Lindley Model Research Articles

Estimation of Lindley constant-stress model via product of spacing with Type-II censored accelerated life data

Accelerated Life Test has been introduced as a tool to aid in obtaining sufficient failure time data of test units quickly and extrapolating lifetime information under use conditions. When the lifetime of units is distributed by the Lindley model, the estimations for the unknown parameters and the reliability function are established based on two frequentist methods and Bayesian method of estimation using Type-II censored data under constant-stress accelerated life test. In the frequentist methods, besides, the conventional likelihood-based estimation, another competitive method, known as the maximum product of spacing method is proposed for estimating the parameters and the reliability function under normal conditions as an alternative approach to the common likelihood method. In Bayesian estimation, both maximum likelihood and maximum product of spacing-based Bayesian estimates are discussed for unknown parameters as well as the reliability function. Moreover, the approximate confidence intervals and highest posterior density credible intervals of the parameters and reliability function are also obtained. Extensive Monte-Carlo simulation studies are conducted to evaluate the performance of the proposed estimates. Finally, to demonstrate the proposed methodology, two real-life accelerated life test data are considered to show the applicabilities of the proposed methods.

A new flexible extension of the Lindley distribution with applications

A new general family of distributions based on the Lindley model was introduced. Some properties of the proposed model, including moments, quantiles, and order statistics are presented. In addition, some reliability measures of this model were investigated. The parameters were estimated using the moments and the maximum likelihood methods for complete and right-censored data. Then, the behavior of the maximum likelihood estimator was investigated in a simulation study. Finally, the model was applied to the analysis of two data sets to demonstrate its usefulness.

Testing the Validity of Lindley Model Based on Informational Energy with Application to Real Medical Data

In this article, a test statistic for testing the validity of the Lindley model based on the informational energy is proposed. Consistency of our test is shown. Through a simulation study, we obtain the critical values of the test statistic and then the power of the test is computed by Monte Carlo method against various alternatives. The performance of the proposed test with some competing tests is compared. Our results show that our test is superior to the classical nonparametric tests and can apply to a testing problem in practice. A real medical data set is presented and analyzed.

Estimating Population Size of Criminals: A New Horvitz–Thompson Estimator under One-Inflated Positive Poisson–Lindley Model

Many crime datasets often display an excess of “1” counts, arises when arrested criminals have the desire and ability to avoid subsequent arrests. In this study, a new Horvitz–Thompson (HT) estimator under one-inflated positive Poisson–Lindley (OIPPL) distribution which allow for one-inflation and the existence of heterogeneity in the data is developed to estimate the hidden population size of criminals. From the simulation study and applications to real crime datasets, the OIPPL is capable to provide an adequate fit to the datasets considered and the proposed HT estimator is found to produce a more precise estimate of the population size with a narrower 95% confidence interval as compared to several other contending estimators considered in this study.

Investigation of the Factors Influencing the Crash Frequency in Expressway Tunnels: Considering Excess Zero Observations and Unobserved Heterogeneity

The existing crash modeling techniques for expressway tunnels must overcome the following difficulties: 1) The collected risk factors contributing to the tunnel crashes include narrow ranges, especially the pavement conditions and weather conditions of the tunnels are rarely taken into account. 2) Most researchers ignored the estimation deviation caused by the excess zero observations of tunnel crash datasets. 3) No existing tunnel crash model can combine the random-parameters approach and spatial-temporal approach to solve the estimation deviation caused by the inter-samples and spatial-temporal heterogeneity. To address these problems, this study presents an investigation of the safety effects of risk factors of tunnel design features, traffic conditions, pavement conditions and weather conditions utilizing a 12-quarter period (3 years) of data as well as five crash frequency models: 1) a fixed parameters negative binomial model (FPNB), 2) a random parameters negative binomial model (RPNB), 3) a random parameters negative binomial Lindley model (RPNBL), 4) a spatial and random parameters negative binomial Lindley model (SP-RPNBL), and 5) a spatial-temporal and random parameters negative binomial Lindley model (ST-RPNBL). The results showed that the ST-RPNBL model solves the deviation that arises from excess zero observations by introducing the Lindley distribution and considers the unobserved heterogeneity by introducing both the random parameters and spatial-temporal parameters that provided better goodness of fit and offered more insights into the factors that contribute to tunnel safety. Furthermore, the ST-RPNBL model detected 16 variables that were significantly correlated with tunnel crash frequency, of which 12 variables were associated with a higher crash frequency and four variables were associated with a lower crash frequency. The random variables of the curvature, the steep downgrade indicator, the proportion of class 5 vehicle and the skidding resistance index (SRI) were identified, and the influence of each significant variable on the crash frequency was analyzed.

A New Family of Extended Lindley Models: Properties, Estimation and Applications

There are many proposed life models in the literature, based on Lindley distribution. In this paper, a unified approach is used to derive a general form for these life models. The present generalization greatly simplifies the derivation of new life distributions and significantly increases the number of lifetime models available for testing and fitting life data sets for biological, engineering, and other fields of life. Several distributions based on the disparity of the underlying weights of Lindley are shown to be special cases of these forms. Some basic statistical properties and reliability functions are derived for the general forms. In addition, comparisons among various forms are investigated. Moreover, the power distribution of this generalization has also been considered. Maximum likelihood estimator for complete and right-censored data has been discussed and in simulation studies, the efficiency and behavior of it have been investigated. Finally, the proposed models have been fit to some data sets.

Exploration of some distributional properties, parameter estimates and applications of the wrapped quasi Lindley distribution

A circular distribution called Wrapped Quasi Lindley distribution with two parameters has been recently proposed, but apart from the expressions for pdf, cdf, their circular representations, characteristic function and the maximum likelihood equations for the proposed distribution, no other properties of the distribution as well as the characteristics of the parameter estimates were explored by the authors. A slight error has also been observed in the expression for pdf of the distribution. Also, the application of the distribution in modeling real life data was not exhibited. Further, the form of the characteristic function in the paper is not compact and there is no closed form of the expression of the trigonometric moments. This paper thus aims to rectify the expression for pdf and explore a few descriptive measures and distributional properties of the Wrapped Quasi Lindley distribution and derive closed form expressions for the characteristic function and hence the trigonometric moments using an identity. it is found that the operations of wrapping and convoluting linear distributions around unit circle are commutative. The maximum likelihood estimates of the parameters of the distribution are shown to be consistent through a simulation study. The utility of the Wrapped Quasi Lindley model to a real-life data set on orientations is shown and the goodness-of-fit of the distribution is assessed and compared to that of the Wrapped Exponential and Wrapped Lindley distribution with the help of the log-likelihood, AIC and BIC measures. Further, the probabilities of the orientations to lie in a certain interval are estimated on the basis of the fitted Wrapped Quasi Lindley distribution. The distribution is found to be more appropriate in modeling the situations where the directions having lower magnitude have higher likelihood of occurrence.

Applying a Correlated Random Parameters Negative Binomial Lindley Model to Examine Crash Frequency Along Highway Tunnels in China

Highway tunnels have a higher risk of crashing than open roads, which require a systematic approach to tunnel safety. However, previous research had the following problems: 1) Studies have largely focused on open roads, with very little research on tunnels. 2) The collected crash contributing factors involve narrow ranges, with very little tunnel crash data including both tunnel design features, traffic conditions and pavement conditions. 3) None of the studies considered both excess zero observations and unobserved heterogeneity with its interactions. To address these issues, this paper first established an appropriate tunnel dataset containing 3 to 5 years of crash data from several highways in China and the influence factors of tunnel design features, traffic conditions and pavement conditions. A correlated random parameters negative binomial Lindley (CRPNB-L) model that considers both excess zero observations and unobserved heterogeneity with its interaction effects was then proposed. Compared to the uncorrelated random parameters negative binomial Lindley (URPNB-L) model, fixed parameters negative binomial Lindley (FPNB-L) model and fixed parameters negative binomial (FPNB) model, the CRPNB-L model solves the deviation that arises from excess zero observations by introducing the Lindley distribution and considers the unobserved heterogeneity with its interactions by introducing correlated random parameters. In the comparisons, the CRPNB-L model achieves the best effects in the goodness-of-fit. Furthermore, the estimated results of the CRPNB-L model showed that segment length, traffic volume, proportion of class 5 vehicle (heavy trucks and trailers), tunnel entrance and exit segments, and steep uphill and downhill segments were associated with higher crash frequency, while curvature, tunnel length, pavement damage condition index (PCI) and skid resistance index (SRI) were associated with lower crash frequency. In addition, the random variables of the curvature, the steep downgrade indicator, the proportion of class 5 vehicle and SRI were identified and their intercorrelations were analyzed.

Odd Hyperbolic Cosine-FG Family of Lifetime Distributions

In the present paper, a new family of lifetime distributions is introduced via odd ratio function, the well-known concept in survival analysis and reliability engineering. Some important properties of the proposed model including survival function, quantile function, hazard function, order statistic are obtained in a general setting. A special case of this new family is taken up by considering exponential and Lindley models as the parent distributions. In addition, estimation of the unknown parameters of the special model will be examined from the perspective of the classic statistics. A simulation study is presented to investigate the bias and mean square error of the maximum likelihood estimators. Moreover, an example of real data set is studied; point and interval estimations of all parameters are obtained by maximum likelihood and bootstrap (parametric and non-parametric) procedures. Finally, the superiority of the proposed model in terms of the parent exponential distribution over other fundamental statistical distributions is shown via the example of real observations.

Optimal lot disposition from Poisson–Lindley count data

Optimal sampling plans based on overdispersed defect counts for screening lots of outgoing and incoming goods are derived by minimizing the required sample size. Best inspection schemes provide appropriate protections to customers and manufacturers. The stochastic distribution of the number of defects per sampled unit is described by Poisson–Lindley models. Optimal frequentist and Bayesian decision rules for lot disposition are found by solving mixed integer nonlinear programming problems through simulation. The suggested criteria are based on likelihood and posterior odds ratios. The asymptotic normality of the quality score statistic is used to deduce explicit and reasonably accurate approximations of the optimal acceptance sampling plans. The Bayesian approach allows the practitioners to reduce the needed sample size for sentencing lots of high-quality products. For illustrative purposes, the proposed methods are applied to the manufacturing of copper wire.

Statistical Inference for the power Lindley model based on record values and inter-record times

A new generalization of the Lindley distribution, called the power Lindley distribution was proposed by Ghitany et al., (2013), which offers a more flexible distribution for modeling lifetime data, such as in reliability. They studied classical inferences for the model based on complete data sets. However, we may deal with record breaking data sets in which only values smaller (or larger) than the current extreme value are reported. In this paper, by using record values and inter-record times, we develop inference procedures for the estimation of the parameters and prediction of future record values for the power Lindley distribution. First, the maximum likelihood estimate of the parameters and their asymptotic confidence intervals are obtained. Next, we consider Bayes estimation under the symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions by using the joint bivariate density function. Since the closed forms of the estimates are not available, we encounter some computational difficulties to evaluate the Bayes estimates of the parameters involved in the model. For this reason, we use Tierney and Kadane’s method as well as Markov Chain Monte Carlo (MCMC) procedure to compute approximate Bayes estimates. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the power Lindley distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data set is analyzed for illustration purposes.

Applying a random parameters Negative Binomial Lindley model to examine multi-vehicle crashes along rural mountainous highways in Malaysia

Road safety in rural mountainous areas is a major concern as mountainous highways represent a complex road traffic environment due to complex topology and extreme weather conditions and are associated with more severe crashes compared to crashes along roads in flatter areas. The use of crash modelling to identify crash contributing factors along rural mountainous highways suffers from limitations in data availability, particularly in developing countries like Malaysia, and related challenges due to the presence of excess zero observations. To address these challenges, the objective of this study was to develop a safety performance function for multi-vehicle crashes along rural mountainous highways in Malaysia. To overcome the data limitations, an in-depth field survey, in addition to utilization of secondary data sources, was carried out to collect relevant information including roadway geometric factors, traffic characteristics, real-time weather conditions, cross-sectional elements, roadside features, and spatial characteristics. To address heterogeneity resulting from excess zeros, three specialized modelling techniques for excess zeros including Random Parameters Negative Binomial (RPNB), Random Parameters Negative Binomial – Lindley (RPNB-L) and Random Parameters Negative Binomial – Generalized Exponential (RPNB-GE) were employed. Results showed that the RPNB-L model outperformed the other two models in terms of prediction ability and model fit. It was found that heavy rainfall at the time of crash and the presence of minor junctions along mountainous highways increase the likelihood of multi-vehicle crashes, while the presence of horizontal curves along a steep gradient, the presence of a passing lane and presence of road delineation decrease the likelihood of multi-vehicle crashes. Findings of this study have significant implications for road safety along rural mountainous highways, particularly in the context of developing countries.

On Reliability in a Multicomponent Stress-Strength Model with Power Lindley Distribution

In this paper we study the reliability of a multicomponent stress-strength model assuming that the components follow power Lindley model. The maximum likelihood estimate of the reliability parameter and its asymptotic confidence interval are obtained. Applying the parametric Bootstrap technique, interval estimation of the reliability is presented. Also, the Bayes estimate and highest posterior density credible interval of the reliability parameter are derived using suitable priors on the parameters. Because there is no closed form for the Bayes estimate, we use the Markov Chain Monte Carlo method to obtain approximate Bayes estimate of the reliability. To evaluate the performances of different procedures, simulation studies are conducted and an example of real data sets is provided.

The inverted Nadarajah–Haghighi distribution: estimation methods and applications

ABSTRACTThe inverted (or inverse) distributions are sometimes very useful to explore additional properties of the phenomenons which non-inverted distributions cannot. We introduce a new inverted model called the inverted Nadarajah–Haghighi distribution which exhibits decreasing and unimodal (right-skewed) density while the hazard rate shapes are decreasing and upside-down bathtub. Our main focus is the estimation (from both frequentist and Bayesian points of view) of the unknown parameters along with some mathematical properties of the new model. The Bayes estimators and the associated credible intervals are obtained using Markov Chain Monte Carlo techniques under squared error loss function. The gamma priors are adopted for both scale and shape parameters. The potentiality of the distribution is analysed by means of two real data sets. In fact, it is found to be superior in its ability to sufficiently model the data as compared to the inverted Weibull, inverted Rayleigh, inverted exponential, inverted gamma, inverted Lindley and inverted power Lindley models.

The Rayleigh–Lindley model: properties and applications

ABSTRACTIn this paper, the Rayleigh–Lindley (RL) distribution is introduced, obtained by compounding the Rayleigh and Lindley discrete distributions, where the compounding procedure follows an approach similar to the one previously studied by Adamidis and Loukas in some other contexts. The resulting distribution is a two-parameter model, which is competitive with other parsimonious models such as the gamma and Weibull distributions. We study some properties of this new model such as the moments and the mean residual life. The estimation was approached via EM algorithm. The behavior of these estimators was studied in finite samples through a simulation study. Finally, we report two real data illustrations in order to show the performance of the proposed model versus other common two-parameter models in the literature. The main conclusion is that the model proposed can be a valid alternative to other competing models well established in the literature.

The Odd Log-Logistic Marshall-Olkin Lindley Model for Lifetime Data

A new three parameter model called the odd log-logistic Marshall-Olkin-Lindley (OLLMO-L) distribution is proposed and studied. This distribution is symmetric, platykurtic, leptokurtic and unimodal. Various structural properties are derived, including explicit expressions for the ordinary and conditional moments, generating function and order statistics. We use maximum likelihood method to estimate the parameters of model. Simulation studies for some particular cases are also run to assess the performance of the maximum likelihood estimators. We prove empirically that the proposed distribution is a very competitive model to other classical models by means of two real data sets.

Statistical Inference for Lindley Model based on Type II Censored Data

In this paper, the moment-based, maximum likelihood and Bayes estimators for the unknown parameter of the Lindley model based on Type II censored data are discussed. The expectation maximization (EM) algorithm and direct maximization methods are used to obtained the maximum likelihood estimator (MLE). Existence and uniqueness of the moment-based and maximum likelihood estimators are discussed and a bias corrected estimator based on parametric bootstrap is developed. For Bayesian estimation, since the Bayes estimator cannot be obtained in an explicit form, two approximations based on Lindley and the importance sampling methods are used. Asymptotic confidence intervals, bootstrap confidence intervals and credible intervals are also proposed. Based on Type II censored data, the prediction of future observations is discussed. The analysis of a real data has been presented for illustrative purposes. Finally, Monte Carlo simulations are performed to compare the performances of the proposed estimation methods.

On Bayesian Premium Estimators for Gamma Lindley Model under Squared Error Loss Function and Linex Loss Function

We consider the Gamma Lindley distribution (GaL) as the conditional distribution of (X|θ,γ), we focus on the estimation of the Bayesian premium under squared error loss function (symmetric) and linear-exponential (Linex) loss function (asymmetric), using informative priors (the Gamma prior). Because of its difficulty and non-linearity, we use a numerical approximation for computing the Bayesian premium. Finally, a simulation and comparative study with varying sample sizes are given.

A comparison of inverse transform and composition methods of data simulation from the Lindley distribution

This study compares the inverse transform and the composition methods for generating data from the Lindley distribution. The expression for the inverse of the distribution function for the Lindley distribution does not exist in closed form. Hence, authors of many empirical studies on the Lindley distribution used methods for generating Lindley variates other than the inverse transform. We generated data from the Lindley distribution using the inverse transform approach by obtaining the Lindley variates numerically; we also generated data from this distribution using the composition approach. Following the generation of the Lindley variates using these two methods, we compare some statistical properties of the estimates of the Lindley model parameters based on the generated data. We conclude that the two methods produce similar results.

Estimation of a change point in the hazard rate of Lindley model under right censoring

ABSTRACTIn this article, we consider a single change point model for a sudden change in the hazard rate of Lindley distribution to model right-censored survival data. We derive the quantile function to generate random numbers from the proposed distribution by using the Lambert function. The maximum likelihood estimation method is used to estimate parameters of the change point model. A simulation study is also carried out to analyze the performance of the estimators. To validate our findings, a dataset on bone marrow transplant for patients of acute lymphoblastic leukemia is analyzed using the proposed model and is compared with the existing exponential single change point model.