Mathematics

Research

14 pages, 333 KiB

Open AccessArticle

A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

by Dan Lascu and Gabriela Ileana Sebe

Mathematics 2021, 9(3), 255; https://doi.org/10.3390/math9030255 - 28 Jan 2021

Cited by 2 | Viewed by 1719

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one [...] Read more.

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions,

θ

-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems. Full article

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

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15 pages, 473 KiB

Open AccessArticle

An Over and Underdispersed Biparametric Extension of the Waring Distribution

by Valentina Cueva-López, María José Olmo-Jiménez and José Rodríguez-Avi

Mathematics 2021, 9(2), 170; https://doi.org/10.3390/math9020170 - 15 Jan 2021

Cited by 4 | Viewed by 1851

Abstract

A new discrete distribution for count data called extended biparametric Waring (

E B W

) distribution is developed. Its name is related to the fact that, in a specific configuration of its parameters, it can be seen as a biparametric version of [...] Read more.

A new discrete distribution for count data called extended biparametric Waring (

E B W

) distribution is developed. Its name is related to the fact that, in a specific configuration of its parameters, it can be seen as a biparametric version of the univariate generalized Waring (

U G W

) distribution, a well-known model for the variance decomposition into three components: randomness, liability and proneness. Unlike the

U G W

distribution, the

E B W

can model both overdispersed and underdispersed data sets. In fact, the

E B W

distribution is a particular case of a

U W G

distribution when its first parameter is positive; otherwise, it is a particular case of a Complex Triparametric Pearson (

C T P

) distribution. Hence, this new model inherits most of their properties and, moreover, it helps to solve the identification problem in the variance components of the

U G W

model. We compare the

E B W

with the

U G W

by a simulation study, but also with other over and underdispersed distributions through the Kullback-Leibler divergence. Additionally, we have carried out a simulation study in order to analyse the properties of the maximum likelihood parameter estimates. Finally, some application examples are included which show that the proposed model provides similar or even better results than other models, but with fewer parameters. Full article

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

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13 pages, 712 KiB

Open AccessArticle

An Information-Theoretic Approach for Multivariate Skew-t Distributions and Applications

by Salah H. Abid, Uday J. Quaez and Javier E. Contreras-Reyes

Mathematics 2021, 9(2), 146; https://doi.org/10.3390/math9020146 - 11 Jan 2021

Cited by 18 | Viewed by 3343

Abstract

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew- [...] Read more.

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented. Full article

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

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25 pages, 600 KiB

Open AccessFeature PaperArticle

Changepoint in Error-Prone Relations

by Michal Pešta

Mathematics 2021, 9(1), 89; https://doi.org/10.3390/math9010089 - 4 Jan 2021

Cited by 2 | Viewed by 2515

Abstract

Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change [...] Read more.

Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change in trend for a randomly spaced time series is a special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A changepoint estimator is also introduced and its consistency is proved. A boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a theoretical basis for the developed methods, a weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through problems coming from calibration and insurance; however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper. Full article

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

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18 pages, 1115 KiB

Open AccessArticle

Properties and Applications of a New Family of Skew Distributions

by Emilio Gómez-Déniz, Barry C. Arnold, José M. Sarabia and Héctor W. Gómez

Mathematics 2021, 9(1), 87; https://doi.org/10.3390/math9010087 - 3 Jan 2021

Cited by 4 | Viewed by 3351

Abstract

We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals [...] Read more.

We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution. Full article

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

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14 pages, 5627 KiB

Open AccessArticle

m-Consecutive-k-out-of-n: F Structures with a Single Change Point

by Ioannis S. Triantafyllou

Mathematics 2020, 8(12), 2203; https://doi.org/10.3390/math8122203 - 10 Dec 2020

Cited by 3 | Viewed by 2080

Abstract

In the present article, we introduce the m-consecutive-k-out-of-n:F structures with a single change point. The aforementioned system consists of n independent components, of which the first n₁ units are identically distributed with common reliability p₁, [...] Read more.

In the present article, we introduce the m-consecutive-k-out-of-n:F structures with a single change point. The aforementioned system consists of n independent components, of which the first n₁ units are identically distributed with common reliability p₁, while the remaining ones share a different functioning probability p₂. The general setup of the proposed reliability structures is presented in detail, while an explicit expression for determining the number of its path sets of a given size is derived. Additionally, closed formulae for the reliability function and mean time to failure of the aforementioned models are also provided. For illustration purposes, several numerical results and comparisons are presented in order to shed light on the performance of the proposed structure. Full article

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

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20 pages, 539 KiB

Open AccessFeature PaperArticle

Use of Correlated Data for Nonparametric Prediction of a Spatial Target Variable

by Pilar García-Soidán and Tomás R. Cotos-Yáñez

Mathematics 2020, 8(11), 2077; https://doi.org/10.3390/math8112077 - 20 Nov 2020

Cited by 1 | Viewed by 1927

Abstract

The kriging methodology can be applied to predict the value of a spatial variable at an unsampled location, from the available spatial data. Furthermore, additional information from secondary variables, correlated with the target one, can be included in the resulting predictor by using [...] Read more.

The kriging methodology can be applied to predict the value of a spatial variable at an unsampled location, from the available spatial data. Furthermore, additional information from secondary variables, correlated with the target one, can be included in the resulting predictor by using the cokriging techniques. The latter procedures require a previous specification of the multivariate dependence structure, difficult to characterize in practice in an appropriate way. To simplify this task, the current work introduces a nonparametric kernel approach for prediction, which satisfies good properties, such as asymptotic unbiasedness or the convergence to zero of the mean squared prediction error. The selection of the bandwidth parameters involved is also addressed, as well as the estimation of the remaining unknown terms in the kernel predictor. The performance of the new methodology is illustrated through numerical studies with simulated data, carried out in different scenarios. In addition, the proposed nonparametric approach is applied to predict the concentrations of a pollutant that represents a risk to human health, the cadmium, in the floodplain of the Meuse river (Netherlands), by incorporating the lead level as an auxiliary variable. Full article

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

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29 pages, 4007 KiB

Open AccessArticle

Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing

by Mukhtar M. Salah, M. El-Morshedy, M. S. Eliwa and Haitham M. Yousof

Mathematics 2020, 8(11), 1949; https://doi.org/10.3390/math8111949 - 4 Nov 2020

Cited by 40 | Viewed by 2255

Abstract

The extreme value theory is expanded by proposing and studying a new version of the Fréchet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, [...] Read more.

The extreme value theory is expanded by proposing and studying a new version of the Fréchet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, different non-Bayesian estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation method, Anderson–Darling estimation method, ordinary least square estimation method, Cramér–von-Mises estimation method, weighted least square estimation method, left-tail Anderson–Darling estimation method, and right-tail Anderson–Darling estimation method. Numerical simulations were performed for comparing the estimation methods using different sample sizes for three different combinations of parameters. The Barzilai–Borwein algorithm was employed via a simulation study. Three applications were presented for measuring the flexibility and the importance of the new model for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. The modified goodness-of-fit statistic test was applied for the right censored real data set, called leukemia free-survival times for autologous transplants. Based on the maximum likelihood estimators on initial data, the modified goodness-of-fit test recovered the loss in information while the grouping data and followed chi-square distributions. All elements of the modified goodness-of-fit criteria tests are explicitly derived and given. Full article

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

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22 pages, 917 KiB

Open AccessArticle

Scale Mixture of Rayleigh Distribution

by Pilar A. Rivera, Inmaculada Barranco-Chamorro, Diego I. Gallardo and Héctor W. Gómez

Mathematics 2020, 8(10), 1842; https://doi.org/10.3390/math8101842 - 20 Oct 2020

Cited by 10 | Viewed by 2724

Abstract

In this paper, the scale mixture of Rayleigh (SMR) distribution is introduced. It is proven that this new model, initially defined as the quotient of two independent random variables, can be expressed as a scale mixture of a Rayleigh and a particular Generalized [...] Read more.

In this paper, the scale mixture of Rayleigh (SMR) distribution is introduced. It is proven that this new model, initially defined as the quotient of two independent random variables, can be expressed as a scale mixture of a Rayleigh and a particular Generalized Gamma distribution. Closed expressions are obtained for its pdf, cdf, moments, asymmetry and kurtosis coefficients. Its lifetime analysis, properties and Rényi entropy are studied. Inference based on moments and maximum likelihood (ML) is proposed. An Expectation-Maximization (EM) algorithm is implemented to estimate the parameters via ML. This algorithm is also used in a simulation study, which illustrates the good performance of our proposal. Two real datasets are considered in which it is shown that the SMR model provides a good fit and it is more flexible, especially as for kurtosis, than other competitor models, such as the slashed Rayleigh distribution. Full article

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

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20 pages, 6088 KiB

Open AccessArticle

Mutually Complementary Measure-Correlate-Predict Method for Enhanced Long-Term Wind-Resource Assessment

by Woochul Nam and Ki-Yong Oh

Mathematics 2020, 8(10), 1795; https://doi.org/10.3390/math8101795 - 15 Oct 2020

Cited by 1 | Viewed by 2624

Abstract

Evaluating the economic feasibility of wind farms via long-term wind-resource assessments is indispensable because short-term data measured at a candidate wind-farm site cannot represent the long-term wind potential. Prediction errors are significant when seasonal and year-on-year variations occur. Moreover, reliable long-term reference data [...] Read more.

Evaluating the economic feasibility of wind farms via long-term wind-resource assessments is indispensable because short-term data measured at a candidate wind-farm site cannot represent the long-term wind potential. Prediction errors are significant when seasonal and year-on-year variations occur. Moreover, reliable long-term reference data with a high correlation to short-term measured data are often unavailable. This paper presents an alternative solution to predict long-term wind resources for a site exhibiting seasonal and year-on-year variations, where long-term reference data are unavailable. An analysis shows that a mutually complementary measure-correlate-predict method can be employed, because several datasets obtained over short periods are used to correct long-term wind resource data in a mutually complementary manner. Moreover, this method is useful in evaluating extreme wind speeds, which is one of the main factors affecting site compliance evaluation and the selection of a suitable wind turbine class based on the International Electrotechnical Commission standards. The analysis also shows that energy density is a more sensitive metric than wind speed for sites with seasonal and year-on-year variations because of the wide distribution of wind speeds. A case study with short-term data measured at Fujeij, Jordan, clearly identifies the factors necessary to perform the reliable and accurate assessment of long-term wind potentials. Full article

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

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19 pages, 329 KiB

Open AccessArticle

Accelerated Life Tests under Pareto-IV Lifetime Distribution: Real Data Application and Simulation Study

by A. M. Abd El-Raheem, M. H. Abu-Moussa, Marwa M. Mohie El-Din and E. H. Hafez

Mathematics 2020, 8(10), 1786; https://doi.org/10.3390/math8101786 - 15 Oct 2020

Cited by 14 | Viewed by 2313

Abstract

In this article, a progressive-stress accelerated life test (ALT) that is based on progressive type-II censoring is studied. The cumulative exposure model is used when the lifetime of test units follows Pareto-IV distribution. Different estimates as the maximum likelihood estimates (MLEs) and Bayes [...] Read more.

In this article, a progressive-stress accelerated life test (ALT) that is based on progressive type-II censoring is studied. The cumulative exposure model is used when the lifetime of test units follows Pareto-IV distribution. Different estimates as the maximum likelihood estimates (MLEs) and Bayes estimates (BEs) for the model parameters are discussed. Bayesian estimates are derived while using the Tierney and Kadane (TK) approximation method and the importance sampling method. The asymptotic and bootstrap confidence intervals (CIs) of the parameters are constructed. A real data set is analyzed in order to clarify the methods proposed through this paper. Two types of the progressive-stress tests, the simple ramp-stress test and multiple ramp-stress test, are compared through the simulation study. Finally, some interesting conclusions are drawn. Full article

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

10 pages, 907 KiB

Open AccessArticle

New Approach for a Weibull Distribution under the Progressive Type-II Censoring Scheme

by Jung-In Seo, Young Eun Jeon and Suk-Bok Kang

Mathematics 2020, 8(10), 1713; https://doi.org/10.3390/math8101713 - 5 Oct 2020

Cited by 2 | Viewed by 2244

Abstract

This paper proposes a new approach based on the regression framework employing a pivotal quantity to estimate unknown parameters of a Weibull distribution under the progressive Type-II censoring scheme, which provides a closed form solution for the shape parameter, unlike its maximum likelihood [...] Read more.

This paper proposes a new approach based on the regression framework employing a pivotal quantity to estimate unknown parameters of a Weibull distribution under the progressive Type-II censoring scheme, which provides a closed form solution for the shape parameter, unlike its maximum likelihood estimator counterpart. To resolve serious rounding errors for the exact mean and variance of the pivotal quantity, two different types of Taylor series expansion are applied, and the resulting performance is enhanced in terms of the mean square error and bias obtained through the Monte Carlo simulation. Finally, an actual application example, including a simple goodness-of-fit analysis of the actual test data based on the pivotal quantity, proves the feasibility and applicability of the proposed approach. Full article

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

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19 pages, 824 KiB

Open AccessArticle

Progressive Type-II Censoring Schemes of Extended Odd Weibull Exponential Distribution with Applications in Medicine and Engineering

by R. Alshenawy, Ali Al-Alwan, Ehab M. Almetwally, Ahmed Z. Afify and Hisham M. Almongy

Mathematics 2020, 8(10), 1679; https://doi.org/10.3390/math8101679 - 1 Oct 2020

Cited by 32 | Viewed by 3786

Abstract

In this paper, the parameters of the extended odd Weibull exponential distribution are estimated under progressive type-II censoring scheme with random removal. The model parameters are estimated using the maximum product spacing and maximum likelihood estimation methods. Further, we explore the asymptotic confidence [...] Read more.

In this paper, the parameters of the extended odd Weibull exponential distribution are estimated under progressive type-II censoring scheme with random removal. The model parameters are estimated using the maximum product spacing and maximum likelihood estimation methods. Further, we explore the asymptotic confidence intervals and bootstrap confidence intervals for the model parameters. Monte Carlo simulations are performed to compare between the proposed estimation methods under progressive type-II censoring scheme. An empirical study using two real datasets form engineering and medicine fields to validate the introduced methods of inference. Based on our study, we can conclude that the maximum product of spacing method outperforms the maximum likelihood method for estimating the extended odd Weibull exponential (EOWE) parameters under a progressive type-II censoring scheme in both numerical and empirical cases. Full article

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

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21 pages, 1305 KiB

Open AccessArticle

Robust Reliability Estimation for Lindley Distribution—A Probability Integral Transform Statistical Approach

by Muhammad Aslam Mohd Safari, Nurulkamal Masseran and Muhammad Hilmi Abdul Majid

Mathematics 2020, 8(9), 1634; https://doi.org/10.3390/math8091634 - 21 Sep 2020

Cited by 6 | Viewed by 3142

Abstract

In the modeling and analysis of reliability data via the Lindley distribution, the maximum likelihood estimator is the most commonly used for parameter estimation. However, the maximum likelihood estimator is highly sensitive to the presence of outliers. In this paper, based on the [...] Read more.

In the modeling and analysis of reliability data via the Lindley distribution, the maximum likelihood estimator is the most commonly used for parameter estimation. However, the maximum likelihood estimator is highly sensitive to the presence of outliers. In this paper, based on the probability integral transform statistic, a robust and efficient estimator of the parameter of the Lindley distribution is proposed. We investigate the relative efficiency of the new estimator compared to that of the maximum likelihood estimator, as well as its robustness based on the breakdown point and influence function. It is found that this new estimator provides reasonable protection against outliers while also being simple to compute. Using a Monte Carlo simulation, we compare the performance of the new estimator and several well-known methods, including the maximum likelihood, ordinary least-squares and weighted least-squares methods in the absence and presence of outliers. The results reveal that the new estimator is highly competitive with the maximum likelihood estimator in the absence of outliers and outperforms the other methods in the presence of outliers. Finally, we conduct a statistical analysis of four reliability data sets, the results of which support the simulation results. Full article

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

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20 pages, 2160 KiB

Open AccessEditor’s ChoiceArticle

A New Extended Two-Parameter Distribution: Properties, Estimation Methods, and Applications in Medicine and Geology

by Hazem Al-Mofleh, Ahmed Z. Afify and Noor Akma Ibrahim

Mathematics 2020, 8(9), 1578; https://doi.org/10.3390/math8091578 - 12 Sep 2020

Cited by 30 | Viewed by 3875

Abstract

In this paper, a new two-parameter generalized Ramos–Louzada distribution is proposed. The proposed model provides more flexibility in modeling data with increasing, decreasing, J-shaped, and reversed-J shaped hazard rate functions. Several statistical properties of the model were derived. The unknown parameters of the [...] Read more.

In this paper, a new two-parameter generalized Ramos–Louzada distribution is proposed. The proposed model provides more flexibility in modeling data with increasing, decreasing, J-shaped, and reversed-J shaped hazard rate functions. Several statistical properties of the model were derived. The unknown parameters of the new distribution were explored using eight frequentist estimation approaches. These approaches are important for developing guidelines to choose the best method of estimation for the model parameters, which would be of great interest to practitioners and applied statisticians. Detailed numerical simulations are presented to examine the bias and the mean square error of the proposed estimators. The best estimation method and ordering performance of the estimators were determined using the partial and overall ranks of all estimation methods for various parameter combinations. The performance of the proposed distribution is illustrated using two real datasets from the fields of medicine and geology, and both datasets show that the new model is more appropriate as compared to the Marshall–Olkin exponential, exponentiated exponential, beta exponential, gamma, Poisson–Lomax, Lindley geometric, generalized Lindley, and Lindley distributions, among others. Full article

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

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14 pages, 263 KiB

Open AccessArticle

Parameter Estimation and Hypothesis Testing of Geographically Weighted Multivariate Generalized Poisson Regression

by Sarni Maniar Berliana, Purhadi, Sutikno and Santi Puteri Rahayu

Mathematics 2020, 8(9), 1523; https://doi.org/10.3390/math8091523 - 7 Sep 2020

Cited by 5 | Viewed by 2716

Abstract

We introduce a new multivariate regression model based on the generalized Poisson distribution, which we called geographically-weighted multivariate generalized Poisson regression (GWMGPR) model, and we present a maximum likelihood step-by-step procedure to obtain parameters for it. We use the maximum likelihood ratio test [...] Read more.

We introduce a new multivariate regression model based on the generalized Poisson distribution, which we called geographically-weighted multivariate generalized Poisson regression (GWMGPR) model, and we present a maximum likelihood step-by-step procedure to obtain parameters for it. We use the maximum likelihood ratio test to examine the significance of the regression parameters and to define their critical region. Full article

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

20 pages, 4507 KiB

Open AccessArticle

On Designing Non-Parametric EWMA Sign Chart under Ranked Set Sampling Scheme with Application to Industrial Process

by Saber Ali, Zameer Abbas, Hafiz Zafar Nazir, Muhammad Riaz, Xingfa Zhang and Yuan Li

Mathematics 2020, 8(9), 1497; https://doi.org/10.3390/math8091497 - 4 Sep 2020

Cited by 24 | Viewed by 2547

Abstract

Statistical process control (SPC) tools are used for the investigation and identification of unnatural variations in the manufacturing, industrial, and service processes. The control chart, the basic and the most famous tool of SPC, is used for process monitoring. Generally, control charts are [...] Read more.

Statistical process control (SPC) tools are used for the investigation and identification of unnatural variations in the manufacturing, industrial, and service processes. The control chart, the basic and the most famous tool of SPC, is used for process monitoring. Generally, control charts are constructed under normality assumption of the quality characteristic of interest, but in practice, it is quite hard to hold the normality assumption. In such situations, parametric charts tend to offer more frequent false alarms and invalid out-of-control performance. To rectify these problems, non-parametric control charts are used, as these have the same in-control run length properties for all the continuous distributions and are known as in-control robust. This study intends to develop a new non-parametric exponentially weighted moving average (NPEWMA) chart based on sign statistics under a ranked set sampling scheme that is hereafter named (NPREWMA-SN). The run-length profiles of the NPREWMA-SN chart are computed using the Monte Carlo simulation method. The proposed scheme is compared with NPEWMA-SN and classical EWMA-

\bar{X}

charts, using different run length measures. The comparison reveals the in-control robustness and superiority of the proposed scheme over its competitors in detecting all kinds of shifts in the process location. A practical application related to the substrate manufacturing process is included to show the demonstration of the proposed chart. Full article

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

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10 pages, 281 KiB

Open AccessArticle

Stochastic Order for a Multivariate Uniform Distributions Family

by Luigi-Ionut Catana and Anisoara Raducan

Mathematics 2020, 8(9), 1410; https://doi.org/10.3390/math8091410 - 23 Aug 2020

Cited by 4 | Viewed by 2633

Abstract

In this article we give sufficient conditions for stochastic order of multivariate uniform distributions on closed convex sets. Full article

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

17 pages, 356 KiB

Open AccessArticle

The Lambert-F Distributions Class: An Alternative Family for Positive Data Analysis

by Yuri A. Iriarte, Mário de Castro and Héctor W. Gómez

Mathematics 2020, 8(9), 1398; https://doi.org/10.3390/math8091398 - 21 Aug 2020

Cited by 8 | Viewed by 3141

Abstract

In this article, we introduce a new probability distribution generator called the Lambert-F generator. For any continuous baseline distribution F, with positive support, the corresponding Lambert-F version is generated by using the new generator. The result is a new class [...] Read more.

In this article, we introduce a new probability distribution generator called the Lambert-F generator. For any continuous baseline distribution F, with positive support, the corresponding Lambert-F version is generated by using the new generator. The result is a new class of distributions with one extra parameter that generalizes the baseline distribution and whose quantile function can be expressed in closed form in terms of the Lambert W function. The hazard rate function of a Lambert-F distribution corresponds to a modification of the baseline hazard rate function, greatly increasing or decreasing the baseline hazard rate for earlier times. Herein, we study the main structural properties of the new class of distributions. Special attention is given to two particular cases that can be understood as two-parameter extensions of the well-known exponential and Rayleigh distributions. We discuss parameter estimation for the proposed models considering the moments and maximum likelihood methods. Finally, two applications were developed to illustrate the usefulness of the proposed distributions in the analysis of data from different real settings. Full article

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

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24 pages, 417 KiB

Open AccessArticle

A New Modified Kies Family: Properties, Estimation Under Complete and Type-II Censored Samples, and Engineering Applications

by Abdulhakim A. Al-Babtain, Mohammed K. Shakhatreh, Mazen Nassar and Ahmed Z. Afify

Mathematics 2020, 8(8), 1345; https://doi.org/10.3390/math8081345 - 12 Aug 2020

Cited by 44 | Viewed by 3089

Abstract

In this paper, we introduce a new family of continuous distributions that is called the modified Kies family of distributions. The main mathematical properties of the new family are derived. A special case of the new family has been considered in more detail; [...] Read more.

In this paper, we introduce a new family of continuous distributions that is called the modified Kies family of distributions. The main mathematical properties of the new family are derived. A special case of the new family has been considered in more detail; namely, the two parameters modified Kies exponential distribution with bathtub shape, decreasing and increasing failure rate function. The importance of the new distribution comes from its ability in modeling positively and negatively skewed real data over some generalized distributions with more than two parameters. The shape behavior of the hazard rate and the mean residual life functions of the modified Kies exponential distribution are discussed. We use the method of maximum likelihood to estimate the distribution parameters based on complete and type-II censored samples. The approximate confidence intervals are also obtained under the two schemes. A simulation study is conducted and two real data sets from the engineering field are analyzed to show the flexibility of the new distribution in modeling real life data. Full article

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

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10 pages, 340 KiB

Open AccessArticle

Estimation of Non-Linear Parameters with Data Collected Using Respondent-Driven Sampling

by Ismael Sánchez-Borrego, María del Mar Rueda and Héctor Mullo

Mathematics 2020, 8(8), 1315; https://doi.org/10.3390/math8081315 - 7 Aug 2020

Cited by 1 | Viewed by 2285

Abstract

Respondent-driven sampling (RDS) is a snowball-type sampling method used to survey hidden populations, that is, those that lack a sampling frame. In this work, we consider the problem of regression modeling and association for continuous RDS data. We propose a new sample weight [...] Read more.

Respondent-driven sampling (RDS) is a snowball-type sampling method used to survey hidden populations, that is, those that lack a sampling frame. In this work, we consider the problem of regression modeling and association for continuous RDS data. We propose a new sample weight method for estimating non-linear parameters such as the covariance and the correlation coefficient. We also estimate the variances of the proposed estimators. As an illustration, we performed a simulation study and an application to an ethnic example. The proposed estimators are consistent and asymptotically unbiased. We discuss the applicability of the method as well as future research. Full article

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

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28 pages, 525 KiB

Open AccessArticle

The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data

by Ahmed Z. Afify, Ahmed M. Gemeay and Noor Akma Ibrahim

Mathematics 2020, 8(8), 1276; https://doi.org/10.3390/math8081276 - 3 Aug 2020

Cited by 60 | Viewed by 6331

Abstract

Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries [...] Read more.

Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions. Full article

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

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22 pages, 919 KiB

Open AccessArticle

Estimation of Beta-Pareto Distribution Based on Several Optimization Methods

by Badreddine Boumaraf, Nacira Seddik-Ameur and Vlad Stefan Barbu

Mathematics 2020, 8(7), 1055; https://doi.org/10.3390/math8071055 - 1 Jul 2020

Cited by 5 | Viewed by 3124

Abstract

This paper is concerned with the maximum likelihood estimators of the Beta-Pareto distribution introduced in Akinsete et al. (2008), which comes from the mixing of two probability distributions, Beta and Pareto. Since these estimators cannot be obtained explicitly, we use nonlinear optimization methods [...] Read more.

This paper is concerned with the maximum likelihood estimators of the Beta-Pareto distribution introduced in Akinsete et al. (2008), which comes from the mixing of two probability distributions, Beta and Pareto. Since these estimators cannot be obtained explicitly, we use nonlinear optimization methods that numerically provide these estimators. The methods we investigate are the method of Newton-Raphson, the gradient method and the conjugate gradient method. Note that for the conjugate gradient method we use the model of Fletcher-Reeves. The corresponding algorithms are developed and the performances of the methods used are confirmed by an important simulation study. In order to compare between several concurrent models, namely generalized Beta-Pareto, Beta, Pareto, Gamma and Beta-Pareto, model criteria selection are used. We firstly consider completely observed data and, secondly, the observations are assumed to be right censored and we derive the same type of results. Full article

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

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23 pages, 969 KiB

Open AccessArticle

Estimation of Uncertainty in Mortality Projections Using State-Space Lee-Carter Model

by Rokas Gylys and Jonas Šiaulys

Mathematics 2020, 8(7), 1053; https://doi.org/10.3390/math8071053 - 30 Jun 2020

Cited by 4 | Viewed by 2857

Abstract

The study develops alternatives of the classical Lee-Carter stochastic mortality model in assessment of uncertainty of mortality rates forecasts. We use the Lee-Carter model expressed as linear Gaussian state-space model or state-space model with Markovian regime-switching to derive coherent estimates of parameters and [...] Read more.

The study develops alternatives of the classical Lee-Carter stochastic mortality model in assessment of uncertainty of mortality rates forecasts. We use the Lee-Carter model expressed as linear Gaussian state-space model or state-space model with Markovian regime-switching to derive coherent estimates of parameters and to introduce additional flexibility required to capture change in trend and non-Gaussian volatility of mortality improvements. For model-fitting, we use a Bayesian Gibbs sampler. We illustrate the application of the models by deriving the confidence intervals of mortality projections using Lithuanian and Swedish data. The results show that state-space model with Markovian regime-switching adequately captures the effect of pandemic, which is present in the Swedish data. However, it is less suitable to model less sharp but more prolonged fluctuations of mortality trends in Lithuania. Full article

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

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16 pages, 2106 KiB

Open AccessArticle

On the Consecutive k₁ and k₂-out-of-n Reliability Systems

by Ioannis S. Triantafyllou

Mathematics 2020, 8(4), 630; https://doi.org/10.3390/math8040630 - 19 Apr 2020

Cited by 10 | Viewed by 2828

Abstract

In this paper we carry out a reliability study of the consecutive-k₁ and k₂-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure’s reliability, while recurrence [...] Read more.

In this paper we carry out a reliability study of the consecutive-k₁ and k₂-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure’s reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, some numerical results and figures are presented and several concluding remarks are deduced. Full article

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

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29 pages, 450 KiB

Open AccessArticle

Asymptotic Results in Broken Stick Models: The Approach via Lorenz Curves

by Gheorghiță Zbăganu

Mathematics 2020, 8(4), 625; https://doi.org/10.3390/math8040625 - 18 Apr 2020

Cited by 1 | Viewed by 2347

Abstract

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the [...] Read more.

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of

{[0, 1]}^{2}

. This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one. Full article

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

18 pages, 1058 KiB

Open AccessArticle

Multi-Partitions Subspace Clustering

by Vincent Vandewalle

Mathematics 2020, 8(4), 597; https://doi.org/10.3390/math8040597 - 15 Apr 2020

Cited by 5 | Viewed by 2636

Abstract

In model based clustering, it is often supposed that only one clustering latent variable explains the heterogeneity of the whole dataset. However, in many cases several latent variables could explain the heterogeneity of the data at hand. Finding such class variables could result [...] Read more.

In model based clustering, it is often supposed that only one clustering latent variable explains the heterogeneity of the whole dataset. However, in many cases several latent variables could explain the heterogeneity of the data at hand. Finding such class variables could result in a richer interpretation of the data. In the continuous data setting, a multi-partition model based clustering is proposed. It assumes the existence of several latent clustering variables, each one explaining the heterogeneity of the data with respect to some clustering subspace. It allows to simultaneously find the multi-partitions and the related subspaces. Parameters of the model are estimated through an EM algorithm relying on a probabilistic reinterpretation of the factorial discriminant analysis. A model choice strategy relying on the BIC criterion is proposed to select to number of subspaces and the number of clusters by subspace. The obtained results are thus several projections of the data, each one conveying its own clustering of the data. Model’s behavior is illustrated on simulated and real data. Full article

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

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13 pages, 427 KiB

Open AccessArticle

Theoretical Aspects on Measures of Directed Information with Simulations

by Thomas Gkelsinis and Alex Karagrigoriou

Mathematics 2020, 8(4), 587; https://doi.org/10.3390/math8040587 - 15 Apr 2020

Cited by 11 | Viewed by 3591

Abstract

Measures of directed information are obtained through classical measures of information by taking into account specific qualitative characteristics of each event. These measures are classified into two main categories, the entropic and the divergence measures. Many times in statistics we wish to emphasize [...] Read more.

Measures of directed information are obtained through classical measures of information by taking into account specific qualitative characteristics of each event. These measures are classified into two main categories, the entropic and the divergence measures. Many times in statistics we wish to emphasize not only on the quantitative characteristics but also on the qualitative ones. For example, in financial risk analysis it is common to take under consideration the existence of fat tails in the distribution of returns of an asset (especially the left tail) and in biostatistics to use robust statistical methods to trim extreme values. Motivated by these needs in this work we present, study and provide simulations for measures of directed information. These measures quantify the information with emphasis on specific parts (or events) of their probability distribution, without losing the whole information of the less significant parts and at the same time by concentrating on the information of the parts we care about the most. Full article

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

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11 pages, 302 KiB

Open AccessArticle

One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems

by Alexandru Amarioarei and Cristian Preda

Mathematics 2020, 8(4), 576; https://doi.org/10.3390/math8040576 - 13 Apr 2020

Cited by 1 | Viewed by 2619

Abstract

The one dimensional discrete scan statistic is considered over sequences of random variables generated by block factor dependence models. Viewed as a maximum of an 1-dependent stationary sequence, the scan statistics distribution is approximated with accuracy and sharp bounds are provided. The longest [...] Read more.

The one dimensional discrete scan statistic is considered over sequences of random variables generated by block factor dependence models. Viewed as a maximum of an 1-dependent stationary sequence, the scan statistics distribution is approximated with accuracy and sharp bounds are provided. The longest increasing run statistics is related to the scan statistics and its distribution is studied. The moving average process is a particular case of block factor and the distribution of the associated scan statistics is approximated. Numerical results are presented. Full article

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

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21 pages, 2189 KiB

Open AccessArticle

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

by Wenguang Yu, Peng Guo, Qi Wang, Guofeng Guan, Qing Yang, Yujuan Huang, Xinliang Yu, Boyi Jin and Chaoran Cui

Mathematics 2020, 8(4), 511; https://doi.org/10.3390/math8040511 - 2 Apr 2020

Cited by 28 | Viewed by 3771

Abstract

In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical [...] Read more.

In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model. Full article

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

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16 pages, 312 KiB

Open AccessArticle

Treating Nonresponse in Probability-Based Online Panels through Calibration: Empirical Evidence from a Survey of Political Decision-Making Procedures

by Antonio Arcos , Maria del Mar Rueda and Sara Pasadas-del-Amo

Mathematics 2020, 8(3), 423; https://doi.org/10.3390/math8030423 - 15 Mar 2020

Cited by 2 | Viewed by 4897

Abstract

The use of probability-based panels that collect data via online or mixed-mode surveys has increased in the last few years as an answer to the growing concern with the quality of the data obtained with traditional survey modes. However, in order to adequately [...] Read more.

The use of probability-based panels that collect data via online or mixed-mode surveys has increased in the last few years as an answer to the growing concern with the quality of the data obtained with traditional survey modes. However, in order to adequately represent the general population, these tools must address the same sources of bias that affect other survey-based designs: namely under coverage and non-response. In this work, we test several approaches to produce calibration estimators that are suitable for survey data affected by non response where auxiliary information exists at both the panel level and the population level. The first approach adjusts the results obtained in the cross-sectional survey to the population totals, while, in the second, the weights are the result of two-step process where different adjusts on the sample, panel, and population are done. A simulation on the properties of these estimators is performed. In light of theory and simulation results, we conclude that weighting by calibration is an effective technique for the treatment of non-response bias when the response mechanism is missing at random. These techniques have also been applied to real data from the survey Andalusian Citizen Preferences for Political Decision-Making Procedures. Full article

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

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18 pages, 863 KiB

Open AccessArticle

Asymptotic Approximations of Ratio Moments Based on Dependent Sequences

by Hongyan Fang, Saisai Ding, Xiaoqin Li and Wenzhi Yang

Mathematics 2020, 8(3), 361; https://doi.org/10.3390/math8030361 - 6 Mar 2020

Cited by 9 | Viewed by 2303

Abstract

The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative m-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain [...] Read more.

The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative m-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain asymptotic expressions for the moments of random ratios. It is pointed out that our random ratios can include some models such as change-point detection. Last, some simulations are illustrated to test our results. Full article

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

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12 pages, 295 KiB

Open AccessArticle

Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample

by Renyu Ye, Xinsheng Liu and Yuncai Yu

Mathematics 2020, 8(2), 176; https://doi.org/10.3390/math8020176 - 2 Feb 2020

Cited by 2 | Viewed by 2055

Abstract

This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density [...] Read more.

This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density estimation is shown as

L^{p}

(

1 \leq p < \infty

) risks over Besov space. To validate the effectiveness of the block thresholding wavelet method, we provide some examples and implement the numerical simulations. The results indicate that our block thresholding wavelet density estimator is superior in terms of the mean squared error (MSE) when comparing with the nonlinear wavelet density estimator. Full article

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

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10 pages, 221 KiB

Open AccessArticle

Optimal Designs for Carry Over Effects the Case of Two Treatment and Four Periods

by Miltiadis S. Chalikias

Mathematics 2019, 7(12), 1179; https://doi.org/10.3390/math7121179 - 3 Dec 2019

Cited by 5 | Viewed by 1818

Abstract

The optimal cross-over experimental designs are derived in experiments with two treatments, four periods, and an experimental unit. The results are given for the values n = 0mod4, 1mod4, 2mod4 and 3mod4. The criterion being the minimization of the variance of the estimated [...] Read more.

The optimal cross-over experimental designs are derived in experiments with two treatments, four periods, and an experimental unit. The results are given for the values n = 0mod4, 1mod4, 2mod4 and 3mod4. The criterion being the minimization of the variance of the estimated carry over effect. Full article

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

24 pages, 1947 KiB

Open AccessArticle

Statistical Properties and Different Methods of Estimation for Type I Half Logistic Inverted Kumaraswamy Distribution

by Ramadan A. ZeinEldin, Christophe Chesneau, Farrukh Jamal and Mohammed Elgarhy

Mathematics 2019, 7(10), 1002; https://doi.org/10.3390/math7101002 - 22 Oct 2019

Cited by 6 | Viewed by 3592

Abstract

In this paper, we introduce and study a new three-parameter lifetime distribution constructed from the so-called type I half-logistic-G family and the inverted Kumaraswamy distribution, naturally called the type I half-logistic inverted Kumaraswamy distribution. The main feature of this new distribution is to [...] Read more.

In this paper, we introduce and study a new three-parameter lifetime distribution constructed from the so-called type I half-logistic-G family and the inverted Kumaraswamy distribution, naturally called the type I half-logistic inverted Kumaraswamy distribution. The main feature of this new distribution is to add a new tuning parameter to the inverted Kumaraswamy (according to the type I half-logistic structure), with the aim to increase the flexibility of the related inverted Kumaraswamy model and thus offering more precise diagnostics in data analyses. The new distribution is discussed in detail, exhibiting various mathematical and statistical properties, with related graphics and numerical results. An exhaustive simulation was conducted to investigate the estimation of the model parameters via several well-established methods, including the method of maximum likelihood estimation, methods of least squares and weighted least squares estimation, and method of Cramer-von Mises minimum distance estimation, showing their numerical efficiency. Finally, by considering the method of maximum likelihood estimation, we apply the new model to fit two practical data sets. In this regards, it is proved to be better than recent models, also derived to the inverted Kumaraswamy distribution. Full article

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

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