Published

2016-01-01

A Two Parameter Discrete Lindley Distribution

Distribución Lindley de dos parámetros

DOI:

https://doi.org/10.15446/rce.v39n1.55138

Keywords:

Characterization, Discretized version, Estimation, Geometric distribution, Mean residual life, Mixture, Negative moments (en)
Caracterización, Estimación, Distribución Geométrica, Momentos negativos, Mixtura, Versión discretizada, Vida media residual (es)

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Authors

  • Muhammad Aslam King Abdulaziz University, Jeddah, Saudi Arabia
  • Munir Ahmad National College of Business Administration and Economics, Lahore, Pakistan

In this article we have proposed and discussed a two parameter discrete Lindley distribution. The derivation of this new model is based on a two step methodology i.e. mixing then discretizing, and can be viewed as a new generalization of geometric distribution. The proposed model has proved itself as the least loss of information model when applied to a number of data sets (in an over and under dispersed structure). The competing models such as Poisson, Negative binomial, Generalized Poisson and discrete gamma distributions are the well known standard discrete distributions. Its Lifetime classification, kurtosis, skewness, ascending and descending factorial moments as well as its recurrence relations, negative moments, parameters estimation via maximum likelihood method, characterization and discretized bi-variate case are presented.

En este artículo propusimos y discutimos la distribución Lindley de dos parámetros. La obtención de este Nuevo modelo está basada en una metodología en dos etapas: mezclar y luego discretizar, y puede ser vista como una generalización de una distribución geométrica. El modelo propuesto demostró tener la menor pérdida de información al ser aplicado a un cierto número de bases de datos (con estructuras de supra y sobredispersión). Los modelos estándar con los que se puede comparar son las distribuciones Poisson, Binomial Negativa, Poisson Generalizado y Gamma discrete.Su clasificación de tiempo de vida, kurtosis, sesgamiento, momentos factorials ascendientes y descendientes, al igual que sus relaciones de recurrencia, momentos negativos, estimación de parámetros via máxima verosimilitud, caracterización y discretización del caso bivariado son presentados.

A Two Parameter Discrete Lindley Distribution

Distribución Lindley de dos parámetros

TASSADDAQ HUSSAIN1, MUHAMMAD ASLAM2, MUNIR AHMAD3

1Government Postgraduate College, Department of Statistics, Rawalakot, Pakistan. Professor. Email: taskho2000@yahoo.com
2King Abdulaziz University, Faculty of Sciences, Department of Statistics, Jeddah, Saudi Arabia. Professor. Email: aslam_ravian@hotmail.com
3National College of Business Administration and Economics, Lahore, Pakistan. Professor. Email: drmunir@ncbae.edu.pk

Abstract

In this article we have proposed and discussed a two parameter discrete Lindley distribution. The derivation of this new model is based on a two step methodology i.e. mixing then discretizing, and can be viewed as a new generalization of geometric distribution. The proposed model has proved itself as the least loss of information model when applied to a number of data sets (in an over and under dispersed structure). The competing models such as Poisson, Negative binomial, Generalized Poisson and discrete gamma distributions are the well known standard discrete distributions. Its Lifetime classification, kurtosis, skewness, ascending and descending factorial moments as well as its recurrence relations, negative moments, parameters estimation via maximum likelihood method, characterization and discretized bi-variate case are presented.

Key words: Characterization, Discretized version, Estimation, Geometric distribution, Mean residual life, Mixture, Negative moments.

Resumen

En este artículo propusimos y discutimos la distribución Lindley de dos parámetros. La obtención de este Nuevo modelo está basada en una metodología en dos etapas: mezclar y luego discretizar, y puede ser vista como una generalización de una distribución geométrica. El modelo propuesto demostró tener la menor pérdida de información al ser aplicado a un cierto número de bases de datos (con estructuras de supra y sobredispersión). Los modelos estándar con los que se puede comparar son las distribuciones Poisson, Binomial Negativa, Poisson Generalizado y Gamma discrete.Su clasificación de tiempo de vida, kurtosis, sesgamiento, momentos factorials ascendientes y descendientes, al igual que sus relaciones de recurrencia, momentos negativos, estimación de parámetros via máxima verosimilitud, caracterización y discretización del caso bivariado son presentados.

Palabras clave: caracterización, estimación, distribución Geométrica, momentos negativos, mixtura, versión discretizada, vida media residual.

Texto completo disponible en PDF

References

1. Abouammoh, A. & Mashhour, A. (1981), 'A note on the unimodality of discrete distributions', Communication in Statistics Theory and Methods 10(13), 1345-1354.

2. Al-Huniti, A. & Al-Dayian, G. (2012), 'Discrete burr type-iii distribution', American Journal of Mathematics and Statistic 2(5), 145-152.

3. Chakraborty, S. & Chakravarty, D. (2012), 'Discrete gamma distributions: properties and parameters estimations', Communication in Statistics Theory and Methods 41(18), 3301-3324.

4. D\breveeniz, E. & Ojeda, E. (2011), 'The discrete lindley distribution: properties and application', Journal of Statistical Computation and Simulation 81(11), 1405-1416.

5. Hussain, T. & Ahmad, M. (2012), Discrete inverse gamma distribution, 'Official Statistics and its Impact on Good Governance and Economy of Pakistan', 9th International Conference on Statistical Sciences, Islamic Countries Society of Statistical Sciences, Lahore, Pakistan, p. 381-394.

6. Hussain, T. & Ahmad, M. (2014), 'Discrete inverse rayleigh distribution', Pakistan Journal of Statistics 3(2), 203-222.

7. Inusah, S. & Kozubowski, J. (2006), 'A discrete analogue of the laplace distribution', Journal of Statistical Planning and Inference 136, 1090-1102.

8. Jazi, M., Lai, C. & Alamatsaz, M. (2010), 'A discrete inverse weibull distribution and estimation of its parameters', Statistical Methodology 7, 121-132.

9. Johnson, N., Kotz, S. & Kemp, A. (1992), Univariate Discrete Distributions, 2 edn, John Wiley and Sons, New York.

10. Kemp, A. (1997), 'Statistical methodology', Journal of Statistical Planning and Inference 63, 223-229.

11. Kemp, A. (2004), 'Classes of discrete lifetime distributions', Communications in Statistics Theory and Methods 33(12), 3069- 3093.

12. Kemp, A. (2006), 'The discrete half-normal distribution', Advances in mathematical and statistical modeling 9, 353-360.

13. Kielson, J. & Gerber, H. (1971), 'Some results for discrete unimodality', Journal of American Statistical Association 66, 386-389.

14. Kozubowski, J. & Inusah, S. (2006), 'A skew laplace distribution on integers', AISM 58, 555-571.

15. Krishna, H. & Pundir, P. (2007), 'Discrete maxwell distribution'. http://interstat.statjournals.net/YEAR/2007/articles/0711003.pdf (accessed in July 2010).

16. Krishna, H. & Pundir, P. (2009), 'Discrete burr and discrete pareto distributions', Statistical Methodology 6, 177-188.

17. Nakagawa, T. & Osaki, S. (1975), 'The discrete weibull distribution', IEEE Transactions on Reliability 24(5), 300-301.

18. Nekoukhou, V., Alamatsaz, M. & Bidram, H. (2012), 'A discrete analog of the generalized exponential distribution', Communication in Statistics Theory and Methods 41(11), 2000-2013.

19. Rainville, E. (1965), Special Functions, The Macmillan Company, New York.

20. Roy, D. (2003), 'Discrete normal distribution', Communication in Statistics Theory and Methods 32(10), 1871-1883.

21. Roy, D. (2004), 'Discrete rayleigh distribution', IEEE Transactions on Reliability 52(2), 255-260.

22. Szablowski, P. (2001), 'Discrete normal distribution and its relationship with jacobi theta functions', Statistics and Probability Letters 52, 289-299.

[Recibido en septiembre de 2014. Aceptado en marzo de 2015]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv39n1a04,
    AUTHOR  = {Hussain, Tassaddaq and Aslam, Muhammad and Ahmad, Munir},
    TITLE   = {{A Two Parameter Discrete Lindley Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2016},
    volume  = {39},
    number  = {1},
    pages   = {45-61}
}

References

Abouammoh, A. & Mashhour, A. (1981), ‘A note on the unimodality of discrete distributions’, Communication in Statistics Theory and Methods 10(13), 1345–1354.

Al-Huniti, A. & Al-Dayian, G. (2012), ‘Discrete burr type-iii distribution’, American Journal of Mathematics and Statistic 2(5), 145–152.

Chakraborty, S. & Chakravarty, D. (2012), ‘Discrete gamma distributions: Properties and parameters estimations’, Communication in Statistics Theory and Methods 41(18), 3301–3324.

Deniz, E. & Ojeda, E. (2011), ‘The discrete lindley distribution: Properties and application’, Journal of Statistical Computation and Simulation 81(11), 1405–1416.

Hussain, T. & Ahmad, M. (2012), Discrete inverse gamma distribution, in M. Ahmad,ed., ‘Official Statistics and its Impact on Good Governance and Economy of Pakistan’, 9th International Conference on Statistical Sciences, Islamic Countries Society of Statistical Sciences, Lahore, Pakistan, pp. 381–394.

Hussain, T. & Ahmad, M. (2014), ‘Discrete inverse rayleigh distribution’, Pakistan Journal of Statistics 3(2), 203–222.

Inusah, S. & Kozubowski, J. (2006), ‘A discrete analogue of the laplace distribution’, Journal of Statistical Planning and Inference 136, 1090–1102.

Jazi, M., Lai, C. & Alamatsaz, M. (2010), ‘A discrete inverse weibull distribution and estimation of its parameters’, Statistical Methodology 7, 121–132.

Johnson, N., Kotz, S. & Kemp, A. (1992), Univariate Discrete Distributions, 2 edn, John Wiley and Sons, New York.

Kemp, A. (1997), ‘Statistical methodology’, Journal of Statistical Planning and Inference 63, 223–229.

Kemp, A. (2004), ‘Classes of discrete lifetime distributions’, Communications in Statistics Theory and Methods 33(12), 3069– 3093.

Kemp, A. (2006), ‘The discrete half-normal distribution’, Advances in mathematical and statistical modeling 9, 353–360.

Kielson, J. & Gerber, H. (1971), ‘Some results for discrete unimodality’, Journal of American Statistical Association 66, 386–389.

Kozubowski, J. & Inusah, S. (2006), ‘A skew laplace distribution on integers’, AISM 58, 555–571.

Krishna, H. & Pundir, P. (2007), ‘Discrete maxwell distribution’. http://interstat.statjournals.net/YEAR/2007/articles/0711003.pdf (accessed in July 2010).

Krishna, H. & Pundir, P. (2009), ‘Discrete burr and discrete pareto distributions’, Statistical Methodology 6, 177–188.

Nakagawa, T. & Osaki, S. (1975), ‘The discrete weibull distribution’, IEEE Transactions on Reliability 24(5), 300–301.

Nekoukhou, V., Alamatsaz, M. & Bidram, H. (2012), ‘A discrete analog of the generalized exponential distribution’, Communication in Statistics Theory and Methods 41(11), 2000–2013.

Rainville, E. (1965), Special Functions, The Macmillan Company, New York.

Roy, D. (2003), ‘Discrete normal distribution’, Communication in Statistics Theory and Methods 32(10), 1871–1883.

Roy, D. (2004), ‘Discrete rayleigh distribution’, IEEE Transactions on Reliability 52(2), 255–260.

Szablowski, P. (2001), ‘Discrete normal distribution and its relationship with jacobi theta functions’, Statistics and Probability Letters 52, 289–299.

How to Cite

APA

Aslam, M. and Ahmad, M. (2016). A Two Parameter Discrete Lindley Distribution. Revista Colombiana de Estadística, 39(1), 45–61. https://doi.org/10.15446/rce.v39n1.55138

ACM

[1]
Aslam, M. and Ahmad, M. 2016. A Two Parameter Discrete Lindley Distribution. Revista Colombiana de Estadística. 39, 1 (Jan. 2016), 45–61. DOI:https://doi.org/10.15446/rce.v39n1.55138.

ACS

(1)
Aslam, M.; Ahmad, M. A Two Parameter Discrete Lindley Distribution. Rev. colomb. estad. 2016, 39, 45-61.

ABNT

ASLAM, M.; AHMAD, M. A Two Parameter Discrete Lindley Distribution. Revista Colombiana de Estadística, [S. l.], v. 39, n. 1, p. 45–61, 2016. DOI: 10.15446/rce.v39n1.55138. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/55138. Acesso em: 21 apr. 2025.

Chicago

Aslam, Muhammad, and Munir Ahmad. 2016. “A Two Parameter Discrete Lindley Distribution”. Revista Colombiana De Estadística 39 (1):45-61. https://doi.org/10.15446/rce.v39n1.55138.

Harvard

Aslam, M. and Ahmad, M. (2016) “A Two Parameter Discrete Lindley Distribution”, Revista Colombiana de Estadística, 39(1), pp. 45–61. doi: 10.15446/rce.v39n1.55138.

IEEE

[1]
M. Aslam and M. Ahmad, “A Two Parameter Discrete Lindley Distribution”, Rev. colomb. estad., vol. 39, no. 1, pp. 45–61, Jan. 2016.

MLA

Aslam, M., and M. Ahmad. “A Two Parameter Discrete Lindley Distribution”. Revista Colombiana de Estadística, vol. 39, no. 1, Jan. 2016, pp. 45-61, doi:10.15446/rce.v39n1.55138.

Turabian

Aslam, Muhammad, and Munir Ahmad. “A Two Parameter Discrete Lindley Distribution”. Revista Colombiana de Estadística 39, no. 1 (January 1, 2016): 45–61. Accessed April 21, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/55138.

Vancouver

1.
Aslam M, Ahmad M. A Two Parameter Discrete Lindley Distribution. Rev. colomb. estad. [Internet]. 2016 Jan. 1 [cited 2025 Apr. 21];39(1):45-61. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/55138

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