Axioms

Research

8 pages, 249 KiB

Open AccessArticle

A Short Note on Generating a Random Sample from Finite Mixture Distributions

by Luai Al-Labadi and Anna Ly

Axioms 2024, 13(5), 307; https://doi.org/10.3390/axioms13050307 - 8 May 2024

Viewed by 741

Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include [...] Read more.

Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include the inverse transform method, acceptance–rejection method, and Box–Muller transformation, all of which rely on sampling from the uniform

(0, 1)

distribution. A significant concept in statistics is the finite mixture model, characterized by a convex combination of multiple probability density functions. In this paper, we introduce a modified version of the composition method, a standard approach for sampling finite mixture models. Our modification offers the advantage of relying on sampling from the uniform

(0, 1)

distribution, aligning with prevalent methods in computational statistics. This alignment simplifies teaching computational statistics courses, as well as having other benefits. We offer several examples to illustrate the approach. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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31 pages, 2699 KiB

Open AccessArticle

Engineering Applications with Stress-Strength for a New Flexible Extension of Inverse Lomax Model: Bayesian and Non-Bayesian Inference

by Salem A. Alyami, I. Elbatal, Amal S. Hassan and Ehab M. Almetwally

Axioms 2023, 12(12), 1097; https://doi.org/10.3390/axioms12121097 - 29 Nov 2023

Cited by 2 | Viewed by 1152

Abstract

In this paper, we suggest a brand new extension of the inverse Lomax distribution for fitting engineering time data. The newly developed distribution, termed the transmuted Topp–Leone inverse Lomax (TTLILo) distribution, is characterized by an additional shape and transmuted parameters. It is critical [...] Read more.

In this paper, we suggest a brand new extension of the inverse Lomax distribution for fitting engineering time data. The newly developed distribution, termed the transmuted Topp–Leone inverse Lomax (TTLILo) distribution, is characterized by an additional shape and transmuted parameters. It is critical to notice that the skewness, kurtosis, and tail weights of the distribution are strongly influenced by these additional characteristics of the extra parameters. The TTLILo model is capable of producing right-skewed, J-shaped, uni-modal, and reversed-J-shaped densities. The proposed model’s statistical characteristics, including the moments, entropy values, stochastic ordering, stress-strength model, incomplete moments, and quantile function, are examined. Moreover, characterization based on two truncated moments is offered. Using Bayesian and non-Bayesian estimating techniques, we estimate the distribution parameters of the suggested distribution. The bootstrap procedure, approximation, and Bayesian credibility are the three forms of confidence intervals that have been created. A simulation study is used to assess the efficiency of the estimated parameters. The TTLILo model is then put to the test by being applied to actual engineering datasets, demonstrating that it offers a good match when compared to alternative models. Two applications based on real engineering datasets are taken into consideration: one on the failure times of airplane air conditioning systems and the other on the active repair times of airborne communication transceivers. Also, we consider the problem of estimating the stress-strength parameter

R = P (Z_{2} < Z_{1})

with engineering application. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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18 pages, 1521 KiB

Open AccessArticle

New Partially Linear Regression and Machine Learning Models Applied to Agronomic Data

by Gabriela M. Rodrigues, Edwin M. M. Ortega and Gauss M. Cordeiro

Axioms 2023, 12(11), 1027; https://doi.org/10.3390/axioms12111027 - 31 Oct 2023

Viewed by 1613

Abstract

Regression analysis can be appropriate to describe a nonlinear relationship between the response variable and the explanatory variables. This article describes the construction of a partially linear regression model with two systematic components based on the exponentiated odd log-logistic normal distribution. The parameters [...] Read more.

Regression analysis can be appropriate to describe a nonlinear relationship between the response variable and the explanatory variables. This article describes the construction of a partially linear regression model with two systematic components based on the exponentiated odd log-logistic normal distribution. The parameters are estimated by the penalized maximum likelihood method. Simulations for some parameter settings and sample sizes empirically prove the accuracy of the estimators. The superiority of the proposed regression model over other regression models is shown by means of agronomic experimentation data. The predictive performance of the new model is compared with two machine learning techniques: decision trees and random forests. These methods achieved similar prediction performance, i.e., none stands out as a better predictor. In this sense, the objective of the research is to choose the best method. If the objective is only predictive, the decision tree can be used due to its simplicity. For inference purposes, the regression model is recommended, which can provide much more information regarding the relationship of the variables under study. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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17 pages, 741 KiB

Open AccessArticle

Fuzzy vs. Traditional Reliability Model for Inverse Weibull Distribution

by Eslam Hussam, Mohamed A. Sabry, M. M. Abd El-Raouf and Ehab M. Almetwally

Axioms 2023, 12(6), 582; https://doi.org/10.3390/axioms12060582 - 12 Jun 2023

Cited by 4 | Viewed by 1270

Abstract

In this paper, fuzzy stress strengths

R_{F} = P (Y ≺ X)

and traditional stress strengths

R = P (Y < X)

are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic [...] Read more.

In this paper, fuzzy stress strengths

R_{F} = P (Y ≺ X)

and traditional stress strengths

R = P (Y < X)

are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic fuzzy set theory is taken into account in the stress–strength inference, it enables the generation of more precise studies on the underlying systems. We discuss estimating both conventional and fuzzy models of stress strength utilizing a maximum product of spacing, maximum likelihood, and Bayesian approaches. Simulations based on the Markov Chain Monte Carlo method are used to produce various estimators of conventional and fuzzy dependability of stress strength for the inverse Weibull model. To generate both conventional and fuzzy models of dependability, we use the Metropolis–Hastings method while performing Bayesian estimation. In conclusion, we will examine a scenario taken from actual life and apply a real-world data application to validate the accuracy of the provided estimators. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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22 pages, 14223 KiB

Open AccessArticle

Statistical Evaluations and Applications for IER Parameters from Generalized Progressively Type-II Hybrid Censored Data

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

Axioms 2023, 12(6), 565; https://doi.org/10.3390/axioms12060565 - 7 Jun 2023

Cited by 4 | Viewed by 1176

Abstract

Generalized progressively Type-II hybrid strategy has been suggested to save both the duration and cost of a life test when the experimenter aims to score a fixed number of failed units. In this paper, using this mechanism, the maximum likelihood and Bayes inferential [...] Read more.

Generalized progressively Type-II hybrid strategy has been suggested to save both the duration and cost of a life test when the experimenter aims to score a fixed number of failed units. In this paper, using this mechanism, the maximum likelihood and Bayes inferential problems for unknown model parameters, in addition to both reliability, and hazard functions of the inverted exponentiated Rayleigh model, are acquired. Applying the observed Fisher data and delta method, the normality characteristic of the classical estimates is taken into account to derive confidence intervals for unknown parameters and several indice functions. In Bayes’ viewpoint, through independent gamma priors against both symmetrical and asymmetrical loss functions, the Bayes estimators of the unknown quantities are developed. Because the Bayes estimators are acquired in complicated forms, a hybrid Monte-Carlo Markov-chain technique is offered to carry out the Bayes estimates as well as to create the related highest posterior density interval estimates. The precise behavior of the suggested estimation approaches is assessed using wide Monte Carlo simulation experiments. Two actual applications based on actual data sets from the mechanical and chemical domains are examined to show how the offered methodologies may be used in real current events. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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26 pages, 504 KiB

Open AccessArticle

Inverse Unit Teissier Distribution: Theory and Practical Examples

by Najwan Alsadat, Mohammed Elgarhy, Kadir Karakaya, Ahmed M. Gemeay, Christophe Chesneau and M. M. Abd El-Raouf

Axioms 2023, 12(5), 502; https://doi.org/10.3390/axioms12050502 - 20 May 2023

Cited by 6 | Viewed by 1693

Abstract

In this paper, we emphasize a new one-parameter distribution with support as

[1, + \infty)

. It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as [...] Read more.

In this paper, we emphasize a new one-parameter distribution with support as

[1, + \infty)

. It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as the mode, quantiles, stochastic dominance, heavy-tailed nature, moments, etc. Among the strengths of the distribution are the following: (i) the closed-form expressions and flexibility of the main functions, and in particular, the probability density function is unimodal and the hazard rate function is increasing or unimodal; (ii) the manageability of the moments; and, more importantly, (iii) it provides a real alternative to the famous Pareto distribution, also with support as

[1, + \infty)

. Indeed, the proposed distribution has different functionalities but also benefits from the heavy-right-tailed nature, which is demanded in many applied fields (finance, the actuarial field, quality control, medicine, etc.). Furthermore, it can be used quite efficiently in a statistical setting. To support this claim, the maximum likelihood, Anderson–Darling, right-tailed Anderson–Darling, left-tailed Anderson–Darling, Cramér–Von Mises, least squares, weighted least-squares, maximum product of spacing, minimum spacing absolute distance, and minimum spacing absolute-log distance estimation methods are examined to estimate the unknown unique parameter. A Monte Carlo simulation is used to compare the performance of the obtained estimates. Additionally, the Bayesian estimation method using an informative gamma prior distribution under the squared error loss function is discussed. Data on the COVID mortality rate and the timing of pain relief after receiving an analgesic are considered to illustrate the applicability of the proposed distribution. Favorable results are highlighted, supporting the importance of the findings. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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17 pages, 612 KiB

Open AccessArticle

A New Probability Distribution: Model, Theory and Analyzing the Recovery Time Data

by Huda M. Alshanbari, Omalsad Hamood Odhah, Zubair Ahmad, Faridoon Khan and Abd Al-Aziz Hosni El-Bagoury

Axioms 2023, 12(5), 477; https://doi.org/10.3390/axioms12050477 - 15 May 2023

Cited by 10 | Viewed by 2356

Abstract

Probability models are frequently used in numerous healthcare, sports, and policy studies. These probability models use datasets to identify patterns, analyze lifetime scenarios, predict outcomes of interest, etc. Therefore, numerous probability models have been studied, introduced, and implemented. In this paper, we also [...] Read more.

Probability models are frequently used in numerous healthcare, sports, and policy studies. These probability models use datasets to identify patterns, analyze lifetime scenarios, predict outcomes of interest, etc. Therefore, numerous probability models have been studied, introduced, and implemented. In this paper, we also propose a novel probability model for analyzing data in different sectors, particularly in biomedical and sports sciences. The probability model is called a new modified exponential-Weibull distribution. The heavy-tailed characteristics along with some other mathematical properties are derived. Furthermore, the estimators of the new modified exponential-Weibull are derived. A simulation study of the new modified exponential-Weibull model is also provided. To illustrate the new modified exponential-Weibull model, a practical dataset is analyzed. The dataset consists of seventy-eight observations and represents the recovery time after the injuries in different basketball matches. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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14 pages, 3490 KiB

Open AccessArticle

The Markov Bernoulli Lomax with Applications Censored and COVID-19 Drought Mortality Rate Data

by Bahady I. Mohammed, Yusra A. Tashkandy, Mohmoud M. Abd El-Raouf, Md. Moyazzem Hossain and Mahmoud E. Bakr

Axioms 2023, 12(5), 439; https://doi.org/10.3390/axioms12050439 - 28 Apr 2023

Viewed by 1372

Abstract

In this article, we present a Markov Bernoulli Lomax (MB-L) model, which is obtained by a countable mixture of Markov Bernoulli and Lomax distributions, with decreasing and unimodal hazard rate function (HRF). The new model contains Marshall- Olkin Lomax and Lomax distributions as [...] Read more.

In this article, we present a Markov Bernoulli Lomax (MB-L) model, which is obtained by a countable mixture of Markov Bernoulli and Lomax distributions, with decreasing and unimodal hazard rate function (HRF). The new model contains Marshall- Olkin Lomax and Lomax distributions as a special case. The mathematical properties, as behavior of probability density function (PDF), HRF, r^th moments, moment generating function (MGF) and minimum (maximum) Markov-Bernoulli Geometric (MBG) stable are studied. Moreover, the estimates of the model parameters by maximum likelihood are obtained. The maximum likelihood estimation (MLE), bias and mean squared error (MSE) of MB-L parameters are inspected by simulation study. Finally, a MB-L distribution was fitted to the randomly censored and COVID-19 (complete) data. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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14 pages, 2383 KiB

Open AccessArticle

Risk Analysis: Changing the Story with the Statistical Stochastic Process and VaR

by Lianghong Wu

Axioms 2023, 12(5), 418; https://doi.org/10.3390/axioms12050418 - 25 Apr 2023

Viewed by 2083

Abstract

With the dramatically increased demand for data analysis, statistical techniques play a key role in modern society for both academics and practitioners. Statistical techniques have been evolving from descriptive statistics to statistical inference in fields that require the evaluation of uncertainty and the [...] Read more.

With the dramatically increased demand for data analysis, statistical techniques play a key role in modern society for both academics and practitioners. Statistical techniques have been evolving from descriptive statistics to statistical inference in fields that require the evaluation of uncertainty and the quantification of risks. With the growing complexity of various fields, such as manufacturing and industrial applications, as well as business decision-making, modeling and quantifying risks has become essential. In this paper, we aimed to use statistical risk analysis and Value at Risk (VaR) to address the decision problem for project portfolios. Traditional economic evaluation criteria used in the management of project portfolios, as they pertain to new product development (NPD), are based on the assumption that pinpoint estimations will remain constant in the future. The assumption that NPD is static, however, is clearly unrealistic due to the inherent uncertainty of NPD projects. In this study, we stress the critical role that uncertainty plays in the selection of NPD portfolios, and clarify the reasons why it must not be overlooked. Using Value at Risk measurements, we show how uncertainty plays a critical role in evaluating and prioritizing NPD portfolios. The implications of this study regarding statistically modeling NPD portfolio decisions are provided for academics and practitioners. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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9 pages, 282 KiB

Open AccessArticle

Non-Parametric Hypothesis Testing for Unknown Aged Class of Life Distribution Using Real Medical Data

by Mahmoud. E. Bakr and Abdulhakim A. Al-Babtain

Axioms 2023, 12(4), 369; https://doi.org/10.3390/axioms12040369 - 11 Apr 2023

Cited by 4 | Viewed by 1226

Abstract

Over the last few decades, the statisticians and reliability analysts have looked at putting exponentiality to the test using the Laplace transform technique. The non-parametric statistical test used in this study, which is based on this technique, evaluates various treatment modalities by looking [...] Read more.

Over the last few decades, the statisticians and reliability analysts have looked at putting exponentiality to the test using the Laplace transform technique. The non-parametric statistical test used in this study, which is based on this technique, evaluates various treatment modalities by looking at failure behavior in the survival data that were gathered. Following use of the suggested strategy, patient survival times are recorded. In this investigation, it was presupposed that the Laplace transform order of (UBAC2) attribute or the constant failure rate would determine how the observed data behave (exponential scenario). If the survival data are exponential, the recommended treatment approach is ineffective. If the survival data are UBAC2L, the technique in use produces a better or a higher expected total present value than an older one governed by an exponential survival function (discussed in the Applications section). The efficiency and critical values of the test are calculated and compared to those of other tests in order to ensure that the suggested statistical test is applied correctly. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

22 pages, 1369 KiB

Open AccessArticle

Comparative Study with Applications for Gompertz Models under Competing Risks and Generalized Hybrid Censoring Schemes

by Laila A. Al-Essa, Ahmed A. Soliman, Gamal A. Abd-Elmougod and Huda M. Alshanbari

Axioms 2023, 12(4), 322; https://doi.org/10.3390/axioms12040322 - 24 Mar 2023

Cited by 2 | Viewed by 1490

Abstract

In reliability and survival analysis, the time-to-failure data play an important role in the development of the reliability and life characteristics of the products. In some cases, these kinds of data are modeled using a competing risks model. The problem of conducting comparative [...] Read more.

In reliability and survival analysis, the time-to-failure data play an important role in the development of the reliability and life characteristics of the products. In some cases, these kinds of data are modeled using a competing risks model. The problem of conducting comparative life testing under a competing risks model when the units come from different lines of production has recently been addressed. In this paper, we address this problem when the life of the unit is distributed using the Gompertz distribution, noting that the units come from two lines of production and two independent causes of failure are activated. The data are collected under a joint generalized type-II hybrid censoring scheme. Maximum likelihood estimators of the unknown parameters are derived, along with the corresponding asymptotic confidence intervals. We also adopt two bootstrap confidence intervals. Using independent gamma priors, the Bayes estimators relative to squared error loss function are obtained with credible intervals. The properties and quality of estimators are measured by performing a Monte Carlo simulation study. Finally, a real-life data set is analyzed to discuss the applicability of the proposed methods to real phenomena. The optimal plan with respect to comments on the numerical results is discussed in the conclusion. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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21 pages, 2628 KiB

Open AccessArticle

Estimations of Modified Lindley Parameters Using Progressive Type-II Censoring with Applications

by Refah Alotaibi, Mazen Nassar and Ahmed Elshahhat

Axioms 2023, 12(2), 171; https://doi.org/10.3390/axioms12020171 - 7 Feb 2023

Cited by 3 | Viewed by 1371

Abstract

This study addresses the estimation problems of the modified Lindley distribution using a progressive Type-II censoring plan. Using the maximum likelihood and maximum product of spacing and Bayesian estimation methods, the unknown parameter, reliability, and hazard rate functions are estimated. Employing the assumption [...] Read more.

This study addresses the estimation problems of the modified Lindley distribution using a progressive Type-II censoring plan. Using the maximum likelihood and maximum product of spacing and Bayesian estimation methods, the unknown parameter, reliability, and hazard rate functions are estimated. Employing the assumption of the gamma prior and a symmetric loss function, Bayes estimators are investigated when the observed data are obtained using the likelihood and product of spacing functions. Additionally, the approximate confidence intervals using both classical methods and the highest posterior density credible intervals are also discussed. To assess the different estimating strategies, a comprehensive simulation experiment that considers various sample sizes and censoring schemes is implemented. Finally, two actual data sets are examined to verify the utility of the modified Lindley distribution and the usefulness of the suggested estimators. The findings demonstrate that, in order to obtain the necessary estimators, the maximum product of the spacing method is preferred over the maximum likelihood method; whereas, when compared to the conventional techniques, the Bayesian approach using the likelihood and product of spacing functions provides more acceptable estimates. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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24 pages, 482 KiB

Open AccessArticle

Modelling Coronavirus and Larvae Pyrausta Data: A Discrete Binomial Exponential II Distribution with Properties, Classical and Bayesian Estimation

by Mohamed S. Eliwa, Abhishek Tyagi, Bader Almohaimeed and Mahmoud El-Morshedy

Axioms 2022, 11(11), 646; https://doi.org/10.3390/axioms11110646 - 16 Nov 2022

Cited by 6 | Viewed by 1672

Abstract

In this article, we propose the discrete version of the binomial exponential II distribution for modelling count data. Some of its statistical properties including hazard rate function, mode, moments, skewness, kurtosis, and index of dispersion are derived. The shape of the failure rate [...] Read more.

In this article, we propose the discrete version of the binomial exponential II distribution for modelling count data. Some of its statistical properties including hazard rate function, mode, moments, skewness, kurtosis, and index of dispersion are derived. The shape of the failure rate function is increasing. Moreover, the proposed model is appropriate for modelling equi-, over- and under-dispersed data. The parameter estimation through the classical point of view has been done using the method of maximum likelihood, whereas, in the Bayesian framework, assuming independent beta priors of model parameters, the Metropolis–Hastings algorithm within Gibbs sampler is used to obtain sample-based Bayes estimates of the unknown parameters of the proposed model. A detailed simulation study is carried out to examine the outcomes of maximum likelihood and Bayesian estimators. Finally, two distinctive real data sets are analyzed using the proposed model. These applications showed the flexibility of the new distribution. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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