Axioms

Research

30 pages, 1356 KiB

Open AccessArticle

Estimation of the Reliability Function of the Generalized Rayleigh Distribution under Progressive First-Failure Censoring Model

by Qin Gong, Rui Chen, Haiping Ren and Fan Zhang

Axioms 2024, 13(9), 580; https://doi.org/10.3390/axioms13090580 - 26 Aug 2024

Cited by 1 | Viewed by 845

Abstract

This study investigates the statistical inference of the parameters, reliability function, and hazard function of the generalized Rayleigh distribution under progressive first-failure censoring samples, considering factors such as long product lifetime and challenging experimental conditions. Firstly, the progressive first-failure model is introduced, and [...] Read more.

This study investigates the statistical inference of the parameters, reliability function, and hazard function of the generalized Rayleigh distribution under progressive first-failure censoring samples, considering factors such as long product lifetime and challenging experimental conditions. Firstly, the progressive first-failure model is introduced, and the maximum likelihood estimation for the parameters, reliability function, and hazard function under this model are discussed. For interval estimation, confidence intervals have been constructed for the parameters, reliability function, and hazard function using the bootstrap method. Next, in Bayesian estimation, considering informative priors and non-information priors, the Bayesian estimation of the parameters, reliability function, and hazard function under symmetric and asymmetric loss functions is obtained using the MCMC method. Finally, Monte Carlo simulation is conducted to compare mean square errors, evaluating the superiority of the maximum likelihood estimation and Bayesian estimation under different loss functions. The performance of the estimation methods used in the study is illustrated through illustrative examples. The results indicate that Bayesian estimation outperforms maximum likelihood estimation. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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31 pages, 2503 KiB

Open AccessArticle

Chen-Burr XII Model as a Competing Risks Model with Applications to Real-Life Data Sets

by Zakiah I. Kalantan, Sulafah M. S. Binhimd, Heba N. Salem, Gannat R. AL-Dayian, Abeer A. EL-Helbawy and Mervat K. Abd Elaal

Axioms 2024, 13(8), 531; https://doi.org/10.3390/axioms13080531 - 5 Aug 2024

Cited by 1 | Viewed by 809

Abstract

In this paper Chen-Burr XII distribution is constructed and graphical description of the probability density function, hazard rate and reversed hazard rate functions of the proposed model is obtained. Also, some statistical characteristics of the Chen-Burr XII distribution are discussed and some new [...] Read more.

In this paper Chen-Burr XII distribution is constructed and graphical description of the probability density function, hazard rate and reversed hazard rate functions of the proposed model is obtained. Also, some statistical characteristics of the Chen-Burr XII distribution are discussed and some new models as sub-models from the Chen-Burr XII distribution are introduced. Moreover, maximum likelihood estimation of the parameters, reliability, hazard rate and reversed hazard rate functions of the Chen-Burr XII distribution are considered. Also, the asymptotic confidence intervals of the distribution parameters, reliability, hazard rate and reversed hazard rate functions are presented. Finally, three real life data sets are applied to prove how the Chen-Burr XII distribution can be applied in real life and to confirm its superiority over some existing distributions. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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10 pages, 240 KiB

Open AccessArticle

Measuring Sphericity in Positive Semi-Definite Matrices

by Dário Ferreira and Sandra S. Ferreira

Axioms 2024, 13(8), 512; https://doi.org/10.3390/axioms13080512 - 29 Jul 2024

Viewed by 912

Abstract

The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional [...] Read more.

The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

20 pages, 334 KiB

Open AccessArticle

Parameter Estimation in Spatial Autoregressive Models with Missing Data and Measurement Errors

by Tengjun Li, Zhikang Zhang and Yunquan Song

Axioms 2024, 13(5), 315; https://doi.org/10.3390/axioms13050315 - 10 May 2024

Viewed by 1816

Abstract

This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a [...] Read more.

This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a combination of inverse probability weighting (IPW) and mean imputation is utilized to mitigate the bias caused by missing data. Under several mild conditions, it is demonstrated that the proposed estimators are consistent and possess oracle properties. The efficacy of the proposed parameter estimation process is assessed through Monte Carlo simulation studies. Finally, the applicability of the proposed method is further substantiated using the Boston Housing Dataset. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

11 pages, 280 KiB

Open AccessArticle

Partial Derivatives Estimation of Multivariate Variance Function in Heteroscedastic Model via Wavelet Method

by Junke Kou and Hao Zhang

Axioms 2024, 13(1), 69; https://doi.org/10.3390/axioms13010069 - 20 Jan 2024

Viewed by 1345

Abstract

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. [...] Read more.

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

27 pages, 1051 KiB

Open AccessArticle

Fisher Information, Asymptotic Behavior, and Applications for Generalized Order Statistics and Their Concomitants Based on the Sarmanov Family

by Mohamed A. Abd Elgawad, Haroon M. Barakat, Islam A. Husseiny, Ghada M. Mansour, Salem A. Alyami, Ibrahim Elbatal and Metwally A. Alawady

Axioms 2024, 13(1), 17; https://doi.org/10.3390/axioms13010017 - 25 Dec 2023

Cited by 2 | Viewed by 1553

Abstract

In this paper, the Fisher information (FI), relevant to m-generalized order statistics (m-GOSs) and their concomitants of the shape-parameter of the Sarmanov family of bivariate distributions, is investigated. In addition, we study the concomitants of m-GOSs from this family. [...] Read more.

In this paper, the Fisher information (FI), relevant to m-generalized order statistics (m-GOSs) and their concomitants of the shape-parameter of the Sarmanov family of bivariate distributions, is investigated. In addition, we study the concomitants of m-GOSs from this family. Furthermore, we look at how those concomitants were distributed collectively. The FI contained in the scale and shape parameters of the exponential and power function distributions, respectively, in concomitants of m-GOSs is obtained. A study of the asymptotic behavior of the concomitants of ordinary order statistics is also provided. Some versatile applications for this study are offered. As a final step, we examined a bivariate real-world data set for illustrative purposes. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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13 pages, 317 KiB

Open AccessArticle

Excess Lifetime Extropy of Order Statistics

by Mansour Shrahili and Mohamed Kayid

Axioms 2023, 12(11), 1024; https://doi.org/10.3390/axioms12111024 - 31 Oct 2023

Cited by 6 | Viewed by 1323

Abstract

This paper explores the concept of residual extropy as an uncertainty measure for order statistics. We specifically derive the residual extropy for the ith-order statistic and establish its relationship with the residual extropy of the ith-order statistic from a random sample [...] Read more.

This paper explores the concept of residual extropy as an uncertainty measure for order statistics. We specifically derive the residual extropy for the ith-order statistic and establish its relationship with the residual extropy of the ith-order statistic from a random sample generated from a uniform distribution. By employing this approach, we obtain a formula for the residual extropy of order statistics applicable to general continuous distributions. In addition, we offer two lower bounds that can be applied in situations where obtaining closed-form expressions for the residual extropy of order statistics in diverse distributions proves to be challenging. Additionally, we investigate the monotonicity properties of the residual extropy of order statistics. Furthermore, we present other aspects of the residual extropy of order statistics, including its dependence on the position of order statistics and various features of the underlying distribution. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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18 pages, 324 KiB

Open AccessArticle

Regression Estimation with Errors in the Variables via the Laplace Transform

by Huijun Guo and Qingqun Bai

Axioms 2023, 12(10), 992; https://doi.org/10.3390/axioms12100992 - 19 Oct 2023

Viewed by 1243

Abstract

This paper considers nonparametric regression estimation with errors in the variables. It is a standard assumption that the characteristic function of the covariate error does not vanish on the real line. This assumption is rather strong. In this paper, we assume the covariate [...] Read more.

This paper considers nonparametric regression estimation with errors in the variables. It is a standard assumption that the characteristic function of the covariate error does not vanish on the real line. This assumption is rather strong. In this paper, we assume the covariate error distribution is a convolution of uniform distributions, the characteristic function of which contains zeros on the real line. Our regression estimator is constructed via the Laplace transform. We prove its strong consistency and show its convergence rate. It turns out that zeros in the characteristic function have no effect on the convergence rate of our estimator. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

13 pages, 5492 KiB

Open AccessArticle

Modified Maximum Likelihood Estimation of the Inverse Weibull Model

by Mohamed Kayid and Mashael A. Alshehri

Axioms 2023, 12(10), 961; https://doi.org/10.3390/axioms12100961 - 12 Oct 2023

Cited by 3 | Viewed by 1608

Abstract

The inverse Weibull model is a simple and flexible model used for survival analysis, reliability theory, and other scientific fields. The main problem in this context is the estimation of the model parameters. In this study, a modified version of the maximum likelihood [...] Read more.

The inverse Weibull model is a simple and flexible model used for survival analysis, reliability theory, and other scientific fields. The main problem in this context is the estimation of the model parameters. In this study, a modified version of the maximum likelihood estimator is presented. The idea behind it is that the likelihood equation for the shape parameters of the model is biased; therefore, an unbiased version was defined. The new estimator is based on the definition of an unbiased likelihood equation. Simulation results show that the new modified estimator for the shape parameter has a smaller mean square error. Finally, the proposed estimator and the maximum likelihood estimator were compared in the analysis of the three real data sets. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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20 pages, 506 KiB

Open AccessArticle

Product of Spacing Estimation of Stress–Strength Reliability for Alpha Power Exponential Progressively Type-II Censored Data

by Mazen Nassar, Refah Alotaibi and Chunfang Zhang

Axioms 2023, 12(8), 752; https://doi.org/10.3390/axioms12080752 - 30 Jul 2023

Cited by 4 | Viewed by 1306

Abstract

The present study focuses on estimating the stress–strength parameter when the parent distribution is the alpha power exponential model and the available data are progressively Type-II censored. As a starting point, the usual maximum likelihood approach is applied to obtain point and interval [...] Read more.

The present study focuses on estimating the stress–strength parameter when the parent distribution is the alpha power exponential model and the available data are progressively Type-II censored. As a starting point, the usual maximum likelihood approach is applied to obtain point and interval estimates of the model parameters, as well as the stress–strength parameter. Another competing strategy employed in this paper is the maximum product of spacing method, which may be thought of as a rival to the maximum likelihood method. The product of spacing approach is used to obtain point and interval estimates for the various parameters. The asymptotic properties of both methods are used to obtain interval estimates of the model parameter and the stress–strength parameter, and the variance of the stress–strength parameter is approximated using the well-known delta method. Two parametric bootstrap confidence intervals are provided based on the suggested classical estimation procedures. A simulation study is also used to assess the performance of various point and interval estimations. For illustrative purposes, the proposed methods are applied to two real data sets, one for the kidney patients’ recurrence times to infection and the other for breaking the strength of jute fibers. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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21 pages, 10003 KiB

Open AccessArticle

Analysis of WE Parameters of Life Using Adaptive-Progressively Type-II Hybrid Censored Mechanical Equipment Data

by Ahmed Elshahhat, Ehab M. Almetwally, Sanku Dey and Heba S. Mohammed

Axioms 2023, 12(7), 690; https://doi.org/10.3390/axioms12070690 - 16 Jul 2023

Cited by 3 | Viewed by 1420

Abstract

A new two-parameter weighted-exponential (WE) distribution, as a beneficial competitor model to other lifetime distributions, namely: generalized exponential, gamma, and Weibull distributions, is studied in the presence of adaptive progressive Type-II hybrid data. Thus, based on different frequentist and Bayesian estimation methods, we [...] Read more.

A new two-parameter weighted-exponential (WE) distribution, as a beneficial competitor model to other lifetime distributions, namely: generalized exponential, gamma, and Weibull distributions, is studied in the presence of adaptive progressive Type-II hybrid data. Thus, based on different frequentist and Bayesian estimation methods, we study the inferential problem of the WE parameters as well as related reliability indices, including survival and failure functions. In frequentist setups, besides the standard likelihood-based estimation, the product of spacing (PS) approach is also taken into account for estimating all unknown parameters of life. Making use of the delta method and the observed Fisher information of the frequentist estimators, approximated asymptotic confidence intervals for all unknown parameters are acquired. In Bayes methodology, from the squared-error loss with independent gamma density priors, the point and interval estimates of the unknown parameters are offered using both joint likelihood and the product of spacings functions. Because a closed solution to the Bayes estimators is not accessible, the Metropolis–Hastings sampler is presented to approximate the Bayes estimates and also to create their associated highest interval posterior density estimates. To figure out the effectiveness of the developed approaches, extensive Monte Carlo experiments are implemented. To highlight the applicability of the offered methodologies in practice, one real-life data set consisting of 30 failure times of repairable mechanical equipment is analyzed. This application demonstrated that the offered WE model provides a better fit compared to the other eight lifetime models. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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17 pages, 337 KiB

Open AccessArticle

Randomly Stopped Sums with Generalized Subexponential Distribution

by Jūratė Karasevičienė and Jonas Šiaulys

Axioms 2023, 12(7), 641; https://doi.org/10.3390/axioms12070641 - 28 Jun 2023

Cited by 2 | Viewed by 1164

Abstract

Let

{ξ_{1}, ξ_{2}, \dots}

be a sequence of independent possibly differently distributed random variables, defined on a probability space

(Ω, F, P)

with distribution functions [...] Read more.

Let

{ξ_{1}, ξ_{2}, \dots}

be a sequence of independent possibly differently distributed random variables, defined on a probability space

(Ω, F, P)

with distribution functions

{F_{ξ_{1}}, F_{ξ_{2}}, \dots}

. Let

η

be a counting random variable independent of sequence

{ξ_{1}, ξ_{2}, \dots}

. In this paper, we find conditions under which the distribution function of randomly stopped sum

S_{η} = ξ_{1} + ξ_{2} + \dots + ξ_{η}

belongs to the class of generalized subexponential distributions. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

22 pages, 453 KiB

Open AccessArticle

Monitoring the Weibull Scale Parameter Based on Type I Censored Data Using a Modified EWMA Control Chart

by Dan Yu, Li Jin, Jin Li, Xixi Qin, Zhichuan Zhu and Jiujun Zhang

Axioms 2023, 12(5), 487; https://doi.org/10.3390/axioms12050487 - 17 May 2023

Cited by 2 | Viewed by 1739

Abstract

In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the [...] Read more.

In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the scale parameter with the shape parameter fixed. A comparative analysis with existing cumulative sum and exponentially weighted moving average results from the literature is provided. The zero-state and steady-state behaviour of the new scheme are considered with regard to the average run length, the standard deviation of the run length, and other performance measures. Our simulation shows stronger power in detecting changes in the censored lifetime data using the modified scheme than that using the traditional exponentially weighted moving average scheme, and the new scheme is superior to the cumulative sum scheme in most situations. A real-data example further demonstrates the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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16 pages, 698 KiB

Open AccessArticle

A Time-Varying Coefficient Double Threshold GARCH Model with Explanatory Variables

by Tongwei Zhang, Lianyan Fu, Dehui Wang and Zhuoxi Yu

Axioms 2023, 12(5), 476; https://doi.org/10.3390/axioms12050476 - 15 May 2023

Cited by 1 | Viewed by 1781

Abstract

In this article, we consider the nonparametric inference for the time-varying coefficient double-threshold generalized autoregressive conditional heteroscedastic models. The quasi-maximum exponential likelihood estimators (QMELEs) of the model’s parameters and the asymptotic properties of the estimators are obtained. The simulation study implies that the [...] Read more.

In this article, we consider the nonparametric inference for the time-varying coefficient double-threshold generalized autoregressive conditional heteroscedastic models. The quasi-maximum exponential likelihood estimators (QMELEs) of the model’s parameters and the asymptotic properties of the estimators are obtained. The simulation study implies that the distribution of the estimators is asymptotically normal. A real data application to stock returns is given. Both the simulations and real data example imply that the model and the QMELE are proper, compatible and accurately fit the financial time series data of the Nikkei 225. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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25 pages, 445 KiB

Open AccessArticle

Reliability Class Testing and Hypothesis Specification: NBRULC−t_∘ Characterizations with Applications for Medical and Engineering Data Modeling

by Hana Alqifari, Mohamed S. Eliwa, Walid B. H. Etman, Mahmoud El-Morshedy, Laila A. Al-Essa and Rashad M. EL-Sagheer

Axioms 2023, 12(5), 414; https://doi.org/10.3390/axioms12050414 - 24 Apr 2023

Cited by 5 | Viewed by 1950

Abstract

Due to the complexity of the data being generated day in and day out in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability called the classes of [...] Read more.

Due to the complexity of the data being generated day in and day out in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability called the classes of life and determinant probability distributions has been presented. This article introduces novel statistical probability models for the reliability class of life test under different reliability processes in the age range

t_{\circ}

. Several probabilistic properties and features were derived and rigorously screened to test the new reliability class. According to the U-statistic, a novel hypothesis test was created to evaluate the exponentiality property. The comparative efficiency of the test according to Pitman’s asymptotic efficiency was examined and compared with other reliability classes. To prove the superiority of the new reliability class, some probability models were utilized, including the Weibull, Makeham, gamma, and linear failure rate models. Moreover, critical point simulations of the null Monte Carlo distribution and some applications of the censored and uncensored data were implemented to validate the class test listed by the reliability analysis. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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