Lomax Distribution Research Articles

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

A Novel Extended Power-Lomax Distribution for Modeling Real-Life Data: Properties and Inference

One of the important features of generalized distribution is its ability and flexibility to model real-life data in several applied fields such as medicine, engineering, and survival analysis, among others. In this paper, a flexible four-parameter Lomax extension called the alpha-power power-Lomax (APPLx) distribution is introduced. The APPLx distribution is analytically tractable, and it can be used quite effectively for real-life data analysis. Key mathematical properties of the APPLx distribution including mode, moments, stress-strength reliability, quantile and generating functions, and order statistics are presented. The APPEx parameters are estimated by using eight classical estimation methods. Extensive simulation studies are provided to explore the performance of the proposed estimation methods and to provide a guideline for practitioners and engineers to choose the best estimation method. Three real-life datasets from applied fields are fitted to assess empirically the flexibility of the APPLx distribution. The APPLx distribution shows greater flexibility as compared to the McDonal–Lomax, Fréchet Topp–Leone Lomax, transmuted Weibull–Lomax, Kumaraswamy–Lomax, beta exponentiated-Lomax, Weibull–Lomax, Burr-X Lomax, Lomax–Weibull, odd exponentiated half-logistic Lomax, and alpha-power Lomax distributions.

A new improved form of the Lomax model: Its bivariate extension and an application in the financial sector

The Lomax model, also known as Pareto Type-II, has broad feasibility, especially in financing. This work introduces a new generalization of the Lomax model called the arc-sine exponentiation Lomax, which helps consider economic phenomena. The arc-sine exponentiation Lomax distribution captures a variety of shapes of density and hazard functions. The estimators of the proposed model’s parameters are derived using the maximum likelihood method. In a simulation study, the accuracy and efficacy of estimators are evaluated by computing their mean square errors and biases.Furthermore, a bivariate extension of the arc-sine exponentiation Lomax model is also introduced. The bivariate extension is introduced using Farlie–Gumble–Morgenstern copula approach. The new bivariate model is called Farlie–Gumble–Morgenstern arc-sine exponentiation Lomax distribution. Finally, a data set of thirty-two observations representing the export of goods demonstrates the arc-sine exponentiation Lomax model. The best-fitting results of the arc-sine exponentiatial Lomax are compared with some prominent extensions of the Lomax distribution.

Properties and Application of the Suggested Exponentiated Lomax Distribution Family

The Exponentiated Lomax Distribution is considered one of the most commonly used continuous distribution which has a major role in analysing and modelling life time data. Therefore, A family was formed for the Exponential Lomax Distribution by introducing two new distributions as special case of the Exponentiated Lomax Distribution: (Modified Exponentiated Lomax Distribution (MELD) and Restricted Exponentiated Lomax Distribution (RELD. Furthermore, to assess the usefulness and flexibility, the two distributions were applied upon simulation study besides real application with real data set. The simulation results clearly shown the flexible performance of the maximum likelihood estimators for the parameter. Also, the real application results are clearly shown that the proposed distributions have outstanding performance than other considered distributions for all information criteria.

Before-after safety analysis of a shared space implementation

Shared spaces aim to reduce the dominance of motor vehicles by promoting pedestrian/cyclist activity and minimizing segregation between road users.Despite the intended scope to improve the safety of vulnerable road users, only few works in the literature focused on before-after safety evaluations, mainly analyzing changes in users’ trajectories and speeds, traffic volumes, and conflict counts, which, while useful, cannot univocally quantify road safety.Here, we propose a more advanced methodology, based on surrogate measures of safety and Extreme Value Theory, to assess road safety before and after the implementation of a shared space. The aim is to produce a crash risk estimation in different scenarios, obtaining a quantitative and comprehensive indicator, useful to practitioners for evaluating the safety of urban design solutions.A real-world case study illustrates the proposed procedure. Video data were collected on two separate days, before and after a shared space implementation, and were semi-automatically processed to extract road users’ trajectories. Analysis of traffic volumes, trajectories, speeds and yield ratios allowed to understand the spatial behavior of road users in the two scenarios. Traffic conflicts, identified with an innovative surrogate measure of safety called “time to avoided collision point” (TTAC), were then used to estimate a Lomax distribution, and therefore to model the probabilistic relationship between conflicts and crashes, eventually retrieving a crash risk estimate. Results show that the analyzed shared space was able to significantly reduce the risk of crashes, and these findings are consistent with the observed changes in users speed and spatial behavior. The analyzed case study and its limitations were useful in highlighting the methodology main features and suggesting practical prescriptions for practitioners.

Kumarswamy Truncated Lomax Distribution with Applications

This paper introduces a new flexible generalized family of distributions, named Kumarswamy Truncated Lomax Distribution. We study its statistical properties including quantile function, skewness, kurtosis, moments, generating functions, incomplete moments and order statistics. Maximum likelihood estimation of the model parameters is investigated. An application is carried out on real data set to illustrate the performance and flexibility of the proposed model.

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

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

Statistical inference for multi stress–strength reliability based on progressive first failure with lifetime inverse Lomax distribution and analysis of transformer insulation data

AbstractThis research aims to develop a multireliability inference method for stress–strength variables based on progressive first failure and an inverse Lomax distribution. This research examines the difficulties associated with estimating the stress–strength reliability function, R, when X, Y, and Z are drawn from three different Inverse Lomax distributions. On the basis of progressive first‐failure censored samples, reliability estimators for multi‐stress–strength Inverse Lomax distributions are estimated using the maximum likelihood, maximum product of spacing, and Bayesian estimation methods. The Bayes estimate of R is obtained using the MCMC method for a symmetric loss function. Monte Carlo simulations and real‐data applications are used to assess and compare the performance of the various suggested estimators.

A Compound Class of Unit Burr XII Model: Theory, Estimation, Fuzzy, and Application

The current research offers an enhanced three-parameter lifetime model that combines the unit Burr XII distribution with a power series distribution. The novel class of distribution is named the unit Burr XII power series (UBXIIPS). This compounding technique allows for the production of flexible distributions with strong physical meanings in domains such as biology and engineering. The UBXIIPS class includes the unit Burr XII Poisson (UBXIIP) distribution, the unit Burr XII binomial distribution, the unit Burr XII geometric distribution, and the unit Burr XII negative binomial distribution. The statistical properties of the class include formulas for the density and cumulative distribution functions, and limiting behaviour, moments and incomplete moments, entropy measures, and quantile function are provided. For estimating population parameters and fuzzy reliability for the UBXIIP model, maximum likelihood and Bayesian approaches are studied by the Metropolis–Hastings algorithm. For maximum likelihood estimators, the length of asymptotic confidence intervals is specified, whereas, for Bayesian estimators, the length of credible confidence intervals is assigned. A simulation investigation of the UBXIIP model was established to evaluate the performance of suggested estimates. In addition, the UBXIIP distribution is explored using real-world data. The UBXIIP distribution appears to offer some benefits in understanding lifetime data when compared to unit Weibull, beta, Kumaraswamy, Kumaraswamy Kumaraswamy, Marshall-Olkin Kumaraswamy, and Topp–Leone Weibull Lomax distributions.

Parameter estimation of the Lomax distribution using genetic algorithm based on the ranked set samples

ABSTRACT Due to the importance of business process automation in industry, hyperautomation and Robotic Process Automation (RPA) attract increasing attention in the scientific field as well. Hyperautomation allows automation to do important tasks performed by business people by merging Artificial Intelligence (AI) technologies with RPA. Lomax distribution has been widely used in engineering science, service times, use and supply times of products in enterprise information systems, income and wealth data, business failure data, medical and biological science, life test and other fields. In this paper, the maximum likelihood estimation of unknown parameters of the Lomax distribution is considered using simple random sample, ranked set sample, median ranked set sample and extreme ranked set sample. Model parameters are estimated by the maximum likelihood estimation method based on Genetic Algorithm. An extensive Monte Carlo simulation study, considering perfect and imperfect ranking, is carried out in order to compare the performance of the genetic algorithm and classical optimisation method from ranked set sample-based designs with corresponding simple random sample estimators. Real-life data has been used to illustrate the application of the results obtained.

A new heavy tailed distribution with actuarial measures

Actuaries are constantly on the lookout for heavy-tailed (HT) distributions in order to model data important to business and actuarial risk problems. In this article, we describe a novel type of heavy-tailed distribution that may be used to model data in the financial disciplines. Our proposed model, identified as a Kavya-Manoharan power Lomax (KMPLo) distribution, is thoroughly studied. Some fundamental characterizations, such as the quantile function and moments, have been developed. The maximum likelihood (ML) and Bayesian (B) analytical techniques are used to acquire estimates of the unknown parameters of the new model. A detailed simulation study is undertaken to evaluate the performance of the ML and B estimators. The KMPLo model's utility is demonstrated by an application to a HT insurance loss data set. The experimental results reveal that the suggested model is more adaptable and affordable than the other seven competing models including (i) the four-parameter model is the power Lomax (PLo) model; (ii) the four-parameter models odd log-logistic modified Weibull (OLLMW), Kumaraswamy Weibull (KW), generalized modified Weibull (GMW), extended odd Weibull Lomax (EOWL), Weibull-Lomax (WL), Marshall–Olkin power Lomax (MOPLo), Marshall–Olkin alpha power Lomax (MOAPLo) and exponentiated generalized alpha power exponential (EGAPEx) distributions (iii) the five-parameter distributions; , Kumaraswamy generalized power Lomax (KGPLo) and Marshall-Olkin alpha power exponentiated Weibull (MOAPEW) distribution. In addition, certain essential actuarial metrics, such like value at risk and conditional value at risk, are determined.

Bivariate power Lomax distribution with medical applications.

In this paper, a bivariate power Lomax distribution based on Farlie-Gumbel-Morgenstern (FGM) copulas and univariate power Lomax distribution is proposed, which is referred to as BFGMPLx. It is a significant lifetime distribution for modeling bivariate lifetime data. The statistical properties of the proposed distribution, such as conditional distributions, conditional expectations, marginal distributions, moment-generating functions, product moments, positive quadrant dependence property, and Pearson's correlation, have been studied. The reliability measures, such as the survival function, hazard rate function, mean residual life function, and vitality function, have also been discussed. The parameters of the model can be estimated through maximum likelihood and Bayesian estimation. Additionally, asymptotic confidence intervals and credible intervals of Bayesian's highest posterior density are computed for the parameter model. Monte Carlo simulation analysis is used to estimate both the maximum likelihood and Bayesian estimators.

On Fitting the Lomax Distribution: A Comparison between Minimum Distance Estimators and Other Estimation Techniques

In this paper, we investigate the performance of a variety of frequentist estimation techniques for the scale and shape parameters of the Lomax distribution. These methods include traditional methods such as the maximum likelihood estimator and the method of moments estimator. A version of the maximum likelihood estimator adjusted for bias is included as well. Furthermore, an alternative moment-based estimation technique, the L-moment estimator, is included, along with three different minimum distance estimators. The finite sample performances of each of these estimators are compared in an extensive Monte Carlo study. We find that no single estimator outperforms its competitors uniformly. We recommend one of the minimum distance estimators for use with smaller samples, while a bias-reduced version of maximum likelihood estimation is recommended for use with larger samples. In addition, the desirable asymptotic properties of traditional maximum likelihood estimators make them appealing for larger samples. We include a practical application demonstrating the use of the described techniques on observed data.

The properties of odd Lomax-Dagum distribution and its application

An Odd Lomax-Dagum distribution has been developed in this study. The distribution has been developed from Odd Lomax-G family of distributions. The structural characteristics of the distribution have been derived with relevant supporting plots of the probability distribution function (PDF), hazard function, survival function etc. The performance of Odd Lomax-Dagum distribution is examined by fitting it to two skewed datasets, along with Weilbull-Dagum distribution (WeiDaD), Dagum distribution (DaD) and Lomax distribution (LomD). The distribution demonstrated great flexibility in modelling the two datasets as compared to other distributions considered in the study.

Stress–Strength Modeling Using Median-Ranked Set Sampling: Estimation, Simulation, and Application

In this study, we look at how to estimate stress–strength reliability models, R1 = P (Y < X) and R2 = P (Y < X), where the strength X and stress Y have the same distribution in the first model, R1, and strength X and stress Z have different distributions in the second model, R2. Based on the first model, the stress Y and strength X are assumed to have the Lomax distributions, whereas, in the second model, X and Z are assumed to have both the Lomax and inverse Lomax distributions, respectively. With the assumption that the variables in both models are independent, the median-ranked set sampling (MRSS) strategy is used to look at different possibilities. Using the maximum likelihood technique and an MRSS design, we derive the reliability estimators for both models when the strength and stress variables have a similar or dissimilar set size. The simulation study is used to verify the accuracy of various estimates. In most cases, the simulation results show that the reliability estimates for the second model are more efficient than those for the first model in the case of dissimilar set sizes. However, with identical set sizes, the reliability estimates for the first model are more efficient than the equivalent estimates for the second model. Medical data are used for further illustration, allowing the theoretical conclusions to be verified.

Analysis of Household Income, Expenditure and Consumption Survey Research Data for North Sinai Governorate in Egypt Using Length Biased Truncated Lomax Distribution

The length biased truncated Lomax distribution is introduced in this study as a weighted form of the truncated Lomax distribution. The length biased truncated Lomax distribution’s essential distributional features are investigated. In the case of complete and type-II censored data, the maximum likelihood method is provided for estimating population parameter. The model parameter asymptotic confidence interval is calculated. To demonstrate the pattern of the estimate, a sample generation algorithm is supplied, as well as a Monte Carlo simulation analysis. We can see from the simulation research that as the censoring level is increased, the mean squared error of parameter estimates decrease’s for all given values. With increasing sample size, the mean squared error and average length of parameter estimates decrease. The estimates get increasingly accurate as the sample size grows higher, suggesting that its asymptotically unbiased. Furthermore, in all cases, the mean squared error diminishes as the sample size grows, indicating that the estimates of parameter are consistent. Modelling to medical data and the percentage of household spending on education out of total household expenditure from the household income, expenditure and consumption survey (HIECS) data are used to show the importance of the new model. The Kumaraswamy, beta, truncated power Lomax, truncated Weibull, and one parameter-beta distributions perform poorly in comparison to the suggested distribution.

On a Novel Extended Lomax Distribution with Asymmetric Properties and Its Statistical Applications

In this article, we highlight a new three-parameter heavy-tailed lifetime distribution that aims to extend the modeling possibilities of the Lomax distribution. It is called the extended Lomax distribution. The considered distribution naturally appears as the distribution of a transformation of a random variable following the logweighted power distribution recently introduced for percentage or proportion data analysis purposes. As a result, its cumulative distribution has the same functional basis as that of the Lomax distribution, but with a novel special logarithmic term depending on several parameters. The modulation of this logarithmic term reveals new types of asymetrical shapes, implying a modeling horizon beyond that of the Lomax distribution. In the first part, we examine several of its mathematical properties, such as the shapes of the related probability and hazard rate functions; stochastic comparisons; manageable expansions for various moments; and quantile properties. In particular, based on the quantile functions, various actuarial measures are discussed. In the second part, the distribution’s applicability is investigated with the use of the maximum likelihood estimation method. The behavior of the obtained parameter estimates is validated by a simulation work. Insurance claim data are analyzed. We show that the proposed distribution outperforms eight well-known distributions, including the Lomax distribution and several extended Lomax distributions. In addition, we demonstrate that it gives preferable inferences from these competitor distributions in terms of risk measures.

Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

<abstract><p>Examining life-testing experiments on a product or material usually requires a long time of monitoring. To reduce the testing period, units can be tested under more severe than normal conditions, which are called accelerated life tests (ALTs). The objective of this study is to investigate the problem of point and interval estimations of the Lomax distribution under constant stress partially ALTs based on progressive first failure type-II censored samples. The point estimates of unknown parameters and the acceleration factor are obtained by using maximum likelihood and Bayesian approaches. Since reliability data are censored, the maximum likelihood estimates (MLEs) are derived utilizing the general expectation-maximization (EM) algorithm. In the process of Bayesian inference, the Bayes point estimates as well as the highest posterior density credible intervals of the model parameters and acceleration factor, are reported. This is done by using the Markov Chain Monte Carlo (MCMC) technique concerning both symmetric (squared error) and asymmetric (linear-exponential and general entropy) loss functions. Monte Carlo simulation studies are performed under different sizes of samples for comparison purposes. Finally, the proposed methods are applied to oil breakdown times of insulating fluid under two high-test voltage stress level data.</p></abstract>

Parametric Modeling Approach to Covid-19 Pandemic Data

The problem of skewness is common among clinical trials and survival data, which has been the research focus derivation and proposition of different flexible distributions. Thus, a new distribution called Extended Rayleigh Lomax distribution is constructed from Rayleigh Lomax distribution to capture the excessiveness of some survival data. We derive the new distribution by using beta logit function proposed by Jones (2004). Some statistical properties of the distribution such as density, cumulative density, reliability rate, hazard rate, reverse hazard rate, moment generating and likelihood functions; skewness, kurtosis and coefficient of variation are obtained. We also performed the expected estimation of model parameters by maximum likelihood; goodness of fit and model selection criteria, including Anderson Darling, CramerVon Misses, Kolmogorov Smirnov (KS), Akaike Information, Bayesian Information, and Consistent Akaike Information Criterion is employed to select the better distribution from those models considered in the work. The results from the statistics criteria show that the intended distribution performs well and has a good representation of the States in Nigeria’s Covid-19 death cases data than other competing models.

Prior Preferences for the Inverse Power Lomax Distribution: Bayesian Method

Prior Preferences for the Inverse Power Lomax Distribution: Bayesian Method