Residual Life Function Research Articles

A New Flexible Extension of the XLindley Distribution with Properties and Application on Income Tax and Carbon Fibers Data

A new and enhanced extension of the XLindley distribution termed the sized-biased XLindley distribution (SBXLD), has been introduced. This new model explores two specific variants: the length-biased XLindley distribution and the area-biased XLindley distribution. Various crucial properties such as moments, moment generating function, quantile function, survival, and hazard functions, mean residual life function, and Rényi entropy have been derived and extensively investigated. For parameter estimation, five distinct methods have been employed to estimate the model parameters. Through a comprehensive simulation study, the most effective estimation method has been identified. The applicability and efficiency of the SBXLD model have been demonstrated using two datasets from different domains. It has been observed that the SBXLD model effectively analyzed these datasets and yielded superior results compared to other competitive distributions under consideration.

Physical aspects of quantile residual lifetime sequence

Abstract The modeling of count data is found in many fields, such as statistical physics, public health, medicine, epidemiology, applied science, sociology, and agriculture. In many physical situations, it has been observed that many times in the real world, the original variables may be continuous in nature, but discrete by observation. In this study, the α {\rm{\alpha }} -quantile residual life function for discrete lifetime models is defined and some attributes are investigated. The relation between this measure and the hazard rate function is studied. We discuss how this measure could be useful for finding the burn-in time of a lifetime dataset. Then, a new stochastic order based on the α {\rm{\alpha }} -quantile residual life is proposed and studied.

On the Topp-Leone Generalized Power Weibull Distribution: Properties, Applications and Regression

Abstract The Topp-Leone generalized power Weibull distribution, which is an extension of generalized power Weibull, is proposed and its properties explored. The failure rate of the proposed distribution exhibits increasing, reversed J, upside-down bathtub, and bathtub shapes. Some statistical properties are obtained: quantile function, moments, moment generating function, incomplete moment, mean and median deviations, mean residual life function, and Lorenz as well as Bonferroni curves. The maximum likelihood estimation approach is deployed to estimate the model parameters. Simulation studies are conducted to evaluate the performance and accuracy of the maximum likelihood estimates of the model parameters. Applications of the model to real datasets are presented. A location-scale regression model is also developed for the proposed model and its application has been demonstrated with a real dataset. Keywords: Bonferroni, Deviation, Mimicked, Simulation, and Upside-down Bathtub.

Constant Stress-Partially Accelerated Life Tests of Vtub-Shaped Lifetime Distribution under Progressive Type II Censoring

In lifetime tests, the waiting time for items to fail may be long under usual use conditions, particularly when the products have high reliability. To reduce the cost of testing without sacrificing the quality of the data obtained, the products are exposed to higher stress levels than normal, which quickly causes early failures. Therefore, accelerated life testing is essential since it saves costs and time. This paper considers constant stress-partially accelerated life tests under progressive Type II censored samples. This is realized under the claim that the lifetime of products under usual use conditions follows Vtub-shaped lifetime distribution, which is also known as log-log distribution. The log–log distribution is highly significant and has several real-world applications since it has distinct shapes of its probability density function and hazard rate function. A graphical description of the log–log distribution is exhibited, including plots of the probability density function and hazard rate. The log–log density has different shapes, such as decreasing, unimodal, and approximately symmetric. Several mathematical properties, such as quantiles, probability weighted moments, incomplete moments, moments of residual life, and reversed residual life functions, and entropy of the log–log distribution, are discussed. In addition, the maximum likelihood and maximum product spacing methods are used to obtain the interval and point estimators of the acceleration factor, as well as the model parameters. A simulation study is employed to assess the implementation of the estimation approaches under censoring schemes and different sample sizes. Finally, to demonstrate the viability of the various approaches, two real data sets are investigated.

A new extension of XLindley distribution with mathematical properties, estimation, and application on the rainfall data

Data analysis is indeed quite popular and crucial in various fields such as meteorology, hydrology, epidemiology, economics, and biology. So, the purpose of this study is to introduce a new probability distribution to analyze rainfall data. A new extension of XLindley distribution is introduced using the alpha power transformation technique. The new distribution is named “alpha power transformed XLindley – (APTXL) distribution”. Mathematical properties of APTXL distribution are derived such as ordinary moments, moment generating function, quantile function, mean residual life function, and order statistics. The model parameters are estimated using five different estimation methods such as maximum likelihood, Anderson Darling, Cramer von Misses, Ordinary least squares, and weighted least squares. A comprehensive simulation study is utilized to check the behavior and efficiency of the derived estimators. It is found that the weighted least square approach efficiently estimates the model parameter than the other methods. This derivation of a new model not only contributes to theoretical advances in statistical methodology but also provides practical methods for rainfall data modeling. The APTXL distribution is used to evaluate rainfall datasets from Saudi Arabia. It is identified that the APTXL distribution provides efficient results as compared to considered competitive distributions.

Binomial Poisson Ailamujia model with statistical properties and application

In this study, an improved extension of the Poisson-Ailamujia distribution is introduced. The new distribution was derived using a binomial mixing approach, and the new model is named the “Binomial Poisson-Ailamujia (Bin-PA)” distribution. Some important statistical properties are derived, including mode, quantile function, moments and their associated measures, actuarial (risk) measures, and reliability features such as survival, hazard (failure) rate, and mean residual life function. The parameters of the proposed distribution are estimated using the maximum likelihood estimation method. A comprehensive simulation study is also carried out to access the behavior-derived maximum likelihood estimators. Furthermore, a new count-regression model was also introduced. Two datasets are utilized to demonstrate the applicability and usefulness of the new model. It is concluded that the Binomial Poisson-Ailamujia distribution is more flexible and efficiently analyzed both datasets as compared to competitive discrete distributions.

On weighted Amarendra distribution with properties and applications

In this paper some statistical properties including nature of hazard function, mean residual life function, and moments based descriptive statistical constants including coefficient of variation, skweness, kurtosis and index of dispersion of the weighted Amarendra distribution has been discussed. Two methods of estimation, namely, maximum likelihood estimation and maximum product spacing estimation have been discussed. The simulation study has been presented to know the consistency of the estimator of parameters given by the two methods of estimation. Confidence intervals of the parameters are given. The goodness of fit of the distribution has been demonstrated with two real lifetime datasets and the goodness of fit shows that weighted Amarendra distribution provides better fit over weighted Pratibha distribution, weighted Komal distribution, weighted Lindley distribution, weighted Garima distribution, weighted Sujatha distribution, weighted Akash distribution and Gamma distribution.

Decreasing Trend Bumped Probability Distribution: its Properties, Simulations and Application

In this paper, a probability model is proposed which engaged the mathematical combination of Lindley distribution and a trigonometric component known as haversine function〖 δ〗_s. The one parameter model prototype sustains the capacity to forecast multimodal decreasing trend sinusoidal outcomes. By nomenclature, the proposed probability model is called Modified Lindley Trigonometric Distribution (MLTD). Some statistical properties studied include the hazard function, mean residual life function, moments, conditional moments and moment generating function, Bonferroni and Lorenz curve, entropy, asymptotic distribution, order statistics, and parameter estimation; where the hazard function specially features a cyclic or periodic bathtub and inverted bathtub shape in chain format. The numerical behavior of the estimates of the average bias and mean square error were examined under Monte Carlo simulation approach; and an applicative simulation is experimented to underscore the parametric behavior of MLTD in data modeling. A real life flood data is used to illustrate the essence of the development.

Equilibrium Reinsurance Strategy and Mean Residual Life Function

Equilibrium Reinsurance Strategy and Mean Residual Life Function

ON THE MOMENTS OF THE 3-PARAMETER GOMPERTZ DISTRIBUTION

Gompertz distribution is a classical probability distribution extensively used in actuarial science, reliability, and survival analysis. Gompertz distribution also plays a role in various fields, such as biology, economics, and marketing analysis. Some extensions of this distribution have been studied and applied to various problems. In this article, we are concerned with some statistical properties of a 3-parameter Gompertz distribution. This extension of the Gompertz distribution introduced has been used in studying competing risk survival analysis. Our main results are the derivation of moments of this distribution and other statistical properties related to moments, such as moment generating function, mean residual life function, mean inactivity time and Lorenz curve. These results will serve as a complement to the theoretical aspect of the analysis of the distribution.

A flexible generalized XLindley distribution with application to engineering

This paper presents a novel three-parameter distribution called the exponentiated generalized XLindley distribution. The distribution’s probability density function can be positively or negatively skewed, reverse-J or symmetric. Both an increasing and decreasing hazard rate function is conceivable. The hazard rate function, moments, residual and reverse residual life function, reliability function, and Rényi entropy are among the mathematical elements that can be ascertained from the new model. To estimate the parameters of the model, the maximum likelihood method is utilized. A simulation study was carried out to assess maximum likelihood estimators’ effectiveness. Two sets of data related to civil engineering were subjected to the new three-parameter model. The model is shown to suit these two symmetric datasets more closely than comparison models shown in this paper.

A study on utilization of two cold standby components to increase reliability of a coherent system

. This article focuses on a specific type of coherent system in which, when the coherent system fails, either no active component remains or the remaining active components are connected in parallel. Roy and Gupta (2023) also considered the same type of coherent systems. We observe the restart of such a coherent system using two cold standby components. In our study, standbys are not placed into action simultaneously. They are put into action one by one. We compute the reliability function of the considered coherent system, which is equipped with two cold standby components. In addition, we show that when we use a stronger standby component (in the sense of failure rate order) at first, we get a longer system lifetime. Also, we compute three different mean residual life functions of the system. We have presented a few numerical results to illustrate the importance of our analysis.

Secant Kumaraswamy Family of Distributions: Properties, Regression Model, and Applications

In this study, Secant Kumaraswamy family of distributions is proposed and studied. This is motivated by the fact that no one distribution can model all types of data from different fields. Therefore, there is the need to develop distributions with desirable properties and flexible enough for modelling data exhibiting different characteristics. Some properties of the new family of distributions, including the quantile function, moments, moment generating function, and mean residual life function, are derived. Five special cases of the family of distributions are presented, and their flexibility is shown by the varying degrees of skewness and kurtosis and nonmonotonic hazard rates. The maximum likelihood estimation method is used to obtain estimators of the family of distributions. Two location-scale regression models are developed for the Secant Kumaraswamy Weibull distribution, which is a special case of the family of distributions. Six different real datasets are used to demonstrate the usefulness of the family of distributions and the regression models. The results show that the family of distributions can be used to model real datasets.

Weighted extropy measures in general Morgenstern family under k-record values with application to medical data

In this paper, we study the marginal distribution of concomitants of k-record (KR) values from generalized Farlie–Gumbel–Morgenstern (GFGM) of bivariate distributions. In addition, the joint distribution of concomitants of KR for this family is obtained. Furthermore, some useful recurrence relations between moments of concomitants are derived. In addition, the hazard rate, the reversed hazard rate, and mean residual life functions of concomitants for this family are obtained. Some recent new measures of information, such as weighted extropy, weighted cumulative past extropy, and weighted cumulative residual extropy, are investigated for the concomitant of KR under the GFGM family. A non-parametric estimator of the proposed measure is provided by combining the empirical method with the concurrent use of KR in the GFGM family. Finally, we analyzed real-world data to examine our findings.

Bivariate Epanechnikov-exponential distribution: statistical properties, reliability measures, and applications to computer science data

<p>One important area of statistical theory and its applications to bivariate data modeling is the construction of families of bivariate distributions with specified marginals. This motivates the proposal of a bivariate distribution employing the Farlie-Gumbel-Morgenstern (FGM) copula and Epanechnikov exponential (EP-EX) marginal distribution, denoted by EP-EX-FGM. The EP-EX distribution is a complementing distribution, not a rival, to the exponential (EX) distribution. Its simple function shape and dependence on a single scale parameter make it an ideal choice for marginals in the suggested new bivariate distribution. The statistical properties of the EP-EX-FGM model are examined, including product moments, coefficient of correlation between the internal variables, moment generating function, conditional distribution, concomitants of order statistics (OSs), mean residual life function, and vitality function. In addition, we calculated reliability and information measures including the hazard function, reversed hazard function, positive quadrant dependence feature, bivariate extropy, bivariate weighted extropy, and bivariate cumulative residual extropy. Estimating model parameters is accomplished by utilizing maximum likelihood, asymptotic confidence intervals, and Bayesian approaches. Finally, the advantage of EP-EX-FGM over the bivariate Weibull FGM distribution, bivariate EX-FGM distribution, and bivariate generalized EX-FGM distribution is illustrated using actual data sets.</p>

Estimation of the Xrama Distribution Parameter Under Complete and Progressive Type-II Censored Schemes

In this paper, a two-component mixture of Exponential and Rama distribution named Xrama distribution is introduced and discussed. The distributional properties such as moment and the related measures including other characteristics namely quantile function, mean residual life function, Bonferroni curve, Lorenz curve, order statistics, stochastic ordering, and entropy were derived. Simulation study for both complete and progressive type-2 censored schemes is used to illustrate and analyze the behavior of the model. The proposed model’s performance is assessed through a comparison with various other lifetime distributions using negative log-likelihood, Akaike Information Criterion, Corrected Akaike Information Criterion, Bayesian Information Criterion, and Hannan–Quinn information criterion on an actual dataset. The results reveal that, when modeling real lifetime data, the suggested model exhibits greater flexibility than the competing models examined in this study.

Kernel estimators for mean residual lifetime in length-biased sampling

In this article, we propose three non parametric kernel estimators for the mean residual life function when the data are selected proportionally to their length. We evaluate the mean squared error of the three estimators and investigate the consistency for all three of them.

Statistical analysis of Saudi Arabia and UK Covid-19 data using a new generalized distribution

In this article, we extended the Topp–Leone distribution to create the Odd-Topp–Leone–Gompertz-G distribution, a new family of distributions. We were motivated by the limitations of classical models in modeling real life datasets hence, the creation of a new flexible distribution. Expansion of density, distribution of order statistics, Rényi entropy, moments, quantile, generating functions, residual life function and reverse residual life function, and reliability model are some of its structural aspects that are obtained. We looked at some special cases of the new distribution and concluded that our new model can handle both highly skewed data and non-monotonic hazard rate functions. The maximum likelihood estimation technique was used to estimate the model parameters. A simulation exercise was carried out to evaluate the performance of the proposed family of distribution. The article used the Saudi Arabia and United Kingdom dataset to demonstrate the utility of the new model and it was found that the new model performs better in modeling these datasets as compared to non-nested models presented in this article.

An Improved XShanker Distribution with Applications to Rainfall and Vinyl Chloride Data

This article is an improvement on the XShanker distribution having a single parameter which is the scale parameter and also in the class of Lindley distributions. It is named Double XShanker following the approach that generated it. The distributional properties which include the non-central moment with the associated statistics, the moment generating function, characteristic function, mean residual life function, stress-strength reliability function, Bonferroni and Lorenz curve functions, odd function, stochastic ordering, distribution of order statistics, and Reny entropy. The parameter was estimated using the method of maximum likelihood. Some visualizations based on theoretical values were presented. Some statistics were computed from theoretical values with convergence behavior recorded. A simulation study was conducted with 1000 samples of different sizes and the behavior observed is that as sample size increases the estimates decrease indicating precision. Data on rainfall and Vinyl chloride were used to validate the usefulness of the suggested distribution.

A quasi Rama distribution

A two-parameter quasi Rama distribution which contains the Rama distribution as particular case has been proposed. Its statistical properties based on moments have been discussed. The hazard rate function, mean residual life function, mean deviations and stochastic ordering of the distribution have been derived and studied. The estimation of parameters using method of moments and maximum likelihood methods has been discussed. A simulation study has been presented to know the performance of maximum likelihood estimates. The goodness of fit of the proposed distribution on two datasets relating to failure times has been presented.