A notable bounded probability distribution for environmental and lifetime data

ABSTRACTThe Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.

This study introduces the Lindley Stochastic Frontier Analysis—LSFA model, a novel approach that incorporates the Lindley distribution to enhance the flexibility and accuracy of efficiency estimation. The LSFA model is compared against traditional SFA models, including the half-normal, exponential, and gamma models, using advanced numerical methods such as the Gauss–Hermite Quadrature, Monte Carlo Integration, and Simulated Maximum Likelihood Estimation for parameter estimation. Simulation studies revealed that the LSFA model outperforms in scenarios involving small sample sizes and complex, skewed distributions, particularly those characterized by gamma distributions. In contrast, traditional models such as the half-normal model perform better in larger samples and simpler settings, while the gamma model is particularly effective under exponential inefficiency distributions. Among the numerical techniques, the Gauss–Hermite Quadrature demonstrates a strong performance for half-normal distributions, the Monte Carlo Integration offers consistent results across models, and the Simulated Maximum Likelihood Estimation shows robustness in handling gamma and Lindley distributions despite higher errors in simpler cases. The application to a banking dataset assessed the performance of 12 commercial banks pre-COVID-19 and during COVID-19, demonstrating LSFA’s superior ability to handle skewed and intricate data structures. LSFA achieved the best overall reliability in terms of the root mean square error and bias, while the gamma model emerged as the most accurate for minimizing absolute and percentage errors. These results highlight LSFA’s potential for evaluating efficiency during economic shocks, such as the COVID-19 pandemic, where data patterns may deviate from standard assumptions. This study highlights the advantages of the Lindley distribution in capturing non-standard inefficiency patterns, offering a valuable alternative to simpler distributions like the exponential and half-normal models. However, the LSFA model’s increased computational complexity highlights the need for advanced numerical techniques. Future research may explore the integration of generalized Lindley distributions to enhance model adaptability with enriched numerical optimization to establish its effectiveness across diverse datasets.

A two-parameter Quasi Lindley distribution (QLD), of which the Lindley distribution (LD) is a particular case, has been introduced. Its moments, failure rate function, mean residual life function and stochastic orderings have been discussed. It is found that the expressions for failure rate function, mean residual life function, and stochastic orderings of the QLD shows its flexibility over Lindley distribution and exponential distribution. Although, the QLD has two parameters, the expressions for coefficients of variation, skewness, and kurtosis depend upon only one parameter. The maximum likelihood method and the method of moments have been discussed for estimating its parameters. The distribution has been fitted to some data-sets to test its goodness of fit to which earlier the Lindley distribution has been fitted by others and it is found that to almost all these data-sets the QLD provides closer fits than those by the Lindley distribution. Key words: Lindley distribution, moments, failure rate function, mean residual life function, stochastic ordering, estimation of parameters, goodness of fit. .

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511-1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.

In this paper, we introduce a new distribution named as the Pseudo Lindley Distribution (PsLD) as a generalization of the Lindley distribution (LD). A full and detailed description are provided in terms of moments, cumulates, characteristic function, failure, rate function, stochastic ordering, distributions of sums, and parameters estimation. Simulations studies and data driven applications are also reported. Keywords: Lindley distribution; Exponential distribution; Gamma distribution; Stochastic ordering; Maximum-likelihood estimation.

There is an increasing interest in expanding the one‐parameter Lindley distribution to two‐parameter, three‐parameter, and five‐parameter. The univariate one‐parameter Lindley distribution is still one of the most applicable distributions in data analysis especially in lifetime data. Modeling dependent random quantities required bivariate parametric probability distributions. This study presents a new bivariate three‐parameter probability distribution called bivariate modified Lindley distribution. The one‐parameter modified Lindley distribution is used as a base line to construct the new model. Its statistical properties including cumulative function, density function, marginals, moments, conditional distributions, and copula are discussed. Simulation is constructed to declare theoretical properties, to show the flexibility of the new model and to investigate the goodness of fit. Two sets of real data, financial data and UEFA Champion’s League data, are used to show the applicability of the proposed model for different types of data.

In this paper, we introduced a new class of lifetime distribution and considered the mathematical properties of one of the sub models called a three parameter generalized Lindley distribution (TPGLD). The new class of distributions generalizes some of the Lindley family of distribution such as the power Lindley distribution, the Sushila distribution, the Lindley-Pareto distribution, the Lindley-half logistic distribution and the classical Lindley distribution. An application of the TPGLD to two real lifetime data sets reveals its superiority over the exponentiated power Lindley distribution, the exponentiated Lindley geometric distribution, the power Lindley distribution, the Lindley-exponential distribution and the classical one parameter Lindley distribution in modeling the lifetime data sets under study.

Recent studies have pointed out the potential of the odd Fréchet family (or class) of continuous distributions in fitting data of all kinds. In this article, we propose an extension of this family through the so-called "Topp-Leone strategy", aiming to improve its overall flexibility by adding a shape parameter. The main objective is to offer original distributions with modifiable properties, from which adaptive and pliant statistical models can be derived. For the new family, these aspects are illustrated by the means of comprehensive mathematical and numerical results. In particular, we emphasize a special distribution with three parameters based on the exponential distribution. The related model is shown to be skillful to the fitting of various lifetime data, more or less heterogeneous. Among all the possible applications, we consider two data sets of current interest, linked to the COVID-19 pandemic. They concern daily cases confirmed and recovered in Pakistan from March 24 to April 28, 2020. As a result of our analyzes, the proposed model has the best fitting results in comparison to serious challengers, including the former odd Fréchet model.

This paper presents a novel extension of the exponentiated inverse Rayleigh distribution called the half-logistic exponentiated inverse Rayleigh distribution. This extension improves the flexibility of the distribution for modeling lifetime data for both monotonic and non-monotonic hazard rates. The statistical properties of the half-logistic exponentiated inverse Rayleigh distribution, such as the quantiles, moments, reliability, and hazard function, are examined. In particular, we provide several techniques to estimate the half-logistic exponentiated inverse Rayleigh distribution parameters: weighted least squares, Cramér-Von Mises, maximum likelihood, maximum product spacings and ordinary least squares methods. Moreover, numerical simulations were performed to evaluate these estimation methods for both small and large samples through Monte Carlo simulations, and the finding reveals that the maximum likelihood estimation was the best among all estimation methods since it comprises small mean square error compared to other estimation methods. We employ real-world lifetime data to demonstrate the performance of the newly generated distribution compared to other distributions through practical application. The results show that the half-logistic exponentiated inverse Rayleigh distribution performs better than alternative versions of the Rayleigh distributions.

Local or district heating by natural gas: Which is better from energetic, environmental and economic point of views?

Term energy efficient building is well know from year 1991, when Austrian physicist Dr. Feist designed and built first passive house, using current physical and practical knowledge. In the next 25 years buildings using principles of energy efficient design have changed dramatically. In a good way. It is mandatory for Slovak Republic as a part of European Union to act according European parliament directives. One of directives concerns lowering total energy consumption and emissions in the building sector – Directive 2010/31/EU on Energy performance of buildings, also known as Directive “20-20-20”. According to this directive, Slovak Republic has agreed to lower total energy consumption in building industry by 20% until year 2020. Plan on lowering total energy consumption has affected creation of new – technical and energy efficient building materials with emphasis on environmental load. It this paper, ultra-low-energy family house located in Košice, Slovakia was assessed from environmental and energy point of view. With help on modern diagnostic methods and thermo- physical simulation software DesignBuilder, we will virtually evaluate energy need of house throughout the reference year, and indoor quality from the environmental point of view, such as CO2 levels and bounded energy using LCA method.

There is a problem arising when Lindley distribution is used to model right–skewed and unimodal data, which lies in its inability to model data with its peak farther from 0. A modification is required to increase the flexibility of this distribution, with “transformed–transformer” being one of the proposed method. This method is done by making a composition of two random variables through their respective distribution functions, with random variable T being the “transformed”, and random variable X being the “transformer”. In this paper, Lindley distribution was chosen to be the “transformed” and exponential distribution was chosen to be the “transformer”, constructing the Lindley–exponential distribution. However, there is a trouble using distribution if the data has a heavier tail. An alternative distribution is required while maintaining the properties of the Lindley-Exponential Distribution. Through transformation of variables method, a new distribution, Lindley-Exponential Slash Distribution is introduced, which is unimodal, right–skewed, heavier tailed. This paper covered some of its statistical characteristics, such as pdf, cdf, survival function, hazard rate, k th moment, mean, variance, skewness, and kurtosis. Parameter estimation was carried out with maximum likelihood estimation through numerical method. An application of distribution was illustrated on maximum annual precipitation data of Durham City.

ABSTRACTThis article addresses two methods of estimation of the probability density function (PDF) and cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE). Since the Lindley distribution is more flexible than the exponential distribution, the same estimators have been found out for the exponential distribution and compared. Monte Carlo simulations and a real data analysis are performed to compare the performances of the proposed methods of estimation.

Modeling and analyzing lifespan data is essential in many application areas, including medicine, engineering, and finance. These types of data have been modeled using various lifetime distributions. The assumed probability model(s) have a significant impact on the efficiency of the procedures used in statistical research..This present work add another novel distribution with two parameter called novel Logistic Exponential extension distribution (NLEED), including the Logistic Exponential distribution as special cases. We provide a complete mathematical treatment of this distribution. We derive the moment generating function and the r-th moment, thereby generalizing some results from the literature. Expressions for the density, moment generating function, entropy and the r-th moment of the order statistic are also obtained. Two real-world data sets are analyzed to explore the applicability, suitability and flexibility of the proposed distribution and we find that the proposed model fits much better than other well-known competing one and two parameters models such as the New Extension Exponential, the Gamma Lindley, the Zeghdoudi , the XLindley, the Xgamma and the Lindley distributions are considered. We hope that this proposed model can be an alternative in the fields of survival analysis, probability theory and applied statistics.

Additive manufacturing is an innovative way of producing complex parts. There is abundant knowledge about mechanical properties and production costs, with many studies in the literature reporting comparisons between different additive manufacturing technologies based on technical or economic criteria. However, there has been only limited environmental analysis of what happens during the manufacturing stage. Most of the studies concern the specific energy consumption and only rely on electrical consumption of the machine during the building process. In this paper, environmental impact is based on life cycle inventory data (energy, material, fluid). To ensure the continuing development of these processes, it is important to develop predictive models of the environmental impact, so that products can be evaluated not only from a technical and economic perspective, but also from an environmental point of view. This paper describes a new methodology for developing predictive models to jointly evaluate technical, economic, and environmental data. First, the authors propose a generic method to study the environmental aspects during the manufacturing stage. Then, these aspects are combined technical and cost information. This methodology can be applied to numerous additive manufacturing processes and will enable manufacturers to choose manufacturing technology or machine based on multiple criteria.