In this paper, we introduce a three-parameter generalization of the Lindley distribution. This includes as special cases the exponential and gamma distributions. The distribution exhibits decreasing, increasing and bathtub hazard rate depending on its parameters. We study various properties of the new distribution and provide numerical examples to show the flexibility of the model. We also derive a bivariate version of the proposed distribution.

A new continuous distribution called Lindley-Lindley distribution is defined and studied. Relevant mathematical properties are derived. We present three characterizations of the new distribution based on the truncated moments of certain functions of the random variable,the hazard function and in terms of the conditional expectation of a function of the random variable. Some new bivariate type distributions using Farlie Gumbel Morgenstern copula, modified Farlie Gumbel Morgenstern copula and Clayton copula are introduced. The main justification of this paper is to show how different frequentist estimators of the new model perform for different sample sizes and different parameter values and to provide a guideline for choosing the best estimation method for the parameters of the proposed model. The unknown parameters of the new distribution are estimated using the maximum likelihood, ordinary least squares, Cramer-Von-Mises, weighted least squares and Bayesian methods. The obtained estimators are compared using Markov Chain Monte Carlo simulations and observed that Bayesian estimators are generally more efficient than the other estimators.

In this study, we introduce a new model called the Extended Exponentiated PowerLindley distribution which extends the Lindley distribution and has increasing, bathtub andupside down shapes for the hazard rate function. It also includes the power Lindley distributionas a special case. Several statistical properties of the distribution are explored, such as thedensity, hazard rate, survival, quantile functions, and moments. Estimation using the maximumlikelihood method and inference on a random sample from this distribution are investigated. Asimulation study is performed to compare the performance of the di® erent parameter estimatesin terms of bias and mean square error. We apply a real data set to illustrate the applicabilityof the new model. Empirical ¯ ndings show that proposed model provides better ¯ ts than otherwell-known extensions of Lindley distributions.

In this paper‎, ‎order statistics and associated inferences are considered from Lindley distribution‎. ‎We derive the exact forms of means‎, ‎variances and covariances as well as the moment generating functions of order statistics‎. ‎These obtained forms allow us to compute the means‎, ‎variances‎, ‎and covariances of the order statistics for various values of the shape parameter‎. ‎These values are then used to compute the coefficients of the best linear unbiased estimators‎, ‎the best linear invariant estimators‎, ‎and the least square estimators of the location and scale parameters‎. ‎The variances and covariances of these estimators are also presented‎. ‎Using the best linear unbiased estimators and best linear invariant estimators we construct confidence intervals for the location and scale parameters through Monte Carlo simulations‎. ‎In addition‎, ‎based on the ordered data‎, ‎we investigate how to obtain the best linear unbiased predictor and the best linear invariant predictor for future order statistics‎. ‎Finally‎, ‎data analysis and Monte Carlo simulation have been performed for illustrative purposes and comparative studies‎, ‎respectively.

The power version of the modi, ed Lindley distribution is introduced in this paper, o, ering a new two-parameter lifetime distri-bution. As a main interest, it provides a motivated alternative to the Weibull and power Lindley distributions. We discuss its main charac-teristics and properties, including shapes of the probability density and hazard rate functions, incomplete moments, crude moments, variance, skewness, kurtosis and order statistics. Then, a statistical study of the model is developed. The parameters are estimated by the maximum likelihood method. A simulation study examines the numerical com-portment of the bias and mean square error of the maximum likelihood estimates of the parameters. Application of the new model to three data sets is presented, showing that the model has a better , t behavior in comparison to some other well-known lifetime models, including the Weibull and power Lindley models.

The problem of distinguishing between distributions is always important. It becomes more complicated when data is contaminated by outliers. Here, we use two well-known Lindley and exponential distributions infected by outliers. The closeness of the Lindley distribution in comparison with the exponential distribution with outliers is discussed in this research. Three ways such as likelihood ratio, asymptotic likelihood ratio tests and minimum Kolmogorov distance are used to select the proper fitted model for a real data set. We perform Monte Carlo simulation to obtain the probability of correct selection for various values of sample sizes and parameters based on the best criteria in the distributions. In general, it has been seen that the Lindley distribution is closer to exponential distribution contaminated by outliers based on the likelihood ratio and Kolmogorov criteria. An actual example of real data is used to see the behaviour of the distributions.

In this study, two parameters of the Lomax-Lindley distribution were developed, which generalized the existing Lindley distribution and has the growing and decreasing properties of the current distribution. The newly suggested Lomax-Lindley distribution parameters were estimated using maximum likelihood estimators (MLEs). Maximum likelihood estimators (MLEs) are biased for small and intermediate sample sizes. The two-parameter Lindley (TPL) distribution is increasingly being utilized to characterize data on lifetime and survival times because distribution can provide a better fit than several existing lifetime models. A real-world industrial system application is also provided to demonstrate how the concepts might be applied. The Mat Lab program was utilized for both the numerical result and the graphical representation.