A Comparative Study Between Two Discrete Lindley Distributions
Keywords:Discretization methods, Lindley distribution, likelihood, series, survival analysis, Monte Carlo simulation
The methods of generate a probability function from a probability density function has long been used in recent years. In general, the discretization process produces probability functions that can be rivals to traditional distributions used in the analysis of count data as the geometric, the Poisson and negative binomial distributions. In this paper, by the method based on an infinite series, we studied an alternative discrete Lindley distribution to those study in Gomez (2011) and Bakouch (2014). For both distributions, a simulation study is carried out to examine the bias and mean squared error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence intervals. For the discrete Lindley distribution obtained by infinite series method we present the analytical expression for bias reduction of the maximum likelihood estimator. Some examples using real data from the literature show the potential of these distributions.
Aghababaei Jazi, M., Lai, C. D., Hossein Alamatsaz, M. (2010). A discrete inverse Weibull distribution and estimation
of its parameters. Statistical Methodology, 7, 121–132.
Bakouch, H. S., Jazi, M. A., Nadarajah, S. (2014). A new discrete distribution. Statistics, 48 (1), 200–240.
Bi, Z., Faloutsos, C., Korn, F. (2001). The DGX distribution for mining massive, skewed data. Em: Proceedings of the
seventh ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, pp. 17–26.
Bracquemond, C., Gaudoin, O. (2003). A survey on discrete lifetime distributions. International Journal of Reliability,
Quality and Safety Engineering, 10 (01), 69–98.
Chakraborty, S. (2015). Generating discrete analogues of continuous probability distributions - a survey of methods
and constructions. Journal of Statistical Distributions and Applications, 2 (1), 1–30.
Chakraborty, S., Chakravarty, D. (2012). Discrete gamma distributions: properties and parameter estimations.
Communications in Statistics-Theory and Methods, 41 (18), 3301–3324.
Collett, D. (2003). Modelling Survival Data in Medical Research, 2o edn. Chapaman and Hall, New York.
Cox, D. R., Snell, E. J. (1968). A general definition of residuals. Journal of the Royal Statistical Society Series B
(Methodological), 30 (2), 248–275.
Deb, P., Trivedi, P. K., et al. (1997). Demand for medical care by the elderly: a finite mixture approach. Journal of
applied Econometrics, 12 (3), 313–336.
Doray, L. G., Luong, A. (1997). Efficient estimators for the good family. Communications in Statistics-Simulation and
Computation, 26 (3), 1075–1088.
Gómez-Déniz, E., Calderı́n-Ojeda, E. (2011). The discrete Lindley distribution: properties and applications. Journal of
Statistical Computation and Simulation, 81 (11), 1405–1416.
Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika,
Haight, F. A. (1957). Queueing with balking. Biometrika, 44 (3/4), 360–369.
Hamada, M. S., Wilson, A. G., Reese, C. S., Martz, H. F. (2008). Bayesian reliability. Springer Series in Statistics,
Springer, New York.
Hussain, T., Ahmad, M. (2014). Discrete inverse Rayleigh distribution. Pak J Statist, 30 (2), 203–222.
Inusah, S., J. Kozubowski, T. (2006). A discrete analogue of the Laplace distribution. Journal of Statistical Planning
and Inference, 136.
Kalbfleisch, J. D., Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, 2o edn. Wiley, New York, NY.
Keilson, J., Gerber, H. (1971). Some results for discrete unimodality. Journal of the American Statistical Association,
Kemp, A. W. (1997). Characterizations of a discrete normal distribution. Journal of Statistical Planning and Inference,
(2), 223 – 229, in Honor of C.R. Rao.
Kemp, A. W. (2004). Classes of discrete lifetime distributions.
Kemp, A. W. (2008). The discrete half-normal distribution. Em: Advances in mathematical and statistical modeling,
Springer, pp. 353–360.
Klein, J. P., Moeschberger, M. L. (1997). Survival Analysis: Techniques for Censored and Truncated Data. Springer-
Verlag, New York.
Kozubowski, T. J., Inusah, S. (2006). A skew Laplace distribution on integers. Annals of the Institute of Statistical
Mathematics, 58 (3), 555–571.
Krishna, H., Pundir, P. S. (2009). Discrete Burr and discrete Pareto distributions. Statistical Methodology, 6 (2),
Kulasekera, K., Tonkyn, D. W. (1992). A new discrete distribution, with applications to survival, dispersal and
dispersion. Communications in Statistics-Simulation and Computation, 21 (2), 499–518.
Lawless, J. F. (2003). Statistical models and methods for lifetime data, 2o edn. Wiley Series in Probability and Statistics,
Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.
Lee, E. T., Wang, J. W. (2003). Statistical methods for survival data analysis, 3o edn. Wiley Series in Probability and
Statistics, Hoboken, NJ.
Liu, W., Cela, J. (2008). Count data models in SAS. Em: SAS Global Forum, Citeseer, vol 317, pp. 1–12.
Long, J. S. (1990). The origins of sex differences in science. Social forces, 68 (4), 1297–1316.
Long, J. S., Freese, J., et al. (2001). Predicted probabilities for count models. Stata Journal, 1 (1), 51–7.
Meeker, W. Q., Escobar, L. A. (1998). Statistical Methods for Reliability Data. John Wiley & Sons, New York.
Nakagawa, T., Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, 5, 300–301.
Nekoukhou, V., Alamatsaz, M. H., Bidram, H. (2012). A discrete analog of the generalized exponential distribution.
Communication in Statistics- Theory and Methods, 41, 2000–2013.
Nekoukhou, V., Alamatsaz, M. H., Bidram, H. (2013). Discrete generalized exponential distribution of a second type.
Statistics, 47, 876–887.
Roy, D. (2003). The discrete normal distribution. Communication in Statistics- Theory and Methods, 32, 1871–1883.
Roy, D. (2004). Discrete Rayleigh distribution. Reliability, IEEE Transactions on, 53 (2), 255–260.
Sato, H., Ikota, M., Sugimoto, A., Masuda, H. (1999). A new defect distribution metrology with a consistent discrete
exponential formula and its applications. Semiconductor Manufacturing, IEEE Transactions on, 12 (4), 409–418.
Siromoney, G. (1964). The general Dirichlet’s series distribution. Journal of the Indian Statistical Association, 2.
How to Cite
To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.
Ethical Guidelines for Journal Publication
The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles.
Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.
Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.
Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.
Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.