Lomax Distribution Research Articles

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

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(Z2<Z1) with engineering application.

Bayesian analysis for the Shannon entropy of the Lomax distribution using noninformative priors

The Lomax distribution is one of the well-known distributions that is used to fit heavy-tailed data. In this paper, we investigate the estimation of Shannon entropy of the Lomax distribution using noninformative priors. Some important priors including Jeffreys prior, reference priors and probability matching priors are presented. We demonstrate that the reference and matching priors for the Lomax entropy do not match the reference and probability matching priors of the Lomax parameters, regardless of which parameter of interest is considered. The propriety and the existence of the expectation of the posterior under each prior are validated, respectively. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian estimates in terms of the mean squared error and coverage probability. Finally, the approach is applied to three real data sets for illustrative purposes.

Statistical Fuzzy Reliability Analysis: An Explanation with Generalized Intuitionistic Fuzzy Lomax Distribution

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.

The Type II Topp-Leone Inverse Power Lomax distribution with Simulation and Applications

In this work, we present a four-parameter lifetime model that can be used to model reliability issues, fatigue life studies, and survival data called the Type II Topp-Leone Inverse Power Lomax distribution. It has the Type II inverse Lomax, Inverse Power Lomax, and Inverse Lomax distributions as sub-models. Some of its statistical properties, including complete and incomplete moments, generating functions, characteristics functions, mean residual life, mean inactivity time, Renyi entropy, Tsallis entropy, order statistics, stress-strength reliability, and weighted probability moment, have formal formulas that we have developed. The model's parameters are estimated using the maximum likelihood estimation technique. The effectiveness of maximum likelihood estimators is evaluated in terms of absolute bias and simulation study standard error. Two lifetime data sets are used to demonstrate how the new model can be applied. Using the same comparative criteria, the proposed distribution offers a better fit than a few well-known distributions.

Review on Various Forms of Lomax Distribution

The purpose of this brief survey is to discover the various forms of Lomax distribution developed from research articles released in recent years. This paper plainly and simply describes the most recent outcomes of different forms of lomax distribution applications. also illustrating critical aspects in the use of lomax distribution in various forms. This paper explores the significance of emphasising lomax distribution and their formulae for various forms.

A new univariate continuous distribution with applications in reliability

In this article, the odd Lomax Gompertz distribution has been introduced, which is derived by modifying the Gompertz distribution to serve as a baseline model in the odd generalized Lomax distribution. The newly proposed model offers enhanced flexibility and provides a promising alternative for modeling lifetime data. This study seeks to establish a solid theoretical foundation for its application through the exploration of several properties, such as non-central moments, stochastic orderings quantile function, and entropy measure, for the new model. Additionally, by conducting simulation analysis, the performance of the various estimation methods is being assessed, which enables the identification of the most reliable approach for estimating the unknown parameters of the newly developed model. The simulation analysis of the two-risk metrics, namely, value at risk and expected shortfall, revealed the ability of the distribution to capture diverse failure rate patterns, which makes it particularly relevant for assessing financial risks. Finally, the suggested model is practiced to two real-life datasets to provide the compelling evidence of superior flexibility and practical versatility compared to existing models in the literature.

Model Uncertainty and Selection of Risk Models for Left-Truncated and Right-Censored Loss Data

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.

Moment Properties of Record Values from Rayleigh Lomax Distribution and Characterization

In this paper we have obtained explicit expressions for single and product moments of upper record values from Rayleigh Lomax distribution. Also, recursive relationships between moments of upper records are developed. The characterization theorems based on recurrence relations and conditional expectation are also presented.

Statistical inference of the inverted exponentiated Lomax distribution using generalized order statistics with application to COVID-19

In this study, the parameters of the inverted exponentiated Lomax distribution via generalized order statistics are assessed using Bayesian and maximum likelihood approaches. The maximum likelihood estimators along with approximate confidence intervals are calculated. Under the squared error loss function, the Bayesian estimator, percentile bootstrap, and bootstrap-t credible periods are produced. Furthermore, the proposed estimators are dedicated to schemes such as type-II censored ordinary order statistics joint density function. A numerical simulation is used to assess the behavior and sensitivity of the estimates for various sample sizes. From the posterior distributions, the Metropolis–Hastings technique is used to generate Markov chain Monte Carlo samples. We utilize this technique to examine a current dataset of interest: daily cases of COVID-19 instances detected in Saudi Arabia from May 31 to October 28, 2020 (inclusive). In the future, the proposed methodology could be useful for analyzing data on COVID-19 instances in other countries for comparative studies.

Reliability analysis of the triple modular redundancy system under step-partially accelerated life tests using Lomax distribution

Triple modular redundancy (TMR) is a robust technique utilized in safety-critical applications to enhance fault-tolerance and reliability. This article focuses on estimating the distribution parameters of a TMR system under step-stress partially accelerated life tests, where each component included in the system follows a Lomax distribution. The study aims to analyze the system’s reliability and mean residual lifetime based on the estimated parameters. Various estimation techniques, including maximum likelihood, percentile, least squares, and maximum product of spacings, are explored. Additionally, the optimal stress change time is determined using two criteria. An illustrative example supported by two actual data sets is presented to showcase the methodology’s application. By conducting Monte Carlo simulations, the assessment of the estimation methods’ effectiveness reveals that the maximum likelihood method outperforms the other three methods in terms of both accuracy and performance, as indicated by the numerical outcomes. This research contributes to the understanding and practical implementation of TMR systems in safety-critical industries, potentially saving lives and preventing catastrophic events.

The Gull Alpha Power Lomax distributions: Properties, simulation, and applications to modeling COVID-19 mortality rates.

The Gull Alpha Power Lomax distribution is a new extension of the Lomax distribution that we developed in this paper (GAPL). The proposed distribution's appropriateness stems from its usefulness to model both monotonic and non-monotonic hazard rate functions, which are widely used in reliability engineering and survival analysis. In addition to their special cases, many statistical features were determined. The maximum likelihood method is used to estimate the model's unknown parameters. Furthermore, the proposed distribution's usefulness is demonstrated using two medical data sets dealing with COVID-19 patients' mortality rates, as well as extensive simulated data applied to assess the performance of the estimators of the proposed distribution.

Bayesian inference using MCMC algorithm of sine truncated Lomax distribution with application

This study makes a significant contribution to the creation of a versatile trigonometric extension of the well-known truncated Lomax distribution. Specifically, we construct a novel one-parameter distribution known as the sine truncated Lomax (STLo) distribution using characteristics from the sine generalized family of distributions. Quantiles, moments, stress–strength reliability, some information measures, residual moments, and reversed residual moments are a few of the crucial elements and characteristics we explored in our research. The flexibility of the STLo distribution in terms of the forms of the hazard rate and probability density functions illustrates how effectively it is able to match many types of data. Maximum likelihood and Bayesian estimation techniques are used to estimate the model parameter. The squared error loss function is employed in the Bayesian approach. To evaluate how various estimates behave, a Monte Carlo simulation study is carried out with the aid of a useful algorithm. Additionally, the STLo distribution has a good fit, making it a viable option when compared to certain other competing models using specific criteria to describe the given dataset.

Classical and Bayesian estimation for the extended odd Weibull power Lomax model with applications

A new continuous distribution called the extended odd Weibull power Lomax (ExOW-POLO) distribution is introduced and studied. Numerous reliability and statistical features are derived. Additionally studied are point estimates using maximum likelihood, maximum product space, least square, weighted least square, and Bayesian estimation techniques. The mean square error and bias of the maximum likelihood and Bayesian parameter estimators are computed using simulation approaches, such as Markov chain Monte Carlo. Two intraocular pressure (IOP) real datasets were conducted between January 2015 and February 2018 on 49 patients (84 eyes) under the age of two who presented with primary congenital glaucoma to the Paediatric Ophthalmology Unit of the Mansoura Ophthalmic Center of Mansoura University in Egypt have been fitted the ExOW-POLO distribution. Comparing the properties of the proposed distribution’s fitting of the data to recognized extensions of the Lomax distribution. The analysis revealed that the most well-known extensions of Lomax distribution were made by the ExOW-POLO distribution outfit. In addition, the correlation measures and independent sample test for the two IOP real datasets are introduced with (A) Levene’s test for equality of variances for the two cases and (B) the t-test for equality of means. For Levene’s test for equality of variances: the null hypothesis is that equal variances are assumed and the alternative hypothesis is that equal variances are not assumed.

Infinite divisibility of the Whittaker distribution

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below.

SINE-LOMAX DISTRIBUTION: PROPERTIES AND APPLICATIONS TO REAL DATA SETS

In this study, a novel distribution called the two-parameter Sine Lomax distribution was introduced. The distribution was developed by combining the Sine generalized family of distributions with the Lomax distribution. Various statistical properties of this new distribution were investigated, including the survival function, hazard function, quantile function, rth moment, entropy, moment generating function, and order statistics. The probability density function (PDF) plot indicated that the distribution is skewed to the right. Additionally, the hazard plot of the Sine Lomax distribution showed both monotonic increase and monotonic decrease. To estimate the parameters of the newly proposed distribution, the maximum likelihood approach was employed. A simulation study was conducted to evaluate the consistency of the estimators. The simulation results indicated that the estimators are consistent, as the bias and mean square error decrease with increasing sample sizes. The performance of the Sine Lomax distribution was compared to other extensions of Lomax distributions and the baseline distribution which is the Lomax distribution using various evaluation criteria, including the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC). The proposed distribution demonstrated the lowest scores among the competing models, indicating its potential for accurately modeling real-world data sets. Based on the results, the proposed Sine Lomax distribution is recommended as a superior alternative to the competing models for modeling certain real-world data sets.

Objective Bayesian inference for the reliability in a bivariate Lomax distribution

Objective Bayesian inference for the reliability in a bivariate Lomax distribution

Classical and Bayesian Inference for the Length Biased Weighted Lomax Distribution under Progressive Censoring Scheme

In this study, the distribution’s reliability and hazard functions, as well as the population parameters, are estimated for the length biased weighted Lomax (LBWLo) based on progressively Type II censored samples. The maximum likelihood and Bayesian methods are implanted to get the proposed estimators. Gamma and Jeffery's priors serve as informative and non-informative priors, respectively, from which the posterior distribution of the LBWLo distribution is constructed. To obtain the Bayesian estimates, the Metropolis-Hasting (MH) algorithm is also used. We obtain asymptotic confidence intervals based on the Fisher information matrix. Using the sample produced by the MH method, we construct the intervals with the highest posterior densities. A numerical simulation research is done to evaluate the effectiveness of the approaches. Through Monte Carlo simulation, we compare the proposed estimators in terms of mean squared error. It is possible to get coverage probability and average interval lengths of 95% .The study's findings supported the idea that, in the majority of cases, Bayes estimates with an informative prior are more appropriate than other estimates.

Flexible Extension of the Lomax Distribution for Asymmetric Data under Different Failure Rate Profiles: Characteristics with Applications for Failure Modeling and Service Times for Aircraft Windshields

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.

Inference and quantile regression for the unit-exponentiated Lomax distribution.

In probability theory and statistics, it is customary to employ unit distributions to explain practical variables having values between zero and one. This study suggests a brand-new distribution for modelling data on the unit interval called the unit-exponentiated Lomax (UEL) distribution. The statistical aspects of the UEL distribution are shown. The parameters corresponding to the proposed distribution are estimated using widely recognized estimation techniques, such as Bayesian, maximum product of spacing, and maximum likelihood. The effectiveness of the various estimators is assessed through a simulated scenario. Using mock jurors and food spending data sets, the UEL regression model is demonstrated as an alternative to unit-Weibull regression, beta regression, and the original linear regression models. Using Covid-19 data, the novel model outperforms certain other unit distributions according to different comparison criteria.

The Modified-Lomax Distribution: Properties, Estimation Methods, and Application

This paper introduces a flexible three-parameter extension of the Lomax model called the odd Lomax–Lomax (OLxLx) distribution. The OLxLx distribution can provide left-skewed, symmetrical, right-skewed, and reversed-J shaped densities and increasing, constant, unimodal, and decreasing hazard rate shapes. Some mathematical properties of the introduced model are derived. The OLxLx density can be expressed as mixture of Lomax densities. The OLxLx parameters are estimated by using eight estimation methods and their performance is explored by using detailed simulation studies. The partial and overall ranks of the mean relative errors, absolute biases, and mean square errors of different estimators are presented to choose the best estimation method. The flexibility and applicability of the OLxLx distribution is shown using real-life medicine data, illustrating the superior fit of the OLxLx distribution over other competing Lomax distributions. The OLxLX distribution outperforms some rival Lomax distributions including the Kumaraswamy–Lomax, McDonald–Lomax, Weibull–Lomax, transmuted Weibull–Lomax, exponentiated-Lomax, Lomax–Weibull, modified Kies–Lomax, Burr X Lomax, beta exponentiated-Lomax, odd exponentiated half-logistic Lomax, and transmuted-Lomax distributions, among others.