Brazilian Journal of Probability and Statistics

A new weighted Lindley distribution with application

A. Asgharzadeh, Hassan S. Bakouch, S. Nadarajah, and F. Sharafi

Full-text: Open access

Abstract

The Lindley distribution has been generalized by many authors in recent years. Here, we introduce a new generalization that provides better fits than the Lindley distribution and all of its known generalizations. The distribution contains Lindley and weighted Lindley (Ghitany et al. (Math. Comput. Simulation 81 (2011) 1190–1201)) distributions as special cases. Also, the distribution can be represented as a mixture of weighted exponential (Gupta and Kundu (Statistics 43 (2009) 621–634)) and weighted gamma distributions, and as a negative mixture of Lindley distributions with different parameters. Various properties of the distribution (including quantiles, moments, moment generating function, hazard rate function, mean residual lifetime, Lorenz curve, Gini index, Rényi entropy and Mathai–Haubold entropy) are derived. Maximum likelihood estimators of the distribution parameters are derived and their behavior is assessed via simulation. Fisher’s information matrix and asymptotic confidence intervals for the distribution parameters are given. Finally, a real data application is presented.

Article information

Source
Braz. J. Probab. Stat., Volume 30, Number 1 (2016), 1-27.

Dates
Received: April 2014
Accepted: June 2014
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1453211800

Digital Object Identifier
doi:10.1214/14-BJPS253

Mathematical Reviews number (MathSciNet)
MR3453512

Zentralblatt MATH identifier
1381.62038

Keywords
Estimation Gini index skewness weighted distributions

Citation

Asgharzadeh, A.; Bakouch, Hassan S.; Nadarajah, S.; Sharafi, F. A new weighted Lindley distribution with application. Braz. J. Probab. Stat. 30 (2016), no. 1, 1--27. doi:10.1214/14-BJPS253. https://projecteuclid.org/euclid.bjps/1453211800


Export citation

References

  • Asai, N., Kubo, I. and Kuo, H. H. (2001). Roles of log-concavity, log-convexity, and growth order in white noise analysis. Infinite Dimensional Analysis Quantum Probability and Related Topics 4, 59–84.
  • Asgharzadeh, A., Nadarajah, S. and Sharafi, F. (2014a). Generalized inverse Lindley distribution. Preprint.
  • Asgharzadeh, A., Sharafi, F. and Nadarajah, S. (2014b). Weibull Lindley distribution. Preprint.
  • Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171–178.
  • Bakouch, H. S., Al-Zahrani, B. M., Al-Shomrani, A. A., Marchi, V. A. A. and Louzada, F. (2012). An extended Lindley distribution. Journal of the Korean Statistical Society 41, 75–85.
  • Barreto-Souza, W. and Bakouch, H. S. (2013). A new lifetime model with decreasing failure rate. Statistics 47, 465–476.
  • Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154–161.
  • Egghe, L. (2002). Development of hierarchy theory for digraphs using concentration theory based on a new type of Lorenz curve. Mathematical and Computer Modelling 36, 587–602.
  • Ferguson, T. S. (1996). A Course in Large Sample Theory. London: Chapman & Hall.
  • Gail, M. H. (2009). Applying the Lorenz curve to disease risk to optimize health benefits under cost constraints. Statistics and Its Interface 2, 117–121.
  • Ghitany, M. E., Al-Mutairi, D. K., Balakrishnan, N. and Al-Enezi, L. J. (2013). Power Lindley distribution and associated inference. Computational Statistics and Data Analysis 64, 20–33.
  • Ghitany, M. E., Alqallaf, F., Al-Mutairi, D. K. and Husain, H. A. (2011). A two-parameter weighted Lindley distribution and its applications to survival data. Mathematics and Computers in Simulation 81, 1190–1201.
  • Ghitany, M. E., Atieh, B. and Nadarajah, S. (2008). Lindley distribution and its application. Mathematics and Computers in Simulation 78, 493–506.
  • Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association 75, 667–672.
  • Groves-Kirkby, C. J., Denman, A. R. and Phillips, P. S. (2009). Lorenz curve and Gini coefficient: Novel tools for analysing seasonal variation of environmental radon gas. Journal of Environmental Management 90, 2480–2487.
  • Gupta, R. D. and Kundu, D. (2009). A new class of weighted exponential distributions. Statistics 43, 621–634.
  • Han, M., Liang, Z. and Li, D. (2011). Sparse kernel density estimations and its application in variable selection based on quadratic Renyi entropy. Neurocomputing 74, 1664–1672.
  • Jelinek, H. F., Tarvainen, M. P. and Cornforth, D. J. (2012). Renyi entropy in identification of cardiac autonomic neuropathy in diabetes. In Proceedings of the 39th Conference on Computing in Cardiology 909–911. Krakow: IEEE.
  • Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley.
  • Koutras, V. P. (2011). Two-level software rejuvenation model with increasing failure rate degradation. In Dependable Computer Systems 101–115. New York: Springer.
  • Kreitmeier, W. and Linder, T. (2011). High-resolution scalar quantization with Renyi entropy constraint. IEEE Transactions on Information Theory 57, 6837–6859.
  • Lai, M.-T. (2013). Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit. International Journal of Reliability and Safety 7, 95–107.
  • Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Ser. B 20, 102–107.
  • Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association 9, 209–219.
  • MacGillivray, H. L. (1986). Skewness and asymmetry: Measures and orderings. The Annals of Statistics 14, 994–1011.
  • Maeda, K. and Nishikawa, M. (2006). Duration of party control in parliamentary and presidential governments: A study of 65 democracies, 1950 to 1998. Comparative Political Studies 39, 352–374.
  • Maldonado, A., Perez-Ocon, R. and Herrera, A. (2007). Depression and cognition: New insights from the Lorenz curve and the Gini index. International Journal of Clinical and Health Psychology 7, 21–39.
  • Mathai, A. M. and Haubold, H. J. (2008). On generalized distributions and pathways. Physics Letters A 372, 2109–2113.
  • Milkie, C. M. and Perakis, A. N. (2004). Statistical methods for planning diesel engine overhauls in the U.S. coast guard. Naval Engineers Journal 31–41.
  • Nadarajah, S. (2009). The skew logistic distribution. Advances in Statistical Analysis 93, 197–203.
  • Ninh, A. and Prekopa, A. (2013). Log-concavity of compound distributions with applications in stochastic optimization. Discrete Applied Mathematics 161, 3017–3027.
  • Popescu, T. D. and Aiordachioaie, D. (2013). Signal segmentation in time-frequency plane using Renyi entropy—Application in seismic signal processing. In Proceedings of the Second International Conference on Control and Fault-Tolerant Systems 312–317. Nice, France: IEEE.
  • Radice, A. (2009). Use of the Lorenz curve to quantify statistical nonuniformity of sediment transport rate. Journal of Hydraulic Engineering 135, 320–326.
  • Rényi, A. (1961). On measures of entropy and information. In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. I 547–561. Berkeley: Univ. California Press.
  • Saidane, S., Babai, M. Z., Aguir, M. S. and Korbaa, O. (2011). Spare parts inventory systems under an increasing failure rate demand interval distribution. In Proceedings of the 41st International Conference on Computers and Industrial Engineering 768–773. Los Angeles, CA.
  • Shakhatreh, M. K. (2012). A two-parameter of weighted exponential distributions. Statistics and Probability Letters 82, 252–261.
  • Sucic, V., Saulig, N. and Boashash, B. (2011). Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Renyi entropy. EURASIP Journal on Advances in Signal Processing 2011, 125.
  • Tsarouhas, P. H. and Arvanitoyannis, I. S. (2010). Reliability and maintainability analysis of bread production line. Critical Reviews in Food Science and Nutrition 50, 327–343.
  • Ugarte, M. D., Militino, A. F. and Arnholt, A. T. (2008). Probability and Statistics with R. London: Chapman & Hall.
  • Woosley, R. L. and Cossman, J. (2007). Drug development and the FDA’s critical path initiative. Clinical Pharmacology and Therapeutics 81, 129–133.
  • Zakerzadeh, H. and Dolati, A. (2009). Generalized Lindley distribution. Journal of Mathematical Extension 3, 13–25.