith, Wilson E. Caimanque, Leonardo Barrios-Blanco and Olayan Albalawi

Axioms 2024, 13(6), 397; https://doi.org/10.3390/axioms13060397 - 14 Jun 2024

Viewed by 531

Abstract

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. [...] Read more.

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. Therefore, it presents an alternative for modeling this type of data. Several mathematical and statistical properties of this distribution have been derived, such as moments, hazard function, the Bonferroni curve, and entropy. Moreover, we investigate the characterizations of the proposed distribution based on its hazard function. Parameter estimation has been performed using both the maximum likelihood method and method of the moments. Because of this, we were able to determine the best critical region and the information matrix, facilitating the calculation of asymptotic confidence intervals. A simulation study is presented to analyze the behavior of the obtained estimators for different sample sizes. To demonstrate the suitability of the proposed distribution, applications and goodness-of-fit tests have been performed on two practical data sets. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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17 pages, 383 KiB

Open AccessArticle

The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

by Wei Yang, Xueting Ji and Jiujun Zhang

Axioms 2024, 13(6), 393; https://doi.org/10.3390/axioms13060393 - 12 Jun 2024

Viewed by 350

Abstract

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses [...] Read more.

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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10 pages, 285 KiB

Open AccessArticle

Application of the Concept of Statistical Causality in Integrable Increasing Processes and Measures

by Dragana Valjarević, Vladica Stojanović and Aleksandar Valjarević

Axioms 2024, 13(2), 124; https://doi.org/10.3390/axioms13020124 - 17 Feb 2024

Viewed by 863

Abstract

In this paper, we investigate an application of the statistical concept of causality, based on Granger’s definition of causality, on raw increasing processes as well as on optional and predictable measures. A raw increasing process is optional (predictable) if the bounded (left-continuous) process [...] Read more.

In this paper, we investigate an application of the statistical concept of causality, based on Granger’s definition of causality, on raw increasing processes as well as on optional and predictable measures. A raw increasing process is optional (predictable) if the bounded (left-continuous) process X, associated with the measure

μ_{A} (X)

, is self-caused. Also, the measure

μ_{A} (X)

is optional (predictable) if an associated process X is self-caused with some additional assumptions. Some of the obtained results, in terms of self-causality, can be directly applied to defining conditions for an optional stopping time to become predictable. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

25 pages, 835 KiB

Open AccessArticle

Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

by Vladica S. Stojanović, Hassan S. Bakouch, Zorica Gajtanović, Fatimah E. Almuhayfith and Kristijan Kuk

Axioms 2024, 13(1), 40; https://doi.org/10.3390/axioms13010040 - 7 Jan 2024

Viewed by 1153

Abstract

This paper presents a novel count time-series model, named integer-valued Split-BREAK process of the first order, abbr. INSB(1) model. This process is examined in terms of its basic stochastic properties, such as stationarity, mean, variance and correlation structure. In addition, the marginal distribution, [...] Read more.

This paper presents a novel count time-series model, named integer-valued Split-BREAK process of the first order, abbr. INSB(1) model. This process is examined in terms of its basic stochastic properties, such as stationarity, mean, variance and correlation structure. In addition, the marginal distribution, over-dispersion and zero-inflation properties of the INSB(1) process are also examined. To estimate the unknown parameters of the INSB(1) process, an estimation procedure based on probability generating functions (PGFs) is proposed. For the obtained estimators, their asymptotic properties, as well as the appropriate simulation study, are examined. Finally, the INSB(1) process is applied in the dynamic analysis of some real-world series, namely, the numbers of serious traffic accidents in Serbia and forest fires in Greece. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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16 pages, 480 KiB

Open AccessArticle

Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data

by Fatimah E. Almuhayfith, Anuresha Krishna, Radhakumari Maya, Muhammad Rasheed Irshad, Hassan S. Bakouch and Munirah Almulhim

Axioms 2024, 13(1), 32; https://doi.org/10.3390/axioms13010032 - 31 Dec 2023

Viewed by 1341

Abstract

INAR models have the great advantage of being able to capture the conditional distribution of a count time series based on their past observations, thus allowing it to be tailored to meet the unique characteristics of count data. This paper reviews the two-parameter [...] Read more.

INAR models have the great advantage of being able to capture the conditional distribution of a count time series based on their past observations, thus allowing it to be tailored to meet the unique characteristics of count data. This paper reviews the two-parameter Poisson extended exponential (PEE) distribution and its corresponding INAR(1) process. Then the INAR of order p (INAR(p)) model that incorporates PEE innovations is proposed, its statistical properties are presented, and its parameters are estimated using conditional least squares and conditional maximum likelihood estimation methods. Two practical data sets are analyzed and compared with competing INAR models in an effort to gauge the performance of the proposed model. It is found that the proposed model performs better than the competitors. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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16 pages, 4693 KiB

Open AccessArticle

An Enhanced Spatial Capture Model for Population Analysis Using Unidentified Counts through Camera Encounters

by Mohamed Jaber, Farag Hamad, Robert D. Breininger and Nezamoddin N. Kachouie

Axioms 2023, 12(12), 1094; https://doi.org/10.3390/axioms12121094 - 29 Nov 2023

Viewed by 907

Abstract

Spatial capture models are broadly used for population analysis in ecological statistics. Spatial capture models for unidentified individuals rely on data augmentation to create a zero-inflated population. The unknown true population size can be considered as the number of successes of a binomial [...] Read more.

Spatial capture models are broadly used for population analysis in ecological statistics. Spatial capture models for unidentified individuals rely on data augmentation to create a zero-inflated population. The unknown true population size can be considered as the number of successes of a binomial distribution with an unknown number of independent trials and an unknown probability of success. Augmented population size is a realization of the unknown number of trials and is recommended to be much larger than the unknown population size. As a result, the probability of success of binomial distribution, i.e., the unknown probability that a hypothetical individual in the augmented population belongs to the true population, can be obtained by dividing the unknown true population size by the augmented population size. This is an inverse problem as neither the true population size nor the probability of success is known, and the accuracy of their estimates strongly relies on the augmented population size. Therefore, the estimated population size in spatial capture models is very sensitive to the size of a zero-inflated population and in turn to the estimated probability of success. This is an important issue in spatial capture models as a typical count model with censored data (unidentified and/or undetected). Hence, in this research, we investigated the sensitivity and accuracy of the spatial capture model to address this problem with the objective of improving the robustness of the model. We demonstrated that the estimated population size using the proposed enhanced capture model was more accurate in comparison with the previous spatial capture model. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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26 pages, 13370 KiB

Open AccessArticle

Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride

by Ahmed Elshahhat, Osama E. Abo-Kasem and Heba S. Mohammed

Axioms 2023, 12(6), 601; https://doi.org/10.3390/axioms12060601 - 16 Jun 2023

Cited by 3 | Viewed by 959

Abstract

A new Type-II generalized progressively hybrid censoring strategy, in which the experiment is ensured to stop at a specified time, is explored when the lifetime model of the test subjects follows a two-parameter alpha-power inverted exponential (Alpha-PIE) distribution. Alpha-PIE’s parameters and reliability indices, [...] Read more.

A new Type-II generalized progressively hybrid censoring strategy, in which the experiment is ensured to stop at a specified time, is explored when the lifetime model of the test subjects follows a two-parameter alpha-power inverted exponential (Alpha-PIE) distribution. Alpha-PIE’s parameters and reliability indices, such as reliability and hazard rate functions, are estimated via maximum likelihood and Bayes estimation methodologies in the presence of the proposed censored data. The estimated confidence intervals of the unknown quantities are created using the normal approximation of the acquired classical estimators. The Bayesian estimators are also produced using independent gamma density priors under symmetrical (squared-error) loss. The Bayes’ estimators and their associated highest posterior density intervals cannot be calculated theoretically since the joint likelihood function is derived in a complicated form, but they can potentially be assessed using Monte Carlo Markov-chain algorithms. We next go through four optimality criteria for identifying the best progressive design. The effectiveness of the suggested estimation procedures is assessed using Monte Carlo comparisons, and certain recommendations are offered. Ultimately, two different applications, one focused on the failure times of electronic tubes and the other on vinyl chloride, are analyzed to illustrate the effectiveness of the proposed techniques that may be employed in real-world scenarios. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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