Search Results (61)

In this paper, a size-biased Lindley (SBL) first-order autoregressive (AR(1)) process is proposed, the so-called SBL-AR(1). Some probabilistic and statistical properties of the proposed process are determined, including the distribution of its innovation process, the Laplace transformation function, multi-step-ahead conditional measures, autocorrelation, and spectral density function. In addition, the unknown parameters of the model are estimated via the conditional least squares and Gaussian estimation methods. The performance and behavior of the estimators are checked through some numerical results by a Monte Carlo simulation study. Additionally, two real-world datasets are utilized to examine the model’s applicability, and goodness-of-fit statistics are used to compare it to several pertinent non-Gaussian AR(1) models. The findings reveal that the proposed SBL-AR(1) model exhibits key theoretical properties, including a closed-form innovation distribution, multi-step conditional measures, and an exponentially decaying autocorrelation structure. Parameter estimation via conditional least squares and Gaussian methods demonstrates consistency and efficiency in simulations. Real-world applications to inflation expectations and water quality data reveal a superior fit over competing non-Gaussian AR(1) models, evidenced by lower values of the AIC and BIC statistics. Forecasting comparisons show that the classical conditional expectation method achieves accuracy comparable to some modern machine learning techniques, underscoring its practical utility for skewed and fat-tailed time series. Full article

This paper presents a new bivariate mixture Lindley power function (BMLPF) distribution that employs a conditional approach with non-identical asymmetric distributions, distinguishing itself by the incorporation of a functional shape parameter. Various structural properties of bivariate distribution are presented, including explicit marginals, cumulative distribution function (CDF), product moments, correlation coefficients, conditional densities, moment generating functions, conditional mean, and variances. The parameters of the proposed distribution are evaluated using the maximum likelihood estimation method. To assess the effectiveness of this estimation approach, an extensive simulation study is carried out. The analysis quantifies these point estimators with their standard errors, RMSE, LCL, and UCL. This research significantly contributes to the development and application of bivariate distributions particularly in modeling and analyzing various types of data. Full article

In this manuscript, a new two-parameter stochastic distribution is proposed and obtained by a continuous half-logistic transformation of the quasi-Lindley (QL) distribution to the unit interval. The resulting distribution, named the quasi-Lindley half-logistic unit (QHU) distribution, is examined in terms of its key stochastic properties, such as asymmetry conditions, shape and modality, moments, etc. In addition, the stochastic dominance of the proposed distribution with respect to its parameters is considered, and it is shown that the QHU distribution, in contrast to the QL distribution that is always positively asymmetric, can have both asymmetric forms. The parameters of the QHU distribution are estimated by the maximum likelihood (ML) method, and the asymptotic properties of thusly obtained estimators are examined. Finally, an application of the proposed distribution in modeling some real-world phenomena is also presented. Full article

The Bivariate Odd Lindley Half-Logistic (BOLiHL) distribution with progressive Type-II censoring provides a powerful statistical tool for analyzing dependent data effectively. This approach benefits society by enhancing engineering systems, improving healthcare decisions, and supporting effective risk management, all while optimizing resources and minimizing experimental burdens. In this paper, the likelihood function derived under progressive Type-II censoring is generalized for the BOLiHL distribution. The well-known maximum likelihood estimation method and Bayesian estimation are applied to evaluate the parameters of the distribution. A study utilizing simulation techniques is performed to evaluate the performance of the estimators, using statistical analysis metrics for censored observations under a progressive Type-II censoring scheme with varying sample sizes, failure times, and censoring schemes. Additionally, a real dataset is studied to validate the proposed model, delivering impactful analyses for practical applications. Full article

In this work, we present a new distribution, which is a slash extension of the distribution of the sum of two independent Lindley random variables. This new distribution is developed using the slash methodology, resulting in a distribution with more flexible kurtosis, i.e., the ability to model atypical data. We study the density function of the new model and some of its properties, such as the cumulative distribution function, moments, and its asymmetry and kurtosis coefficients. The parameters are estimated by the maximum likelihood method with the EM algorithm. Finally, we apply the proposed model to two real datasets with high kurtosis, showing that it provides a better fit than two distributions known in the literature. Full article

This study explores accelerated life tests to examine the durability of highly reliable products. These tests involve applying higher stress levels, such as increased temperature, voltage, or pressure, that cause early failures. The power half-logistic (PHL) distribution is utilized due to its flexibility in modeling the probability density and hazard rate functions, effectively representing various data patterns commonly encountered in practical applications. The step stress partially accelerated life testing model is analyzed under an adaptive type II progressive censoring scheme, with samples drawn from the PHL distribution. The maximum likelihood method estimates model parameters and calculates asymptotic confidence intervals. Bayesian estimates are also obtained using Lindley’s approximation and the Markov Chain Monte Carlo (MCMC) method under different loss functions. Additionally, D- and A-optimality criteria are applied to determine the optimal stress-changing time. Simulation studies are conducted to evaluate the performance of the estimation methods and the optimality criteria. Finally, real-world datasets are analyzed to demonstrate the practical usefulness of the proposed model. Full article

In this paper, we propose a weighted Lindley (NWLi) model for the analysis of extreme historical insurance claims. It extends the classical Lindley distribution by incorporating a weight parameter, enabling more flexibility in modeling insurance claim severity. We provide a comprehensive theoretical overview of the new model and explore two practical applications. First, we investigate the mean-of-order P (MOOP_(P)) approach for quantifying the expected claim severity based on the NWLi model. Second, we implement a peaks over a random threshold (PORT) analysis using the value-at-risk metric to assess extreme claim occurrences under the new model. Further, we provide a simulation study to evaluate the accuracy of the estimators under various methods. The proposed model and its applications provide a versatile tool for actuaries and risk analysts to analyze and predict extreme insurance claim severity, offering insights into risk management and decision-making within the insurance industry. Full article

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. Full article

We introduce and investigate the properties of new families of univariate and bivariate distributions based on the survival function of the Lindley distribution. The univariate distribution, to reflect the nature of its construction, is called a power Lindley survival distribution. The basic distributional properties of this model are described. Maximum likelihood estimates of the parameters of the distribution are studied and the corresponding information matrix is identified. A bivariate power Lindley survival distribution is introduced using the technique of conditional specification. The corresponding joint density and marginal and conditional densities are derived. The product moments of the distribution are obtained, together with bounds on the range of correlations that can be exhibited by the model. Estimation of the parameters of the model is implemented by maximizing the corresponding pseudo-likelihood function. The asymptotic variance–covariance matrix of these estimates is investigated. A simulation study is performed to illustrate the performance of these parameter estimates. Finally some examples of model fitting using real-world data sets are described. Full article

In this article, we consider the statistical analysis of the parameter estimation of the Marshall–Olkin extended generalized extreme value under liner normalization distribution (MO-GEVL) within the context of progressively type-II censored data. The progressively type-II censored data are considered for three specific distribution patterns: fixed, discrete uniform, and binomial random removal. The challenge lies in the computation of maximum likelihood estimations (MLEs), as there is no straightforward analytical solution. The classical numerical methods are considered inadequate for solving the complex MLE equation system, leading to the necessity of employing artificial intelligence algorithms. This article utilizes the genetic algorithm (GA) to overcome this difficulty. This article considers parameter estimation through both maximum likelihood and Bayesian methods. For the MLE, the confidence intervals of the parameters are calculated using the Fisher information matrix. In the Bayesian estimation, the Lindley approximation is applied, considering LINEX loss functions and square error loss, suitable for both non-informative and informative contexts. The effectiveness and applicability of these proposed methods are demonstrated through numerical simulations and practical real-data examples. Full article

In this study, we introduce two novel discrete counterparts for the Rayleigh–Lindley mixture, constructed through the application of survival and hazard rate preservation techniques. These two-parameter discrete models demonstrate exceptional adaptability across various data types, including skewed, symmetric, and monotonic datasets. Statistical analyses were conducted using maximum likelihood estimation and Bayesian approaches to assess these models. The Bayesian analysis, in particular, was implemented with the squared error and LINEX loss functions, incorporating a modified Lwin Prior distribution for parameter estimation. Through simulation studies and numerical methods, we evaluated the estimators’ performance and compared the effectiveness of the two discrete adaptations of the Rayleigh–Lindley distribution. The simulations reveal that Bayesian methods are especially effective in this setting due to their flexibility and adaptability. They provide more precise and dependable estimates for the discrete Rayleigh–Lindley model, especially when using the hazard rate preservation method. This method is a compelling alternative to the traditional survival discretization approach, showcasing its significant potential in enhancing model accuracy and applicability. Furthermore, two real data sets are analyzed to assess the performance of each analog. Full article

Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the improved second-degree Lindley distribution. This distribution, called the Poisson-improved second-degree Lindley distribution, is capable of effectively modeling standard count data with overdispersion. However, if the frequency of the unobserved events is unknown, the proposed distribution cannot be directly used to describe the events. To address this limitation, we propose a modification by truncating the distribution to zero. This results in a tractable zero-truncated distribution that encompasses all types of dispersions. Due to the unknown frequency of unobserved events, the population size as a whole becomes unknown and requires estimation. To estimate the population size, we develop a Horvitz–Thompson-like estimator utilizing truncated distribution. Both the untruncated and truncated distributions exhibit desirable statistical properties. The estimators for both distributions, as well as the population size, are asymptotically unbiased and consistent. The current study demonstrates that both the truncated and untruncated distributions adequately explain the considered medical datasets, which are the number of dicentric chromosomes after being exposed to different doses of radiation and the number of positive Salmonella. Moreover, the proposed population size estimator yields reliable estimates. Full article

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resource allocation, evaluation, and performance measurement. This paper focuses on a one-parameter discrete distribution obtained by compounding the Poisson and new X-Lindley distributions. The probability-generating function, moments, skewness, kurtosis, and other properties are derived in the closed form. The maximum likelihood method, method of moments, least squares method, and weighted least squares method are used for parameter estimation. A simulation study is carried out. The proposed distribution is applied as the innovation in an INAR(1) process. The importance of the proposed model is confirmed through the analysis of two real datasets. Full article

This work aims to study the factors that explain the COVID-19 vaccination rate through a generalized odd log-logistic Lindley regression model with a shape systematic component. To accomplish this, a dataset of the vaccination rate of 254 counties in the state of Texas, US, was used, and simulations were performed to investigate the accuracy of the maximum likelihood estimators in the proposed regression model. The mathematical properties investigated provide important information about the characteristics of the distribution. Diagnostic analysis and deviance residuals are addressed to examine the fit of the model. The proposed model shows effectiveness in identifying the key variables of COVID-19 vaccination rates at the county level, which can contribute to improving vaccination campaigns. Moreover, the findings corroborate with prior studies, and the new distribution is a suitable alternative model for future works on different datasets. Full article

We present the truncated Lindley-G (TLG) model, a novel class of probability distributions with an additional shape parameter, by composing a unit distribution called the truncated Lindley distribution with a parent distribution function $G\left(x\right)$. The proposed model’s characteristics including critical points, moments, generating function, quantile function, mean deviations, and entropy are discussed. Also, we introduce a regression model based on the truncated Lindley–Weibull distribution considering two systematic components. The model parameters are estimated using the maximum likelihood method. In order to investigate the behavior of the estimators, some simulations are run for various parameter settings, censoring percentages, and sample sizes. Four real datasets are used to demonstrate the new model’s potential. Full article