The discrete Lindley distribution: properties and applications
24 Feb 2011-Journal of Statistical Computation and Simulation (Taylor & Francis)-Vol. 81, Iss: 11, pp 1405-1416
TL;DR: In this article, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution, which is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model.
Abstract: Modelling count data is one of the most important issues in statistical research. In this paper, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution. The model obtained is over-dispersed and competitive with the Poisson distribution to fit automobile claim frequency data. After revising some of its properties a compound discrete Lindley distribution is obtained in closed form. This model is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model. The new compound distribution fades away to zero much more slowly than the classical compound Poisson distribution, being therefore suitable for modelling extreme data.
Citations
More filters
TL;DR: In this paper, a two-parameter Lindley distribution for modeling waiting and survival times data has been introduced and its moments, failure rate function, mean residual life function, and stochastic orderings have been discussed.
Abstract: In this paper, a two-parameter Lindley distribution, of which the one parameter Lindley distribution (LD) is a particular case, for modeling waiting and survival times data 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 two-parameter LD shows flexibility over one-parameter LD and exponential distribution. 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 relating to waiting times and survival times to test its goodness of fit to which earlier the one parameter LD has been fitted by others and it is found that to almost all these data-sets the two parameter LD distribution provides closer fits than those by the one parameter LD.
115 citations
TL;DR: A comprehensive survey of the different methods of generating discrete probability distributions as analogues of continuous probability distributions is presented along with their applications in construction of new discrete distributions.
Abstract: In this paper a comprehensive survey of the different methods of generating discrete probability distributions as analogues of continuous probability distributions is presented along with their applications in construction of new discrete distributions. The methods are classified based on different criterion of discretization.
105 citations
TL;DR: In this article, a new probability density function with bounded domain is presented, which is based on the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010).
Abstract: In this paper a new probability density function with bounded domain is presented. The new distribution arises from the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010). This new distribution that depends on two parameters can be considered as an alternative to the classical beta distribution. It presents the advantage of not including any special function in its formulation. After studying its most important properties, some useful results regarding insurance and inventory management applications are obtained. In particular, in insurance, we suggest a special class of distorted premium principles based on this distribution and we compare it with the well-known power dual premium principle. Since the mean of the new distribution can be normalized to give a simple parameter, this new model is appropriate to be used as a regression model when the response is bounded, being therefore an alternative to the beta regression model recently proposed in the statistical literature.
95 citations
DOI•
30 Apr 2013
TL;DR: In this article, a two-parameter Quasi Lindley distribution (QLD) has been introduced and its moments, failure rate function, mean residual life function and stochastic orderings have been discussed.
Abstract: 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.
.
91 citations
Cites background from "The discrete Lindley distribution: ..."
...A discrete version of this distribution has been suggested by Deniz and Ojeda (2011) having its applications in count data related to insurance....
[...]
01 Jan 2015
TL;DR: In this paper, a new one parameter lifetime distribution named "Akash distribution" for modeling lifetime data has been introduced, which has been discussed using maximum likelihood estimation and method of moments, the usefulness and applicability of the proposed distribution have been discussed and illustrated with two real lifetime data sets from medical science and engineering.
Abstract: A new one parameter lifetime distribution named "Akash distribution" for modeling lifetime data has been introduced. Some important mathematical properties of the proposed distribution including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability have been discussed. The condition under which Akash distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along with the conditions under which exponential and Lindley distributions are over-dispersed, equi-dispersed and under-dispersed. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. The usefulness and the applicability of the proposed distribution have been discussed and illustrated with two real lifetime data sets from medical science and engineering.
78 citations
References
More filters
Book•
01 Jan 1950
TL;DR: In this paper, the authors present an approach for estimating the average risk of a risk-optimal risk maximization algorithm for a set of risk-maximization objectives, including maximalaxity and admissibility.
Abstract: Preface to the Second Edition.- Preface to the First Edition.- List of Tables.- List of Figures.- List of Examples.- Table of Notation.- Preparations.- Unbiasedness.- Equivariance.- Average Risk Optimality.- Minimaxity and Admissibility.- Asymptotic Optimality.- References.- Author Index.- Subject Index.
4,382 citations
Book•
29 Jan 1998
TL;DR: In this paper, the authors present an inventory of continuous and discrete time-ruiner models for complete and modified data sets, as well as a comprehensive inventory of discrete and continuous distributions for complete data sets.
Abstract: Preface. Acknowledgments. PART I: INTRODUCTION. 1. Modeling. PART II: ACTUARIAL MODELS. 2. Random Variables. 3. Basic Distributional Quantities. 4. Classifying and Creating Distributions. 5. Frequency and Severity with Coverage Modifications. 6. Aggregate Loss Models. 7. Discrete Time Ruin Models. 8. Continuous Time Ruin Models. PART III: CONSTRUCTION OF EMPIRICAL MODELS. 9. Review of Mathematical Statistics. 10. Estimation for Complete Data. 11. Estimation for Modified Data. PART IV: PARAMETRIC STATISTICAL METHODS. 12. Parameter Estimation. 13. Model Selection. 14. Five Examples. PART V: ADJUSTED ESTIMATES AND SIMULATION. 15. Interpolation and Smoothing. 16. Credibility. 17. Simulation. Appendix A: An Inventory of Continuous Distributions. Appendix B: An Inventory of Discrete Distributions. Appendix C: Frequency and Severity Relationships. Appendix D: The Recursive Formula. Appendix E: Discretization of the Serverity Distribution. Appendix F: Numerical Optimization and Solution of Systems. References. Index.
1,276 citations
"The discrete Lindley distribution: ..." refers background in this paper
...Proof By assuming that the claim amount follows an exponential distribution with parameter γ > 0, the nth fold convolution of exponential distribution has a closed form and it is given by [13,17]...
[...]
TL;DR: In this article, a new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used twoparameter families of life distributions as the Weibull, gamma and lognormal distributions.
Abstract: SUMMARY A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.
1,016 citations
"The discrete Lindley distribution: ..." refers methods in this paper
...Some of those works are by Nakagawa and Osaki [1], where the discrete Weibull distribution is obtained; Roy [2] studied the discrete Rayleigh distribution; in Kemp [3] the discrete half-normal distribution is examined, in Krishna and Pundir [4] the Burr discrete distribution and the Pareto discrete distribution as a particular case of the former are analysed and more recently, Gómez-Déniz [5] derived a new generalization of the geometric distribution obtained from the generalized exponential distribution of Marshall and Olkin [6]....
[...]
994 citations
862 citations
"The discrete Lindley distribution: ..." refers methods in this paper
...This new distribution is generated by discretizing the continuous survival function of the Lindley distribution [7,8], which is given by S(x) = e −θx(1 + θ + xθ) 1 + θ , θ > 0, x > 0....
[...]