SpringerLink
Log in

Efficient Estimation of the PDF and the CDF of a Generalized Logistic Distribution

  • Published:
Annals of Data Science Aims and scope Submit manuscript

Abstract

The generalized logistic distribution is a useful extension of the logistic distribution, allowing for increasing and bathtub shaped hazard rates and has been used to model the data with a unimodal density. Here, we consider estimation of the probability density function and the cumulative distribution function of the generalized logistic distribution. The following estimators are considered: maximum likelihood estimator, uniformly minimum variance unbiased estimator (UMVUE), least square estimator, weighted least square estimator, percentile estimator, maximum product spacing estimator, Cramér–von-Mises estimator and Anderson–Darling estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies are also carried out to show that the maximum-likelihood estimator is better than the UMVUE and that the UMVUE is better than others. Finally, a real data set has been analyzed for illustrative purposes.

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
EUR 32.99 /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

Similar content being viewed by others

References

  1. Ahuja JC, Nash SW (1967) The generalized Gompertz-Verhulst family of distributions. Sankhyã: Indian J Stat, Series A (1961–2002) 29(2):141–156

    Google Scholar 

  2. Alizadeh M, Bagheri SF, Baloui Jamkhaneh E, Nadarajah S (2015a) Estimates of the PDF and the CDF of the exponentiated Weibull distribution. Braz J Probab Stat 29:695–716

    Article  Google Scholar 

  3. Alizadeh M, Rezaei S, Bagheri SF (2015b) On the estimation for the Weibull distribution. Ann Data Sci. doi:10.1007/s40745-015-0046-8

    Google Scholar 

  4. Alkasasbeh MR, Raqab MZ (2009) Estimation of the generalized logistic distribution parameters: comparative study. Stat Methodol 6:262–279

    Article  Google Scholar 

  5. Anderson TW, Darling DA (1952) Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes. Ann Math Stat 23:193–212

    Article  Google Scholar 

  6. Asrabadi BR (1990) Estimation in the Pareto distribution. Metrika 37:199–205

    Article  Google Scholar 

  7. Badar MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S (eds) Prog Sci Eng Compos. Tokyo, ICCM-IV, pp 1129–1136

    Google Scholar 

  8. Bagheri SF, Alizadeh M, Nadarajah S (2016) Efficient estimation of the PDF and the CDF of the exponentiated Gumbel distribution. J Commun Stat Simul Comput 45:339–361

    Article  Google Scholar 

  9. Bagheri SF, Alizadeh M, Nadarajah S, Deiri E (2014) Efficient estimation of the PDF and the CDF of the Weibull extension model. J Commun Stat Simul Comput. doi:10.1080/03610918.2014.894059

    Google Scholar 

  10. Balakrishnan N, Leung MY (1988) Means, variances and covariances of order statistics, BLUE’s for the Type-I generalized logistic distribution, and some applications. J Commun Stat Simul Comput 17:51–84

    Article  Google Scholar 

  11. Balakrishnan N (1990) Approximate maximum likelihood estimation for a generalized logistic distribution. J Stat Plan Inference 26:221–236

    Article  Google Scholar 

  12. Balakrishnan N (1992) Handbook of the logistic distribution, statistics: textbooks and monographs, vol 123. Marcel Dekker, New York

    Google Scholar 

  13. Balakrishnan N, Lee SK (1998) Order statistics from the Type III generalized logistic distribution and applications. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 17. North-Holland, Amsterdam, pp 127–155

    Google Scholar 

  14. Bratpvrbajgyran S (2012) A test of goodness of fit for Rayleigh distribution via cumulative residual entropy. In: Proceedings of the 8th world congress in probability and statistics

  15. Cheng RCH and Amin NAK (1979) Maximum product of spacings estimation with application to the lognormal distribution. University of Wales Institute of Science and Technology, Cardiff, Math. Report 79-1

  16. Cheng RCH, Amin NAK (1983) Estimating parameters in continuous univariate distributions with a shifted origin. J R Stat Soc Ser B 45:394–403

    Google Scholar 

  17. Dattner I, Reiser B (2013) Estimation of distribution functions in measurement error models. J Stat Plan Inference 143:479–493

    Article  Google Scholar 

  18. Dixit UJ, Jabbari Nooghabi M (2010) Efficient estimation in the Pareto distribution. Stat Methodol 7:687–691

    Article  Google Scholar 

  19. Dixit UJ, Jabbari Nooghabi M (2011) Efficient estimation in the Pareto distribution with the presence of outliers. Stat Methodol 8:340–355

    Article  Google Scholar 

  20. Durot C, Huet S, Koladjo F, Robin S (2013) Least-squares estimation of a convex discrete distribution. Comput Stat Data Anal 67:282–298

    Article  Google Scholar 

  21. Duval C (2013) Density estimation for compound Poisson processes from discrete data. Stoch Process Their Appl 123:3963–3986

    Article  Google Scholar 

  22. Er GK (1998) A method for multi-parameter PDF estimation of random variables. Struct Saf 20:25–36

    Article  Google Scholar 

  23. Gastwirth JL (1972) The estimation of the Lorenz curve and Gini index. Rev Econ Stat 54:306316

    Article  Google Scholar 

  24. Hampel D (2008) Estimation of differential entropy for positive random variables and its application in computational neuroscience. In: Bellomo N (ed) Modeling and simulation in science. engineering and technology, vol 2. Birkhuser, Boston, p 213224

    Google Scholar 

  25. Hartley HO (1945) Note on the calculation of the distribution of the estimate of mean deviation in normal samples. Biometrika 33:257258

    Article  Google Scholar 

  26. Hsieh HK (1990) Estimating the critical time of the inverse Gaussian hazard rate. IEEE Trans Reliab 39:342345

    Article  Google Scholar 

  27. Hurvich CM, Shumway R, Tsai CL (1990) Improved estimators of Kullback–Leibler information for autoregressive model selection in small samples. Biometrika 77:709719

    Google Scholar 

  28. Jabbari Nooghabi M, Jabbari Nooghabi H (2010) Efficient estimation of PDF, CDF and rth moment for the exponentiated Pareto distribution in the presence of outliers. Statistics 44:1–20

    Article  Google Scholar 

  29. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distribution, vol 1, 2nd edn. Wiley, New York

    Google Scholar 

  30. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd edn. Wiley, New York

    Google Scholar 

  31. Kao JHK (1958) Computer methods for estimating Weibull parameters in reliability studies. Trans IRE-Reliab Qual Control 13:15–22

    Article  Google Scholar 

  32. Kao JHK (1959) A graphical estimation of mixed Weibull parameters in life testing electron tubes. Technometrics 1:389–407

    Article  Google Scholar 

  33. Kayal S, Kumar S, Vellaisamy P (2013) Estimating the Rnyi entropy of several exponential populations. Braz J Probab Stat 29(1):94111

    Google Scholar 

  34. Lin CT, Wu SJS, Balakrishnan N (2003) Parameter estimation for the linear hazard rate distribution based on records and inter-record times. Commun Stat-Theory Methods 32:729–748

    Article  Google Scholar 

  35. Mann NR, Schafer RE, Singpurwalla ND (1974) Methods for statistical analysis of reliability and life data. Wiley, New York

    Google Scholar 

  36. Mielniczuk J, Wojtys M (2010) Estimation of Fisher information using model selection. Metrika 72:163187

    Article  Google Scholar 

  37. Nilsson M, Kleijn WB (2007) On the estimation of differential entropy from data located on embedded manifolds. IEEE Trans Inf Theory 53:2330–2341

    Article  Google Scholar 

  38. Olapade AK (2004) On extended Type-I generalized logistic distribution. Int J Math Math Sci 57:3069–3074

    Article  Google Scholar 

  39. Przybilla C, Fernandez-Canteli A, Castillo E (2013) Maximum likelihood estimation for the three-parameter Weibull cdf of strength in presence of concurrent flaw populations. J Eur Ceram Soc 33:1721–1727

    Article  Google Scholar 

  40. Saleh Ehsanes Md AK, Hassanein KM, Masoom Ali M (1988) Estimation and testing of hypotheses about the quantile function of the normal distribution. J Inf Optim Sci 9:8598

    Google Scholar 

  41. Saunders SC, Myhre JM (1983) Maximum likelihood estimation for two-parameter decreasing hazard rate distributions using censored data. J Am Stat Assoc 78:664–673

    Article  Google Scholar 

  42. Suzuki G (1965) A consistent estimator for the mean deviation of the Pearson type distribution. Ann Inst Stat Math 17:271–285

    Article  Google Scholar 

  43. Swain J, Venkatraman S, Wilson J (1988) Least squares estimation of distribution function in Johnson’s translation system. J Stat Comput Simul 29:71–297

    Article  Google Scholar 

  44. Wang QJ (1990) Unbiased estimation of probability weighted moments and partial probability weighted moments from systematic and historical flood information and their application to estimating the GEV distribution. J Hydrol 120:115–124

    Article  Google Scholar 

  45. Woo JS, Yoon GE (2001) Estimations of Lorenz curve and Gini index in a Pareto distribution. Commun Stat Appl Methods 8:249–256

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the referees for careful reading and for valuable comments that greatly improved the article. The research work of Yogesh Mani Tripathi is partly supported by a grant SR/S4/MS/785/12 from Department of Science & Technology, India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Patna, Bihta, 801103, India

    Yogesh Mani Tripathi & Amulya Kumar Mahto

  2. Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, 793001, India

    Sanku Dey

Authors
  1. Yogesh Mani Tripathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amulya Kumar Mahto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sanku Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yogesh Mani Tripathi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tripathi, Y.M., Mahto, A.K. & Dey, S. Efficient Estimation of the PDF and the CDF of a Generalized Logistic Distribution. Ann. Data. Sci. 4, 63–81 (2017). https://doi.org/10.1007/s40745-016-0093-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40745-016-0093-9

Keywords

Mathematics Subject Classification

Search

Navigation