Skip to main content
Springer Nature Link
Log in

Use of the heuristic optimization in the parameter estimation of generalized gamma distribution: comparison of GA, DE, PSO and SA methods

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

The generalized gamma distribution (GGD) is a popular distribution because it is extremely flexible. Due to the density function structure of GGD, estimating the parameters of the GGD family by statistical point estimation techniques is a complicated task. In other words, for the parameter estimation, the maximizing likelihood function of GGD is a problematic case. Hence, alternative approaches can be used to obtain estimators of GGD parameters. This paper proposes an alternative parameter estimation method for GGD by using the heuristic optimization approaches such as Genetic Algorithms (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), and Simulated Annealing (SA). A comparison between different modern heuristic optimization methods applied to maximize the likelihood function for parameter estimation is presented in this paper. The paper also investigates both the performance of heuristic methods and estimation of GGD parameters. Simulations show that heuristic approaches provide quite accurate estimates. In most of the cases, DE has better performance than other heuristics in terms of bias values of parameter estimations. Besides, the usefulness of an alternative parameter estimation method for GGD using the heuristic optimization approach is illustrated with a real data set.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Aksoy H (2000) Use of gamma distribution in hydrological analysis. Turk J Eng Environ Sci 24(6):419–428

    Google Scholar 

  • Bai J, Jakeman AJ, McAleer M (1991) A new approach to maximum likelihood estimation of the three-parameter gamma and Weibull distributions. Aust J Stat 33:397–410

    MATH  Google Scholar 

  • Bain L, Antle C (1967) Estimation of parameters in the Weibull distribution. Technometrics 9(3):621–627

    MathSciNet  MATH  Google Scholar 

  • Bard Y (1970) Comparison of gradient methods for the solution of nonlinear parameter estimation problems. SIAM J Numer Anal 7(1):157–186

    MathSciNet  MATH  Google Scholar 

  • Barnett VD (1966) Evaluation of the maximum likelihood estimators when the likelihood equation has multiple roots. Biometrika 53:151–165

    MathSciNet  MATH  Google Scholar 

  • Bowman KO, Shenton LR (1988) Properties of estimators for the gamma distribution. Marcel Dekker, Inc., New York

    MATH  Google Scholar 

  • Brownlee J (2011) Clever algorithms: nature-inspired programming recipes. Creative Commons, Melbourne

    Google Scholar 

  • Chandrasekar K, Ramana NV (2012) Performance comparison of GA, DE, PSO and SA approaches in enhancement of total transfer capability using facts devices. J Electr Eng Technol 7(4):493–500

    Google Scholar 

  • Cohen AC, Whitten BJ (1988) Parameter estimation in reliability and life span models. Marcel Dekker, New York

    MATH  Google Scholar 

  • Comtois P (2000) The gamma distribution as the true aerobiological probability density function (PDF). Aerobiologia 16:171–176

    Google Scholar 

  • Das S, Abraham A, Konar A (2008) Particle swarm optimization and differential evolution algorithms: technical analysis, applications and hybridization perspectives. Comput Intell 116:1–38

    Google Scholar 

  • Dubey S (1965) Asymptotic properties of several estimators of Weibull parameters. Technometrics 7(3):423–434

    MATH  Google Scholar 

  • Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of 6th international symposium on micro machine and human science, Nagoya, Japan, IEEE Service Center, Piscataway NJ, pp 39–43

  • Eberhart R, Shi Y (2001) Particle swarm optimization: developments, applications and resources. Proc IEEE Int Cong Evolut Comput 1:81–86

    Google Scholar 

  • Fisher RA (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc Lond Ser A 222:309–368

    MATH  Google Scholar 

  • Gaskó N, Dumitrescu D, Lung RI (2011) evolutionary detection of berge and nash equilibria. In: Pelta DA, Krasnogor N, Dumitrescu D, Chira C, Lung R (eds) Nature inspired cooperative strategies for optimization (NICSO 2011). Studies in computational intelligence, vol 387. Springer, Berlin

    Google Scholar 

  • Ghitany ME, Atieh B, Nadarajah S (2008) Lindley distribution and its application. Math Comput Simul 78(4):493–506

    MathSciNet  MATH  Google Scholar 

  • Goldberg DE (1989) Genetic algorithms in search optimization and machine learning. Addison-Wesley, Boston

    MATH  Google Scholar 

  • Gomes O, Combes C, Dussauchoy A (2008) Parameter estimation of the generalized gamma distribution. Math Comput Simul 79:955–963

    MathSciNet  MATH  Google Scholar 

  • Hager HW, Bain LJ (1970) Inferential procedures for the generalized gamma distribution. J Am Stat Assoc 65:1601–1609

    MATH  Google Scholar 

  • Hager HW, Bain LJ, Antle CE (1971) Reliability estimation for the generalized gamma distribution and robustness of the Weibull model. Technometrics 13:547–557

    MATH  Google Scholar 

  • Hasanien HM, Muyeen SM (2012) Design optimization of controller parameters used in variable speed wind energy conversion system by genetic algorithms. IEEE Trans Sustain Energy 3(2):200–208

    Google Scholar 

  • Holland J (1975) Adaptation in natural and artificial systems. Michigan Press, Michigan

    Google Scholar 

  • Huang PH, Hwang TY (2006) On new moment estimation of parameters of the generalized gamma distribution using it’s characterization. Taiwan J Math 10:1083–1093

    MathSciNet  MATH  Google Scholar 

  • Husak GJ, Michaelsen J, Funk C (2007) Use of the gamma distribution to represent monthly rainfall in Africa for drought monitoring applications. Int J Climatol 27(7):935–944

    Google Scholar 

  • Jones B, Waller WG, Feldman A (1978) Root isolation using function values. BIT 18:311–319

    MathSciNet  MATH  Google Scholar 

  • Kasprzyk I, Walanus A (2014) Gamma, Gaussian and logistic distribution models for airborne pollen grains and fungal spore season dynamics. Aerobiologia 30:369–383

    Google Scholar 

  • Kennedy J, Eberhart R (1995) Particle swarm optimization. Proc IEEE Int Conf Neural Netw 4:1942–1948

    Google Scholar 

  • Kennedy J, Mendes R (2002) Population structure and particle swarm performance. In: Proceedings of the 2002 congress on evolutionary computation, 2002. CEC’02, vol 2. IEEE, pp 1671–1676. https://doi.org/10.1109/CEC.2002.100449

  • Khodabin M, Ahmadabadi A (2010) Some properties of generalized gamma distribution. J Math Sci 4:9–28

    MathSciNet  MATH  Google Scholar 

  • Kirkpatrick S (1984) Optimization by simulated annealing-quantitative studies. J Stat Phys 34:975–986

    MathSciNet  Google Scholar 

  • Kirkpatrick S, Gerlatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680

    MathSciNet  MATH  Google Scholar 

  • Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, New Jersey

    MATH  Google Scholar 

  • Lawless JF (1980) Inference in the generalized gamma and log gamma distributions. Technometrics 22:409–419

    MathSciNet  MATH  Google Scholar 

  • Lawless JE (1982) Statistical models and methods for lifetime data. Wiley, New York

    MATH  Google Scholar 

  • Lehman E (1962) Shapes, moments and estimators of the Weibull distribution. IEEE Trans Reliab 11(3):32–38

    Google Scholar 

  • Marini F, Walczak B (2015) Particle swarm optimization (PSO): a tutorial. Chemom Intell Lab Syst 149:153–165

    Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1091

    MATH  Google Scholar 

  • Örkcü HH, Aksoy E, Dogan MI (2015a) Estimating the parameters of 3-p Weibull distribution through differential evolution. Appl Math Comput 251:211–224

    MathSciNet  MATH  Google Scholar 

  • Örkcü HH, Özsoy VS, Aksoy E, Dogan MI (2015b) Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison. Appl Math Comput 268:201–226

    MathSciNet  MATH  Google Scholar 

  • Özsoy VS, Örkcü HH, Bal H (2018) Particle swarm optimization applied to parameter estimation of the four-parameter burr III distribution. Iran J Sci Technol Trans A Sci 42(2):895–909

    MathSciNet  MATH  Google Scholar 

  • Parr VB, Webster JT (1965) A method for discriminating between failure density functions used in reliability predictions. Technometrics 7:1–10

    Google Scholar 

  • Puthenpura S, Sinha NK (1986) Modified maximum likelihood method for the robust estimation of system parameters from very noisy data. Automatica 22:231–235

    MATH  Google Scholar 

  • Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. In Proceedings of the 14th annual conference on computer graphics and interactive techniques, pp 25–34

  • Ross R (1994) Graphical methods for plotting and evaluating Weibull distributed data. In: Proceedings of the 4th international conference on properties and applications dielectric materials, Brisbane, Australia, pp 250–253

  • Shanker R, Shukla KK (2016) On modelling of lifetime data using three-parameter generalized lindley and generalized gamma distributions. Biometrics Biostat Int J 4(5):00107

    Google Scholar 

  • Shi YH, Eberhart RC (1998) A modified particle swarm optimizer. IEEE international conference on evolutionary computation, Anchorage Alaska, pp 69–73

  • Singh A, Singh AK, Iaci JR (2002) Estimation of the exposure point concentration term using a gamma distribution. Technology Support Center Issue EPA/600/R-02/084

  • Stacy EW, Mihram GA (1965) Parameter estimation for a generalized gamma distributions. Technometrics 7:349–358

    MathSciNet  MATH  Google Scholar 

  • Storn R, Price K (1995) Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical report TR-95-012. International Computer Science Institute, Berkeley

    Google Scholar 

  • Storn R, Price K (1997) Differential evolution a simple and effcient heuristic for global optimization over continuous spaces. J Global Optim 11(341):359

    MATH  Google Scholar 

  • Tadikamalla PR (1979) Random sampling from the generalized gamma distribution. Computing 23:199–203

    MathSciNet  MATH  Google Scholar 

  • Vaughan DC (1992) On the Tiku-Suresh method of estimation. Commun Stat Theory Methodol 21:451–469

    MathSciNet  MATH  Google Scholar 

  • Von Neumann J (1951) Various techniques used in connection with random digits. Paper No. 13 in “Monte Carlo method”. NBS Appl Math Series No. 12 U.S. Government Printing Office

  • Wang F (2014) Using BBPSO algorithm to estimate the Weibull parameters with censored data. Commun Stat Simul Comput 43:2614–2627

    MathSciNet  MATH  Google Scholar 

  • White JS (1969) The moments of log-Weibull order statistic. Technometrics 11(2):373–386

    MathSciNet  MATH  Google Scholar 

  • Wingo DR (1987) Computing maximum-likelihood parameter estimates of the generalized gamma distribution by numerical root isolation. IEEE Trans Reliab 5:586–590

    MATH  Google Scholar 

  • Yilmaz H, Sazak HS (2014) Double-looped maximum likelihood estimation for the parameters of the generalized gamma distribution. Math Comput Simul 98:18–30

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Finance, Banking and Insurance, Aksaray University, Aksaray, Turkey

    Volkan Soner Özsoy

  2. Department of Statistics, Faculty of Art and Sciences, Uşak University, Uşak, Turkey

    Mehmet Güray Ünsal

  3. Department of Statistics, Faculty of Sciences, Gazi University, Ankara, Turkey

    H. Hasan Örkcü

Authors
  1. Volkan Soner Özsoy
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mehmet Güray Ünsal
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. H. Hasan Örkcü
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Volkan Soner Özsoy.

Additional information

Publisher's Note

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

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 620 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Özsoy, V.S., Ünsal, M.G. & Örkcü, H.H. Use of the heuristic optimization in the parameter estimation of generalized gamma distribution: comparison of GA, DE, PSO and SA methods. Comput Stat 35, 1895–1925 (2020). https://doi.org/10.1007/s00180-020-00966-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-020-00966-4

Keywords