Search Results (49)

In recent years, joint censoring schemes have gained significant attention in lifetime experiments and reliability analysis. A refined approach, known as the balanced joint progressive censoring scheme, has been introduced in statistical studies. This research focuses on statistical inference for two Lomax populations under this censoring framework. Maximum likelihood estimation is employed to derive parameter estimates, and asymptotic confidence intervals are constructed using the observed Fisher information matrix. From a Bayesian standpoint, posterior estimates of the unknown parameters are obtained under informative prior assumptions. To evaluate the effectiveness and precision of these estimators, a numerical study is conducted. Additionally, a real dataset is analyzed to demonstrate the practical application of these estimation methods. Full article

The Lomax distribution has important applications in survival analysis, reliability engineering, insurance, finance, and other fields. Middle-censoring is an important censoring scheme, and data with middle-censoring will produce censoring in random intervals. This paper studies the parameter estimation of the Lomax distribution based on middle-censored data. The expectation–maximization algorithm is employed to compute the maximum likelihood estimates of the two unknown parameters of the Lomax distribution. After processing the data using the midpoint approach estimation, the parameter estimates are obtained by two computational methods: the Newton–Raphson iteration method and the fixed-point method. Moreover, the calculation methods for the asymptotic confidence intervals of the two parameters are provided, with the confidence interval coverage rate serving as one of the criteria for evaluating the estimation performance. In the Bayesian estimation aspect, the shape parameter is estimated using a Gamma prior distribution, and the Gibbs sampling method is employed for the solution. Finally, both simulation data and real data are used to compare the accuracy of the various estimation methods. Full article

In this paper, the stress–strength reliability of complex systems with diverse component types is investigated based on the theoretical framework of survival signatures. Assuming that both the strength and stress of components of the same type follow the Lomax distribution, the maximum likelihood estimation (MLE), maximum spacing estimation (MSPE), and Bayesian estimation for the stress–strength model are derived under the condition that components of the same type have common scale parameters. Subsequently, the 95% Bootstrap-p and Highest Posterior Density confidence intervals for the stress–strength reliability were derived using Monte Carlo simulation. Additionally, since stress cycles are represented by a Poisson process, a dynamic stress–strength model for the system subjected to periodic stresses over the interval $(0,t]$ was developed, together with an approximate computational algorithm for this model. Finally, a simulation experiment was conducted using a system consisting of a total of nine components of three different types to analyze these estimation methods. The findings reveal that MLE exhibits the lowest estimation error, registering merely 0.001 in the case of small-sized samples. Compared with the previous two methods, Bayesian estimation has a relatively larger error. However, in the case of large samples, the error is 0.0112. In addition, the performance and accuracy of the dynamic model were verified through the proposed algorithm. The results indicate that compared with the static model at $t=0$, the error of the algorithm is 0.0464. Overall, the model evaluation results are satisfactory. Full article

Given the increasing number of phenomena that demand interpretation and investigation, developing new distributions and families of distributions has become increasingly essential. This article introduces a novel family of distributions based on the exponentiated reciprocal of the hazard rate function named the new Lomax-G family of distributions. We demonstrate the family’s flexibility to predict a wide range of lifetime events by deriving its cumulative and probability density functions. The new Lomax–Weibull distribution (NLW) is studied as a sub-model, with analytical and graphical evidence indicating its efficiency for reliability analysis and complex data modeling. The NLW density encompasses a variety of shapes, such as symmetrical, semi-symmetrical, right-skewed, left-skewed, and inverted J shapes. Furthermore, its hazard function exhibits a broad range of asymmetric forms. Five estimation techniques for determining the parameters of the proposed NLW distribution include the maximum likelihood, percentile, least squares, weighted least squares, and Cramér–von Mises methods. The performance of the estimators of the studied inferential methods is investigated through a comparative Monte Carlo simulation study and numerical demonstration. Additionally, the effectiveness of the NLW is validated by means of four real-world datasets. The results indicate that the NLW distribution provides a more accurate fit than several competing models. Full article

This paper focuses on applying the Marshall-Olkin approach to generate a new bivariate distribution. The distribution is called the bivariate exponentiated Lomax distribution, and its marginal distribution is the exponentiated Lomax distribution. Numerous attributes are examined, including the joint reliability and hazard functions, the bivariate probability density function, and its marginals. The joint probability density function and joint cumulative distribution function can be stated analytically. Different contour plots of the joint probability density function and joint reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are given. The unknown parameters and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are estimated using the maximum likelihood method. Also, the Bayesian technique is applied to derive the Bayes estimators and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution. In addition, maximum likelihood and Bayesian two-sample prediction are considered to predict a future observation from a future sample of the bivariate exponentiated Lomax distribution. A simulation study is presented to investigate the theoretical findings derived in this paper and to evaluate the performance of the maximum likelihood and Bayes estimates and predictors. Furthermore, the real data set used in this paper comprises the scoring times from 42 American Football League matches that took place over three consecutive independent weekends in 1986. The results of utilizing the real data set approve the practicality and flexibility of the bivariate exponentiated Lomax distribution in real-world situations, and the bivariate exponentiated Lomax distribution is suitable for modeling this bivariate data set. Full article

This research presents a closed-form mathematical framework for assessing the performance of a wireless communication system in the presence of multipath fading channels with an instantaneous signal-to-noise ratio (SNR) subjected to the inverse power Lomax (IPL) distribution. It is demonstrated that depending on the channel parameters, such a model can describe both severe and light fading covering most cases of the well-renowned simplified models (i.e., Rayleigh, Rice, Nakagami-m, Hoyt, $\alpha -\mu$, Lomax, etc.). This study provides the exact results for a basic statistical description of an IPL channel, including the PDF, CDF, MGF, and raw moments. The derived representation was further used to assess the performance of a communication link. For this purpose, the exact expression and their high signal-to-noise ratio (SNR) asymptotics were derived for the amount of fading (AoF), outage probability (OP), average bit error rate (ABER), and ergodic capacity (EC). The closed-form and numerical hyper-Rayleigh analysis of the IPL channel is performed, identifying the boundaries of weak, strong, and full hyper-Rayleigh regimes (HRRs). An in-depth analysis of the system performance was carried out for all possible fading channel parameters’ values. The practical applicability of the channel model was supported by comparing it with real-world experimental results. The derived expressions were tested against a numerical analysis and statistical simulation and demonstrated a high correspondence. Full article

This paper introduces the Lomax-exponentiated odds ratio–G (L-EOR–G) distribution, a novel framework designed to adeptly navigate the complexities of modern datasets. It blends theoretical rigor with practical application to surpass the limitations of traditional models in capturing complex data attributes such as heavy tails, shaped curves, and multimodality. Through a comprehensive examination of its theoretical foundations and empirical data analysis, this study lays down a systematic theoretical framework by detailing its statistical properties and validates the distribution’s efficacy and robustness in parameter estimation via Monte Carlo simulations. Empirical evidence from real-world datasets further demonstrates the distribution’s superior modeling capabilities, supported by compelling various goodness-of-fit tests. The convergence of theoretical precision and practical utility heralds the L-EOR–G distribution as a groundbreaking advancement in statistical modeling, significantly enhancing precision and adaptability. The new model not only addresses a critical need within statistical modeling but also opens avenues for future research, including the development of more sophisticated estimation methods and the adaptation of the model for various data types, thereby promising to refine statistical analysis and interpretation across a wide array of disciplines. Full article

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns. Full article

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

In this paper, we extend the Lomax–Rayleigh distribution to increase its kurtosis. The construction of this distribution is based on the idea of the Slash distribution, that is, its representation is based on the quotient of two independent random variables, one being a random variable with a Lomax–Rayleigh distribution and the other a beta$(q,1)$. Based on the representation of this family, we study its basic properties, such as moments, coefficients of skewness, and kurtosis. We perform statistical inference using the methods of moments and maximum likelihood. To illustrate this methodology, we apply it to two real data sets. Full article

To illustrate data uncertainty, intuitionistic fuzzy sets simply use membership and non-membership degrees. However, in some cases, a more complex strategy is required to deal with imprecise data. One of these techniques is generalized intuitionistic fuzzy sets (GIFSs), which provide a comprehensive framework by adding extra factors that provide a more realistic explanation for uncertainty. GIFSs contain generalized membership, non-membership, and hesitation degrees for establishing symmetry around a reference point. In this paper, we applied a generalized intuitionistic fuzzy set approach to investigate ambiguity in the parameter of the Lomax life distribution, seeking a more symmetric assessment of the reliability measurements. Several reliability measurements and associated cut sets for a novel L-R type fuzzy sets are derived after establishing the scale parameter as a generalized intuitionistic fuzzy number. Additionally, the study includes a range of reliability measurements, such as odds, hazards, reliability functions, etc., that are designed for the Lomax distribution within the framework of generalized intuitionistic fuzzy sets. These reliability measurements are an essential tool for evaluating the reliability characteristics of various types of complex systems. For the purpose of interpretation and application, the results are visually displayed and compared across different cut set values using a numerical example. Full article

In this paper, we present a new method to construct new classes of distortion functions. A distortion function maps the unit interval to the unit interval and has the characteristics of a cumulative distribution function. The method is based on the transformation of an existing non-negative random variable whose distribution function, named the generating distribution, may contain more than one parameter. The coherency of the resulting risk measures is ensured by restricting the parameter space on which the distortion function is concave. We studied cases when the generating distributions are exponentiated exponential and Gompertz distributions. Closed-form expressions for risk measures were derived for uniform, exponential, and Lomax losses. Numerical and graphical results are presented to examine the effects of the parameter values on the risk measures. We then propose a simple plug-in estimate of risk measures and conduct simulation studies to compare and demonstrate the performance of the proposed estimates. The plug-in estimates appear to perform slightly better than the well-known L-estimates, but also suffer from biases when applied to heavy-tailed losses. Full article

Insurance loss data are usually in the form of left-truncation and right-censoring due to deductibles and policy limits, respectively. This paper investigates the model uncertainty and selection procedure when various parametric models are constructed to accommodate such left-truncated and right-censored data. The joint asymptotic properties of the estimators have been established using the Delta method along with Maximum Likelihood Estimation when the model is specified. We conduct the simulation studies using Fisk, Lognormal, Lomax, Paralogistic, and Weibull distributions with various proportions of loss data below deductibles and above policy limits. A variety of graphic tools, hypothesis tests, and penalized likelihood criteria are employed to validate the models, and their performances on the model selection are evaluated through the probability of each parent distribution being correctly selected. The effectiveness of each tool on model selection is also illustrated using well-studied data that represent Wisconsin property losses in the United States from 2007 to 2010. Full article

A novel four-parameter lifetime Lomax model is presented and investigated within the scope of this paper. The failure rate of the innovative model can be “monotonically decreasing failure rate,” “monotonically increasing failure rate,” or “constant failure rate,” and the density of the model can be “asymmetric right skewed,” “symmetric,” “asymmetric left skewed,” or “uniform density”. The new density is expressed as a blend of the Lomax densities that have been multiplied by an exponent. New bivariate Lomax types were created for our research. The maximum likelihood technique was utilized. We performed simulated experiments for the purpose of evaluating the finite sample behavior of maximum likelihood estimators, using “biases” and “mean squared errors” as our primary metrics of analysis. The novel distribution was evaluated based on a number of pertinent Lomax models, including Lomax extensions that were generated on the basis of odd log-logistic, Kumaraswamy, beta, gamma, and Topp–Leone families, among others. The newly developed extension demonstrated its relevance by predicting the service and failure times of datasets pertaining to aircraft windshields. Full article

In this article, four new families named as Weibull extended exponentiated-X (WEE-X), Lomax extended exponentiated-X (LEE-X), Logistic extended exponentiated-X (LGCEE-X), and Burr III extended exponentiated-X (BIIIEE-X) with their quantile functions are proposed. The expressions for distribution function and density function of BIIIEE-X family are written in terms of linear combinations of the exponentiated densities based to parent model. New models, i.e., Weibul extended exponentiated Weibull (WEEW), Lomax extended exponentiated Weibull (LEEW), Logistic extended exponentiated Weibull (LGCEEW), and Burr III extended exponentiated-Weibull (BIIIEEW) distributions are derived, were plotted for functions of probability density and hazard rate at different levels of parameters. Some mathematical properties of the BIIIEEW model are disclosed. The maximum likelihood method for the BIIIEEW model are described. Numerical applications of the BIIIEEW model to disease of cancer datasets are provided. Full article