Skip to main content
SpringerLink
Log in

The Pareto–Poisson Distribution: Characteristics, Estimations and Engineering Applications

  • Published:
Sankhya A Aims and scope Submit manuscript

A Correction to this article was published on 12 January 2023

This article has been updated

Abstract

A new three-parameter lifetime distribution based on compounding Pareto and Poisson distributions is introduced and discussed. Various statistical and reliability properties of the proposed distribution including: quantiles, ordinary moments, median, mode, quartiles, mean deviations, cumulants, generating functions, entropies, mean residual life, order statistics and stress-strength reliability are obtained. In presence of data collected under Type-II censoring, from frequentist and Bayesian points of view, the model parameters are estimated. Using independent gamma priors, Bayes estimators against the squared-error, linear-exponential and general-entropy loss functions are developed. Based on asymptotic properties of the classical estimators, asymptotic confidence intervals of the unknown parameters are constructed using observed Fisher’s information. Since the Bayes estimators cannot be obtained in closed-form, Markov chain Monte Carlo techniques are considered to approximate the Bayes estimates and to construct the highest posterior density intervals. A Monte Carlo simulation study is conducted to examine the performance of the proposed methods using various choices of effective sample size. To highlight the perspectives of the utility and flexibility of the new distribution, two numerical applications using real engineering data sets are investigated and showed that the proposed model fits well compared to other eleven lifetime models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8

Similar content being viewed by others

Change history

References

  • Aarset, M.V. (1987). How to identify a bathtub hazard rate. IEEE Trans. Reliab. 36, 1, 106–108.

    Article  MATH  Google Scholar 

  • Adamidis, K. and Loukas, S. (1998). A lifetime distribution with decreasing failure rate. Stat. Probab. Lett. 39, 35–42.

    Article  MATH  Google Scholar 

  • Amigó, J.M., Balogh, S.G. and Hernández, S. (2018). A brief review of generalized entropies. Entropy 20, 11, 813.

    Article  Google Scholar 

  • Asgharzadeh, A., Bakouch, H.S. and Esmaeili, L. (2013). Pareto Poisson–Lindley distribution with applications. J. Appl. Stat. 40, 8, 1717–1734.

    Article  MATH  Google Scholar 

  • Barreto-Souza, W. and Cribari-Neto, F. (2009). A generalization of the exponential-Poisson distribution. Stat. Probab. Lett. 79, 24, 2493–2500.

    Article  MATH  Google Scholar 

  • Berger, J.O. (2013). Statistical Decision Theory and Bayesian Analysis. Springer Science and Business Media.

  • Chen, M.H. and Shao, Q.M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat. 8, 69–92.

    Google Scholar 

  • De Morais, A.L. (2009). A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series. Dissertação de mestrado–Universidade Federal de Pernambuco. CCEN.

  • Elbatal, I., Zayed, M., Rasekhi, M. and Butt, N.S. (2017). The exponential Pareto power series distribution: Theory and applications. Pak. J. Stat. Oper. Res., 603–615.

  • Elshahhat, A., Aljohani, H.M. and Afify, A.Z. (2021). Bayesian and classical inference under type-II censored samples of the extended inverse Gompertz distribution with engineering applications. Entropy 23, 12, 1578.

    Article  Google Scholar 

  • Gelman, A. and Rubin, D.B. (1992). Inference from iterative simulation using multiple sequence. Stat. Sci. 7, 457–511.

    Article  MATH  Google Scholar 

  • Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A. and Rubin, D.B. (2004). Bayesian Data Analysis, 2nd edn. Chapman and Hall/CRC, USA.

    MATH  Google Scholar 

  • Glaser, R.E. (1980). Bathtub and related failure rate characterizations. J. Amer. Stat. Assoc. 75, 667–672.

    Article  MATH  Google Scholar 

  • Gupta, R.C., Gupta, R.D. and Gupta, P.L. (1998). Modeling failure time data by Lehman alternatives. Commun. Stat.-Theory Methods 27, 4, 887–904.

    Article  MATH  Google Scholar 

  • Gupta, R.D. and Kundu, D. (2001). Generalized exponential distribution: different method of estimations. J. Stat. Comput. Simul. 69, 4, 315–337.

    Article  MATH  Google Scholar 

  • Henningsen, A. and Toomet, O. (2011). maxlik: A package for maximum likelihood estimation in R. Comput. Stat. 26, 3, 443–458.

    Article  MATH  Google Scholar 

  • Johnson, N., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd edn. Wiley, New York.

    MATH  Google Scholar 

  • Jorgensen, B. (2012). Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer, New York.

    Google Scholar 

  • Kundu, D. (2008). Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics 50, 2, 144–154.

    Article  Google Scholar 

  • Lawless, J.F. (2003). Statistical Models and Methods For Lifetime Data, 2nd edn. Wiley, New Jersey.

    MATH  Google Scholar 

  • Kuş, C. (2007). A new lifetime distribution. Comput. Stat. Data Anal.51, 9, 4497–4509.

    Article  MATH  Google Scholar 

  • Lu, W. and Shi, D. (2012). A new compounding life distribution: The Weibull–Poisson distribution. J. Appl. Stat. 39, 1, 21–38.

    Article  MATH  Google Scholar 

  • Mahdavi, A. and Kundu, D. (2017). A new method for generating distributions with an application to exponential distribution. Commun. Stat.-Theory Methods46, 13, 6543–6557.

    Article  MATH  Google Scholar 

  • Marinho, P.R.D., Silva, R.B., Bourguignon, M., Cordeiro, G.M. and Nadarajah, S. (2019). AdequacyModel: an R package for probability distributions and general purpose optimization. PLoS ONE. https://doi.org/10.1371/journal.pone.0221487.

  • Maurya, S.K. and Nadarajah, S. (2021). Poisson generated family of distributions: A review. Sankhya B 83, 2, 484–540.

    Article  MATH  Google Scholar 

  • Murthy, D.N.P., Xie, M. and Jiang, R. (2004). Weibull models Wiley series in probability and statistics. Wiley, Hoboken.

    Google Scholar 

  • Nadarajah, S. (2005). Exponentiated Pareto distributions. Statistics39, 255–260.

    Article  MATH  Google Scholar 

  • Nadarajah, S., Cancho, V.G. and Ortega, E.M. (2013). The geometric exponential Poisson distribution. JISS 22, 3, 355–380.

    Article  MATH  Google Scholar 

  • Nassar, M. and Nada, N. (2013). A new generalization of the Pareto–geometric distribution. J. Egypt. Math. Soc. 21, 2, 148–155.

    Article  MATH  Google Scholar 

  • Plummer, M., Best, N., Cowles, K. and Vines, K. (2006). CODA: Convergence diagnosis and output analysis for MCMC. R news. 6, 7–11.

    Google Scholar 

  • Ristić, M.M. and Nadarajah, S. (2014). A new lifetime distribution. J. Stat. Comput. Simul. 84, 1, 135–150.

    Article  MATH  Google Scholar 

  • Subhradev, S.E.N., Korkmaz, M.C. and Yousof, H.M. (2018). The quasi xgamma-Poisson distribution: Properties and Application. Istatistik Journal of The Turkish Statistical Association 11, 3, 65–76.

    MATH  Google Scholar 

  • Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18, 3, 293–297.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technology and Development, Zagazig University, Zagazig, 44519, Egypt

    Ahmed Elshahhat

  2. Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613, Egypt

    EL-Sayed A. El-Sherpieny & Amal S. Hassan

Authors
  1. Ahmed Elshahhat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. EL-Sayed A. El-Sherpieny
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amal S. Hassan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Elshahhat.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: This article was originally published with misplaced figures.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elshahhat, A., El-Sherpieny, ES.A. & Hassan, A.S. The Pareto–Poisson Distribution: Characteristics, Estimations and Engineering Applications. Sankhya A 85, 1058–1099 (2023). https://doi.org/10.1007/s13171-022-00302-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-022-00302-6

Keywords and phrases

AMS (2000) subject classification

Search

Navigation