Axioms

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15 pages, 1616 KiB

Open AccessArticle

Pretest Estimation for the Common Mean of Several Normal Distributions: In Meta-Analysis Context

by Peter M. Mphekgwana, Yehenew G. Kifle and Chioneso S. Marange

Axioms 2024, 13(9), 648; https://doi.org/10.3390/axioms13090648 (registering DOI) - 22 Sep 2024

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary [...] Read more.

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary test estimator for the common mean

(μ)

with unknown and unequal variances. When there exists prior information regarding the population mean with consideration that

μ

might be equal to the reference value for the population mean, a hypothesis test can be conducted:

H_{0} : μ = μ_{0}

versus

H_{1} : μ \neq μ_{0}

. The initial sample is used to test

H_{0}

, and if

H_{0}

is not rejected, we become more confident in using our prior information (after the test) to estimate

μ

. However, if

H_{0}

is rejected, the prior information is discarded. Our simulations indicate that the proposed preliminary test estimator significantly decreases the mean squared error (MSE) values compared to unbiased estimators such as the Garybill-Deal (GD) estimator, particularly when

μ

closely aligns with the hypothesized mean (

μ_{0}

). Furthermore, our analysis indicates that the proposed test estimator outperforms the existing method, particularly in cases with minimal sample sizes. We advocate for its adoption to improve the accuracy of common mean estimation. Our findings suggest that through careful application to real meta-analyses, the proposed test estimator shows promising potential. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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14 pages, 467 KiB

Open AccessArticle

Transfer Learning for Logistic Regression with Differential Privacy

by Yiming Hou and Yunquan Song

Axioms 2024, 13(8), 517; https://doi.org/10.3390/axioms13080517 - 30 Jul 2024

Viewed by 591

Abstract

Transfer learning, as a machine learning approach enhancing model generalization across different domains, has extensive applications in various fields. However, the risk of privacy leakage remains a crucial consideration during the transfer learning process. Differential privacy, with its rigorous mathematical foundation, has been [...] Read more.

Transfer learning, as a machine learning approach enhancing model generalization across different domains, has extensive applications in various fields. However, the risk of privacy leakage remains a crucial consideration during the transfer learning process. Differential privacy, with its rigorous mathematical foundation, has been proven to offer consistent and robust privacy protection. This study delves into the logistic regression transfer learning problem supported by differential privacy. In cases where transferable sources are known, we propose a two-step transfer learning algorithm. For scenarios with unknown transferable sources, a non-algorithmic, cross-validation-based transferable source detection method is introduced, to mitigate adverse effects from non-informative sources. The effectiveness of the proposed algorithm is validated through simulations and experiments with real-world data. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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16 pages, 603 KiB

Open AccessArticle

An Extension of the Fréchet Distribution and Applications

by Yolanda M. Gómez, Inmaculada Barranco-Chamorro, Jaime S. Castillo and Héctor W. Gómez

Axioms 2024, 13(4), 253; https://doi.org/10.3390/axioms13040253 - 11 Apr 2024

Cited by 1 | Viewed by 976

Abstract

This paper presents the Slash-Exponential-Fréchet distribution, which is an expanded version of the Fréchet distribution. Through its stochastic representation, probability distribution function, moments and other relevant features are obtained. Evidence supports that the updated model displays a lighter right tail than the Fréchet [...] Read more.

This paper presents the Slash-Exponential-Fréchet distribution, which is an expanded version of the Fréchet distribution. Through its stochastic representation, probability distribution function, moments and other relevant features are obtained. Evidence supports that the updated model displays a lighter right tail than the Fréchet model and is more flexible as for skewness and kurtosis. Results on maximum likelihood estimators are given. Our proposition’s applicability is demonstrated through a simulation study and the evaluation of two real-world datasets. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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19 pages, 339 KiB

Open AccessArticle

Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

by Jūratė Karasevičienė and Jonas Šiaulys

Axioms 2024, 13(2), 85; https://doi.org/10.3390/axioms13020085 - 27 Jan 2024

Viewed by 906

Abstract

In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped [...] Read more.

In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped sums considered in the authors’ previous work. In this work, as in the previous one, the primary random variables are supposed to be independent and real-valued, but not necessarily identically distributed. The counting random variable describing the stopping moment of random structures is supposed to be nonnegative, integer-valued and not degenerate at zero. In addition, it is supposed that counting random variable and the sequence of the primary random variables are independent. At the end of the paper, it is demonstrated how randomly stopped structures can be applied to the construction of new generalized subexponential distributions. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

13 pages, 306 KiB

Open AccessArticle

Analysis of Correlation Bounds for Uniformly Expanding Maps on [0, 1]

by Mohamed Abdelkader

Axioms 2023, 12(12), 1072; https://doi.org/10.3390/axioms12121072 - 23 Nov 2023

Viewed by 865

Abstract

In this paper, we provide the decay of correlations for random dynamical systems. Precisely, we consider the uniformly

C^{2}

piecewise expanding maps defined on the unit interval satisfying [...] Read more.

In this paper, we provide the decay of correlations for random dynamical systems. Precisely, we consider the uniformly

C^{2}

piecewise expanding maps defined on the unit interval satisfying

λ (T_{ω}^{'}) = inf | T_{ω}^{'} | > 2

. As a principal tool of these studies, we use a coupling method for analyzing the coupling time of observables with bounded variation. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

16 pages, 463 KiB

Open AccessArticle

A Least Squares Estimator for Gradual Change-Point in Time Series with m-Asymptotically Almost Negatively Associated Errors

by Tianming Xu and Yuesong Wei

Axioms 2023, 12(9), 894; https://doi.org/10.3390/axioms12090894 - 20 Sep 2023

Viewed by 832

Abstract

As a new member of the NA (negative associated) family, the m-AANA (m-asymptotically almost negatively associated) sequence has many statistical properties that have not been developed. This paper mainly studies its properties in the gradual change point model. Firstly, we [...] Read more.

As a new member of the NA (negative associated) family, the m-AANA (m-asymptotically almost negatively associated) sequence has many statistical properties that have not been developed. This paper mainly studies its properties in the gradual change point model. Firstly, we propose a least squares type change point estimator, then derive the convergence rates and consistency of the estimator, and provide the limit distributions of the estimator. It is interesting that the convergence rates of the estimator are the same as that of the change point estimator for independent identically distributed observations. Finally, the effectiveness of the estimator in limited samples can be verified through several sets of simulation experiments and an actual hydrological example. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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16 pages, 316 KiB

Open AccessArticle

Moderate Deviation Principle for Linear Processes Generated by Dependent Sequences under Sub-Linear Expectation

by Peiyu Sun, Dehui Wang, Xue Ding, Xili Tan and Yong Zhang

Axioms 2023, 12(8), 781; https://doi.org/10.3390/axioms12080781 - 11 Aug 2023

Viewed by 925

Abstract

We are interested in the linear processes generated by dependent sequences under sub-linear expectation. Using the Beveridge–Nelson decomposition of linear processes and the inequalities, the moderate deviation principle for linear processes produced by an m-dependent sequence is established. We also prove the upper [...] Read more.

We are interested in the linear processes generated by dependent sequences under sub-linear expectation. Using the Beveridge–Nelson decomposition of linear processes and the inequalities, the moderate deviation principle for linear processes produced by an m-dependent sequence is established. We also prove the upper bound of the moderate deviation principle for linear processes produced by negatively dependent sequences via different methods from m-dependent sequences. These conclusions promote and improve the corresponding results from the traditional probability space to the sub-linear expectation space. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

35 pages, 1209 KiB

Open AccessArticle

Sampling Plan for the Kavya–Manoharan Generalized Inverted Kumaraswamy Distribution with Statistical Inference and Applications

by Najwan Alsadat, Amal S. Hassan, Mohammed Elgarhy, Christophe Chesneau and Ahmed R. El-Saeed

Axioms 2023, 12(8), 739; https://doi.org/10.3390/axioms12080739 - 27 Jul 2023

Cited by 2 | Viewed by 1159

Abstract

In this article, we introduce the Kavya–Manoharan generalized inverse Kumaraswamy (KM-GIKw) distribution, which can be presented as an improved version of the generalized inverse Kumaraswamy distribution with three parameters. It contains numerous referenced lifetime distributions of the literature and a large panel of [...] Read more.

In this article, we introduce the Kavya–Manoharan generalized inverse Kumaraswamy (KM-GIKw) distribution, which can be presented as an improved version of the generalized inverse Kumaraswamy distribution with three parameters. It contains numerous referenced lifetime distributions of the literature and a large panel of new ones. Among the essential features and attributes covered in our research are quantiles, moments, and information measures. In particular, various entropy measures (Rényi, Tsallis, etc.) are derived and discussed numerically. The adaptability of the KM-GIKw distribution in terms of the shapes of the probability density and hazard rate functions demonstrates how well it is able to fit different types of data. Based on it, an acceptance sampling plan is created when the life test is truncated at a predefined time. More precisely, the truncation time is intended to represent the median of the KM-GIKw distribution with preset factors. In a separate part, the focus is put on the inference of the KM-GIKw distribution. The related parameters are estimated using the Bayesian, maximum likelihood, and maximum product of spacings methods. For the Bayesian method, both symmetric and asymmetric loss functions are employed. To examine the behaviors of various estimates based on criterion measurements, a Monte Carlo simulation research is carried out. Finally, with the aim of demonstrating the applicability of our findings, three real datasets are used. The results show that the KM-GIKw distribution offers superior fits when compared to other well-known distributions. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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33 pages, 1233 KiB

Open AccessArticle

Bayesian and Non-Bayesian Estimation for a New Extension of Power Topp–Leone Distribution under Ranked Set Sampling with Applications

by Naif Alotaibi, A. S. Al-Moisheer, Ibrahim Elbatal, Mansour Shrahili, Mohammed Elgarhy and Ehab M. Almetwally

Axioms 2023, 12(8), 722; https://doi.org/10.3390/axioms12080722 - 25 Jul 2023

Cited by 2 | Viewed by 1129

Abstract

In this article, we intend to introduce and study a new two-parameter distribution as a new extension of the power Topp–Leone (PTL) distribution called the Kavya–Manoharan PTL (KMPTL) distribution. Several mathematical and statistical features of the KMPTL distribution, such as the quantile function, [...] Read more.

In this article, we intend to introduce and study a new two-parameter distribution as a new extension of the power Topp–Leone (PTL) distribution called the Kavya–Manoharan PTL (KMPTL) distribution. Several mathematical and statistical features of the KMPTL distribution, such as the quantile function, moments, generating function, and incomplete moments, are calculated. Some measures of entropy are investigated. The cumulative residual Rényi entropy (CRRE) is calculated. To estimate the parameters of the KMPTL distribution, both maximum likelihood and Bayesian estimation methods are used under simple random sample (SRS) and ranked set sampling (RSS). The simulation study was performed to be able to verify the model parameters of the KMPTL distribution using SRS and RSS to demonstrate that RSS is more efficient than SRS. We demonstrated that the KMPTL distribution has more flexibility than the PTL distribution and the other nine competitive statistical distributions: PTL, unit-Gompertz, unit-Lindley, Topp–Leone, unit generalized log Burr XII, unit exponential Pareto, Kumaraswamy, beta, Marshall-Olkin Kumaraswamy distributions employing two real-world datasets. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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25 pages, 2788 KiB

Open AccessArticle

On Estimation of Reliability Functions for the Extended Rayleigh Distribution under Progressive First-Failure Censoring Model

by Mahmoud Hamed Abu-Moussa, Najwan Alsadat and Ali Sharawy

Axioms 2023, 12(7), 680; https://doi.org/10.3390/axioms12070680 - 10 Jul 2023

Cited by 3 | Viewed by 1037

Abstract

When conducting reliability studies, the progressive first-failure censoring (PFFC) method is useful in situations in which the units of the life testing experiment are separated into groups consisting of k units each with the intention of seeing only the first failure in each [...] Read more.

When conducting reliability studies, the progressive first-failure censoring (PFFC) method is useful in situations in which the units of the life testing experiment are separated into groups consisting of k units each with the intention of seeing only the first failure in each group. Using progressive first-failure censored samples, the statistical inference for the parameters, reliability, and hazard functions of the extended Rayleigh distribution (ERD) are investigated in this study. The asymptotic normality theory of maximum likelihood estimates (MLEs) is used in order to acquire the maximum likelihood estimates (MLEs) together with the asymptotic confidence intervals (Asym. CIs). Bayesian estimates (BEs) of the parameters and the reliability functions under different loss functions may be produced by using independent gamma informative priors and non-informative priors. The Markov chain Monte Carlo (MCMC) approach is used so that Bayesian computations are performed with ease. In addition, the MCMC method is used in order to create credible intervals (Cred. CIs) for the parameters, which may be used for either informative or non-informative priors. Additionally, computations for the reliability functions are carried out. A Monte Carlo simulation study is carried out in order to provide a comparison of the behaviour of the different estimations that were created for this work. At last, an actual data set is dissected for the purpose of providing an example. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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19 pages, 345 KiB

Open AccessArticle

A Strong Limit Theorem of the Largest Entries of a Sample Correlation Matrices under a Strong Mixing Assumption

by Haozhu Zhao and Yong Zhang

Axioms 2023, 12(7), 657; https://doi.org/10.3390/axioms12070657 - 2 Jul 2023

Viewed by 888

Abstract

We are interested in an n by p matrix

X_{n}

where the n rows are strictly stationary

α

-mixing random vectors and each of the p columns is an independent and identically distributed random vector;

p = p_{n}

goes to infinity [...] Read more.

We are interested in an n by p matrix

X_{n}

where the n rows are strictly stationary

α

-mixing random vectors and each of the p columns is an independent and identically distributed random vector;

p = p_{n}

goes to infinity as

n \to \infty

, satisfiying

0 < c_{1} \leq p_{n} / n^{τ} \leq c_{2} < \infty

, where

τ > 0

,

c_{2} \geq c_{1} > 0

. We obtain a logarithmic law of

L_{n} = {max}_{1 \leq i < j \leq p_{n}} | ρ_{i j} |

using the Chen–Stein Poisson approximation method, where

ρ_{i j}

denotes the sample correlation coefficient between the ith column and the jth column of

X_{n}

. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

18 pages, 445 KiB

Open AccessArticle

An Exponentiated Skew-Elliptic Nonlinear Extension to the Log–Linear Birnbaum–Saunders Model with Diagnostic and Residual Analysis

by Guillermo Martínez-Flórez, Yolanda M. Gómez and Osvaldo Venegas

Axioms 2023, 12(7), 624; https://doi.org/10.3390/axioms12070624 - 23 Jun 2023

Viewed by 927

Abstract

In this paper, we propose a nonlinear regression model with exponentiated skew-elliptical errors distributed, which can be fitted to datasets with high levels of asymmetry and kurtosis. Maximum likelihood estimation procedures in finite samples are discussed and the information matrix is deduced. We [...] Read more.

In this paper, we propose a nonlinear regression model with exponentiated skew-elliptical errors distributed, which can be fitted to datasets with high levels of asymmetry and kurtosis. Maximum likelihood estimation procedures in finite samples are discussed and the information matrix is deduced. We carried out a diagnosis of the influence for the nonlinear model. To analyze the sensitivity of the maximum likelihood estimators of the model’s parameters to small perturbations in distribution assumptions and parameter estimation, we studied the perturbation schemes, the case weight, and the explanatory and response variables of perturbations; we also carried out a residual analysis of the deviance components. Simulation studies were performed to assess some properties of the estimators, showing the good performance of the proposed estimation procedure in finite samples. Finally, an application to a real dataset is presented. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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27 pages, 613 KiB

Open AccessArticle

A Compound Class of Inverse-Power Muth and Power Series Distributions

by Leonardo Barrios-Blanco, Diego I. Gallardo, Héctor J. Gómez and Marcelo Bourguignon

Axioms 2023, 12(4), 383; https://doi.org/10.3390/axioms12040383 - 16 Apr 2023

Cited by 1 | Viewed by 1519

Abstract

This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability [...] Read more.

This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability density, survival and hazard functions are studied, as well as their moments. Using the stochastic representation of the model, the maximum-likelihood estimators are implemented by the use of the expectation-maximization algorithm, while standard errors are calculated using Oakes’ method. Monte Carlo simulation studies are conducted to show the performance of the maximum-likelihood estimators in finite samples. Two applications to real datasets are shown, where our proposal is compared with some models based on power series compositions. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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17 pages, 4004 KiB

Open AccessArticle

A Modified Gamma Model: Properties, Estimation, and Applications

by Mashael A. Alshehri and Mohamed Kayid

Axioms 2023, 12(3), 262; https://doi.org/10.3390/axioms12030262 - 3 Mar 2023

Viewed by 1455

Abstract

Statistical methods are essential for describing, predicting, and modeling natural phenomena in numerous application areas. These methods are helpful for modeling and predicting data in medicine, reliability engineering, actuarial science, and other fields. This paper presents a novel, simple, and fully flexible modified [...] Read more.

Statistical methods are essential for describing, predicting, and modeling natural phenomena in numerous application areas. These methods are helpful for modeling and predicting data in medicine, reliability engineering, actuarial science, and other fields. This paper presents a novel, simple, and fully flexible modified gamma model. The new model provides various forms of densities, including symmetric, asymmetric, unimodal, and reversed-J shapes, as well as a bathtub-shaped failure rate, which is suitable for modeling the lifespan of patients with an increased risk of death. Some basic and dynamic properties of the model are examined. Four methods for estimating its parameters are discussed, and a simulation study is used to examine the consistency and efficiency of these estimators. Finally, the usefulness of the proposed model is demonstrated in the analysis of some data sets. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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22 pages, 1007 KiB

Open AccessArticle

Survival Analysis of Type-II Lehmann Fréchet Parameters via Progressive Type-II Censoring with Applications

by Ahmed Elshahhat, Ritwik Bhattacharya and Heba S. Mohammed

Axioms 2022, 11(12), 700; https://doi.org/10.3390/axioms11120700 - 7 Dec 2022

Cited by 4 | Viewed by 1313

Abstract

A new three-parameter Type-II Lehmann Fréchet distribution (LFD-TII), as a reparameterized version of the Kumaraswamy–Fréchet distribution, is considered. In this study, using progressive Type-II censoring, different estimation methods of the LFD-TII parameters and its lifetime functions, namely, reliability and hazard functions, are considered. [...] Read more.

A new three-parameter Type-II Lehmann Fréchet distribution (LFD-TII), as a reparameterized version of the Kumaraswamy–Fréchet distribution, is considered. In this study, using progressive Type-II censoring, different estimation methods of the LFD-TII parameters and its lifetime functions, namely, reliability and hazard functions, are considered. In a frequentist setup, both the likelihood and product of the spacing estimators of the considered parameters are obtained utilizing the Newton–Raphson method. From the normality property of the proposed classical estimators, based on Fisher’s information and the delta method, the asymptotic confidence interval for any unknown parametric function is obtained. In the Bayesian paradigm via likelihood and spacings functions, using independent gamma conjugate priors, the Bayes estimators of the unknown parameters are obtained against the squared-error and general-entropy loss functions. Since the proposed posterior distributions cannot be explicitly expressed, by combining two Markov-chain Monte-Carlo techniques, namely, the Gibbs and Metropolis–Hastings algorithms, the Bayes point/interval estimates are approximated. To examine the performance of the proposed estimation methodologies, extensive simulation experiments are conducted. In addition, based on several criteria, the optimum censoring plan is proposed. In real-life practice, to show the usefulness of the proposed estimators, two applications based on two different data sets taken from the engineering and physics fields are analyzed. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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