A negative binomial Lindley approach considering spatiotemporal effects for modeling traffic crash frequency with excess zeros.

Statistical analysis of traffic crash frequency is significant for figuring out the distribution pattern of crashes, predicting the development trend of crashes, formulating traffic crash prevention measures, and improving traffic safety planning systems. In recent years, the theory and practice for traffic safety management have shown that road crash data have characteristics such as spatial correlation, temporal correlation, and excess zeros. If these characteristics are ignored in the modeling process, it may seriously affect the fitting performance and prediction accuracy of traffic crash frequency models and even lead to incorrect conclusions. In this research, traffic crash data from rural two-way two-lane from four counties in Pennsylvania, USA was modeled considering the spatiotemporal effects of crashes. First, a negative binomial Lindley spatiotemporal effect model of crash frequency was constructed at the micro level; Simultaneously, the characteristics and problems of excess zeros and potential heterogeneity of the crash data were resolved; Finally, the effects of road characteristics on crash frequency were analyzed. The results indicate a significant spatial correlation between the crash frequency of adjacent road sections. Compared with the negative binomial model, the negative binomial Lindley model can better handle the excess zeros characteristics in traffic crash data. The model that considers both spatial correlation and temporal conditional autoregressive effects has the best fit for the observed data. In addition, for road sections that allow passing and have a speed limitation of not less than 50 miles per hour, the crash frequency corresponding to these sections is lower due to their good visibility and road conditions. The increase in average turning angle and intersection density on the horizontal curve of the road section corresponds to an increase in crash frequency.

A Lindley-binomial model for analyzing the proportions with sparseness and excessive zeros.

Proportional data arise frequently in a wide variety of fields of study. Such data often exhibit extra variation such as over/under dispersion, sparseness and zero inflation. For example, the hepatitis data present both sparseness and zero inflation with 19 contributing non-zero denominators of 5 or less and with 36 having zero seropositive out of 83 annual age groups. The whitefly data consists of 640 observations with 339 zeros (53%), which demonstrates extra zero inflation. The catheter management data involve excessive zeros with over 60% zeros averagely for outcomes of 193 urinary tract infections, 194 outcomes of catheter blockages and 193 outcomes of catheter displacements. However, the existing models cannot always address such features appropriately. In this paper, a new two-parameter probability distribution called Lindley-binomial (LB) distribution is proposed to analyze the proportional data with such features. The probabilistic properties of the distribution such as moment, moment generating function are derived. The Fisher scoring algorithm and EM algorithm are presented for the computation of estimates of parameters in the proposed LB regression model. The issues on goodness of fit for the LB model are discussed. A limited simulation study is also performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates. The proposed model is illustrated through three aforementioned proportional datasets.

Addressing overdispersion and zero-inflation for clustered count data via new multilevel heterogenous hurdle models.

Unobserved heterogeneity causing overdispersion and the excessive number of zeros take a prominent place in the methodological development on count modeling. An insight into the mechanisms that induce heterogeneity is required for better understanding of the phenomenon of overdispersion. When the heterogeneity is sourced by the stochastic component of the model, the use of a heterogenous Poisson distribution for this part encounters as an elegant solution. Hierarchical design of the study is also responsible for the heterogeneity as the unobservable effects at various levels also contribute to the overdispersion. Zero-inflation, heterogeneity and multilevel nature in the count data present special challenges in their own respect, however the presence of all in one study adds more challenges to the modeling strategies. This study therefore is designed to merge the attractive features of the separate strand of the solutions in order to face such a comprehensive challenge. This study differs from the previous attempts by the choice of two recently developed heterogeneous distributions, namely Poisson-Lindley (PL) and Poisson-Ailamujia (PA) for the truncated part. Using generalized linear mixed modeling settings, predictive performances of the multilevel PL and PA models and their hurdle counterparts were assessed within a comprehensive simulation study in terms of bias, precision and accuracy measures. Multilevel models were applied to two separate real world examples for the assessment of practical implications of the new models proposed in this study.

Inferential properties with a novel two parameter Poisson generalized Lindley distribution with regression and application to INAR(1) process.

Based on the well-known Poisson (P) distribution and the new generalized Lindley distribution (NGLD) developed by using gamma (α,θ) and gamma (α-1,θ) distributions, a new compound two-parameter Poisson generalized Lindley (TPPGL) distribution is proposed in this paper and thereon systematically explores the mathematical properties. Closed form expressions are assembled for such properties including the probability generating function, moments, skewness, kurtosis, etc. The likelihood-based method is used for estimating the parameters followed by a broad Monte Carlo simulation study. To further motivate the proposed model, a count regression model and a first order integer valued autoregressive process are constructed based on the novel TPPGL distribution. The empirical importance of the proposed models is confirmed through application to four real datasets.

Generalization of the Lindley distribution with application to COVID-19 data.

Creating new distributions with more desired and flexible qualities for modeling lifetime data has resulted in a concentrated effort to modify or generalize existing distributions. In this paper, we propose a new distribution called the power exponentiated Lindley (PEL) distribution by generalizing the Lindley distribution using the power exponentiated family of distributions, that can fit lifetime data. Then the main statistical properties such as survival function, hazard function, reverse hazard function, moments, quantile function, stochastic ordering, MRL, order statistics, etc., of the newly proposed distribution have been derived. The parameters of the distribution are estimated using the MLE method. Then, a Monte Carlo simulation study is used to check the consistency of the parameters of the PEL distribution in terms of MSE, RMSE, and bias. Finally, we implement the PEL distribution as a statistical lifetime model for the COVID-19 case fatality ratio (in %) in China and India, and the new cases of COVID-19 reported in Delhi. Then we check whether the new distribution fits the data sets better than existing well-known distributions. Different statistical measures such as the value of the log-likelihood function, K-S statistic, AIC, BIC, HQIC, and p-value are used to assess the accuracy of the model. The suggested model seems to be superior to its base model and other well-known and related models when applied to the COVID-19 data set.

Inverse Lindley power series distributions: a new compounding family and regression model with censored data.

This paper introduces a new class of distributions by compounding the inverse Lindley distribution and power series distributions which is called compound inverse Lindley power series (CILPS) distributions. An important feature of this distribution is that the lifetime of the component associated with a particular risk is not observable, rather only the minimum lifetime value among all risks is observable. Further, these distributions exhibit an unimodal failure rate. Various properties of the distribution are derived. Besides, two special models of the new family are investigated. The model parameters of the two sub-models of the new family are obtained by the methods of maximum likelihood, least square, weighted least square and maximum product of spacing and compared them using the Monte Carlo simulation study. Besides, the log compound inverse Lindley regression model for censored data is proposed. Three real data sets are analyzed to illustrate the flexibility and importance of the proposed models.

Ratio of Two Independent Lindley Random Variables.

The distribution of the ratio of two independently distributed Lindley random variables X and Y , with different parameters, is derived. The associated distributional properties are provided. Furthermore, the proposed ratio distribution is fitted to two applications data (COVID-19 and Bladder Cancer Data), and compared it with some well-known right-skewed variations of Lindley distribution, namely; Lindley distribution, new generalized Lindley distribution, new quasi Lindley distribution and a three parameter Lindley distribution. The numerical result of the study reveals that the proposed distribution of two independent Lindley random variables fits better to the above said data sets than the compared distribution.

A new statistical approach to model the counts of novel coronavirus cases.

This study proposes new statistical tools to analyze the counts of the daily coronavirus cases and deaths. Since the daily new deaths exhibit highly over-dispersion, we introduce a new two-parameter discrete distribution, called discrete generalized Lindley, which enables us to model all kinds of dispersion such as under-, equi-, and over-dispersion. Additionally, we introduce a new count regression model based on the proposed distribution to investigate the effects of the important risk factors on the counts of deaths for OECD countries. Three data sets are analyzed with proposed models and competitive models. Empirical findings show that air pollution, the proportion of obesity, and smokers in a population do not affect the counts of deaths for OECD countries. The interesting empirical result is that the countries with having higher alcohol consumption have lower counts of deaths.

Bivariate Discrete Poisson-Lindley Distributions.

Two families of bivariate discrete Poisson-Lindley distributions are introduced. The first is derived by mixing the common parameter in a bivariate Poisson distribution by different models of univariate continuous Lindley distributions. The second is obtained by generalizing a bivariate binomial distribution with respect to its exponent when it follows any of five different univariate discrete Poisson-Lindley distributions with one or two parameters. The use of probability-generating functions is mainly employed to derive some general properties for both families and specific characteristics for each one of their members. We obtain expressions for probabilities, moments, conditional distributions, regression functions, as well as characterizations for certain bivariate models and their marginals. An attractive property of all bivariate individual models is that they contain only two or three parameters, and one of them is readily estimated by simple ratios of their sample means. This feature, and since all marginal distributions are over-dispersed, strongly suggests their potential use to describe bivariate dependent count data in many different areas.

Nonproportional hazards model with a frailty term for modeling subgroups with evidence of long-term survivors: Application to a lung cancer dataset.

With advancements in medical treatments for cancer, an increase in the life expectancy of patients undergoing new treatments is expected. Consequently, the field of statistics has evolved to present increasingly flexible models to explain such results better. In this paper, we present a lung cancer dataset with some covariates that exhibit nonproportional hazards (NPHs). Besides, the presence of long-term survivors is observed in subgroups. The proposed modeling is based on the generalized time-dependent logistic model with each subgroup's effect time and a random term effect (frailty). In practice, essential covariates are not observed for several reasons. In this context, frailty models are useful in modeling to quantify the amount of unobservable heterogeneity. The frailty distribution adopted was the weighted Lindley distribution, which has several interesting properties, such as the Laplace transform function on closed form, flexibility in the probability density function, among others. The proposed model allows for NPHs and long-term survivors in subgroups. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation studies were conducted to evaluate the estimators' performance. We exemplify this model's use by applying data of patients diagnosed with lung cancer in the state of São Paulo, Brazil.

Bayesian Analysis of Dynamic Cumulative Residual Entropy for Lindley Distribution.

Dynamic cumulative residual (DCR) entropy is a valuable randomness metric that may be used in survival analysis. The Bayesian estimator of the DCR Rényi entropy (DCRRéE) for the Lindley distribution using the gamma prior is discussed in this article. Using a number of selective loss functions, the Bayesian estimator and the Bayesian credible interval are calculated. In order to compare the theoretical results, a Monte Carlo simulation experiment is proposed. Generally, we note that for a small true value of the DCRRéE, the Bayesian estimates under the linear exponential loss function are favorable compared to the others based on this simulation study. Furthermore, for large true values of the DCRRéE, the Bayesian estimate under the precautionary loss function is more suitable than the others. The Bayesian estimates of the DCRRéE work well when increasing the sample size. Real-world data is evaluated for further clarification, allowing the theoretical results to be validated.

Weighted Lindley frailty model: estimation and application to lung cancer data.

In this paper, we propose a novel frailty model for modeling unobserved heterogeneity present in survival data. Our model is derived by using a weighted Lindley distribution as the frailty distribution. The respective frailty distribution has a simple Laplace transform function which is useful to obtain marginal survival and hazard functions. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions. A classical inference procedure based on the maximum likelihood method is presented. Extensive simulation studies are further performed to verify the behavior of maximum likelihood estimators under different proportions of right-censoring and to assess the performance of the likelihood ratio test to detect unobserved heterogeneity in different sample sizes. Finally, to demonstrate the applicability of the proposed model, we use it to analyze a medical dataset from a population-based study of incident cases of lung cancer diagnosed in the state of São Paulo, Brazil.

A random parameters with heterogeneity in means and Lindley approach to analyze crash data with excessive zeros: A case study of head-on heavy vehicle crashes in Queensland.

This study performed statistical analyses to identify likely crash contributing factors for Head-on Fatal and Serious Injury (FSI) collisions involving heavy vehicles (HVs) on the Queensland state road network. Head-on HV collisions are associated with the largest number of fatalities compared to other crash types in Queensland. However, there is limited relevant literature regarding this type of crashes. Recent studies on road safety research have focused on variants of random parameters models to capture unobserved heterogeneity that may influence the occurrence of crashes. Among such models, random parameters with heterogeneity in means has recently provided better results and has become popular. However, this study illustrates a potential limitation regarding the use of these models without explicitly factoring for excessive zero crash observations. To address this potential limitation, a random parameters with heterogeneity in means and a Lindley distribution is introduced in this study to factor for the unobserved heterogeneity using additional variables as well as site-specific variation from excessive zero crash observations. Results showed that a Poisson model with random parameters and heterogeneity in means using a Lindley distribution outperformed multiple alternative state-of-the-art specifications in terms of fit as well as overall prediction ability. The analyses using the proposed modelling approach revealed factors likely to affect the likelihood of Head-on FSI crashes involving HVs in Queensland including volume, segment length, period of analysis, terrain type being rolling, curve (moderate/sharp/very sharp) longer than 50% of the corresponding segment length, rural single carriageway with high (>=100 kph) and medium (>=50 and <100 kph) speed limits, and urban single carriageway. Unobserved heterogeneity regarding the parameter for road curvature was explained using rolling terrain type as an explanatory variable. This study has explained variation in the means of random parameters for a road attribute using the effect of a geometric variable, in which several stakeholders are primarily interested.

A new Lindley-Burr XII power series distribution: model, properties and applications.

A new generalized class of distributions called the Lindley-Burr XII Power Series (LBXIIPS) distribution is proposed and explored. This new class of distributions contain some special cases such as Lindley-Burr XII Poisson (LBXIIP), Lindley-Burr XII Logarithmic (LBXIIL), Lindley-Burr XII Binomial (LBXIIB) and their sub-models among others. Some structural properties of the new distribution including moments, probability weighted moments, distribution of the order statistics and entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators is presented and finally, an application to a real data set in order to illustrate the usefulness of the new distribution is given.

Unit-Lindley mixed-effect model for proportion data.

Recently, unit-Lindley distribution and its associated regression models have been developed as an alternative to Beta regression model for which continuous outcome in the unit interval ( 0 , 1 ) . Proportion data usually occur in clinical trials, economics and social studies with hierarchical structures. In this study, unit-Lindley mixed-effect model is proposed and the appropriate likelihood analysis methods for parameter estimation are investigated. In the case of clustered or longitudinal proportion data in mixed-effect models, the full-likelihood function does not have a closed form. Parameter estimations of unit-Lindley mixed-effect model are obtained with Laplace and adaptive Gaussian quadrature approximation methods in this study. We analyzed a dataset on the proportion of households with insufficient water supply and sewage with some sociodemographic variables in the cities of Brazil by using unit-Lindley mixed-effect model including a random intercept as federative states of Brazil. Analysis results indicate that the proposed unit-Lindley mixed-effect model provides better fit than unit-Lindley regression model and beta mixed model. Also, in the simulation study the accuracy of the estimates of approximation methods are evaluated and compared via Monte Carlo simulation study in terms of bias and mean square error.

Load-Sharing Model under Lindley Distribution and Its Parameter Estimation Using the Expectation-Maximization Algorithm.

A load-sharing system is defined as a parallel system whose load will be redistributed to its surviving components as each of the components fails in the system. Our focus is on making statistical inference of the parameters associated with the lifetime distribution of each component in the system. In this paper, we introduce a methodology which integrates the conventional procedure under the assumption of the load-sharing system being made up of fundamental hypothetical latent random variables. We then develop an expectation maximization algorithm for performing the maximum likelihood estimation of the system with Lindley-distributed component lifetimes. We adopt several standard simulation techniques to compare the performance of the proposed methodology with the Newton-Raphson-type algorithm for the maximum likelihood estimate of the parameter. Numerical results indicate that the proposed method is more effective by consistently reaching a global maximum.

A new two-parameter exponentiated discrete Lindley distribution: properties, estimation and applications.

This paper introduces a new two-parameter exponentiated discrete Lindley distribution. A wide range of its structural properties are investigated. This includes the shape of the probability mass function, hazard rate function, moments, skewness, kurtosis, stress-strength reliability, mean residual lifetime, mean past lifetime, order statistics and L-moment statistics. The hazard rate function can be increasing, decreasing, decreasing-increasing-decreasing, increasing-decreasing-increasing, unimodal, bathtub, and J-shaped depending on its parameters values. Two methods are used herein to estimate the model parameters, namely, the maximum likelihood, and the proportion. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood and proportion estimators. The flexibility of the proposed model is explained by using four distinctive data sets. It can serve as an alternative model to other lifetime distributions in the existing statistical literature for modeling positive real data in many areas.

Classical methods of estimation on constant stress accelerated life tests under exponentiated Lindley distribution.

Accelerated life testing is adopted in several fields to obtain adequate failure time data of test units in a much shorter time than testing at normal operating conditions. The lifetime of a product at constant level of stress is assumed to have an exponentiated Lindley distribution. In this paper, besides maximum likelihood method, eight other frequentist methods of estimation, namely, method of least square estimation, method of weighted least square estimation, method of maximum product of spacing estimation, method of minimum spacing absolute distance estimation, method of minimum spacing absolute-log distance estimation, method of Cramér-von-Mises estimation, method of Anderson-Darling estimation and Right-tail Anderson-Darling estimation are considered to estimate the parameters of the exponentiated Lindley distribution under constant stress accelerated life testing. Moreover, shape parameter and the reliability function under usual conditions are estimated based on aforementioned methods of estimation. To evaluate the performance of the proposed methods, a simulation study is carried out. The performances of the estimators have been compared in terms of their mean squared error using small, medium and large sample sizes. As an illustration, the model and the proposed methods are applied to two accelerated life test data sets.

Discussion comments on 'On moments of the unit Lindley distribution'.

In a note about the paper titled 'On the one parameter unit-Lindley distribution and its associated regression model for proportion data', Mazucheli et al. [J. Appl. Stat. 46 (2019), pp. 700-714] and Nadarajah and Chan [On moments of the unit Lindley distribution, J. Appl. Stat. (under review)] claim that 'The expressions given for the moments and incomplete moments are not correct and not in closed form'. While we agree that they are not in closed form and observe a typo in the expressions for µ k ' and T k ( t ) , k = 1 , … , the expressions for µ 1 ' , µ 2 ' , µ 3 ' and µ 4 ' are, however, entirely correct.

Statistical inference based on generalized Lindley record values.

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.