Discrimination Between Logistic and Gumbel Distribution

S. A. Al-Subh, M. T. Alodat


When two distributions have ,approximately ,the  same characteristics, it is often difficult to discriminate between them. In this study, we use the ratio of likelihoods for selecting between the logistic and Gumbel distributions for describing a set of data. The parameters for the logistic and Gumbel distributions are estimated by using maximum likelihood (ML), moments (MOM) and order statistic (OS) methods.  In addition, by using Monte Carlo simulations, discriminating between the two distributions is investigated in terms of the probability of correct selection (PCS) as found based on the different methods of estimation. In general, it is found that the method of ML outperforms all the other methods when the estimators considered are compared in term of efficiency.


Logistic distribution; Gumbel distribution; Discriminating; Maximum likelihood; Moment; Order statistic.

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A. Alzaid and K.S. Sultan." Discriminating between gamma and lognormal distributions with applications". Journal of King Saud University (Science), vol. 21, pp. 99-108, 2009.

A. K. Dey and D. Kundu. "Discriminating between the log-normal and log-logistic distributions ". Communications in Statistics- Theory and Methods, vol. 39, pp. 280-292, 2010.

D.R. Cox. "Tests of separate families of hypotheses. Proceedings of the Fourth Berkeley Symposium in Mathematical Statistics and Probability". University of California Press, Berkeley, pp. 105–123, 1961.

D. R. Cox." Further results on tests of separate families of hypotheses". J. of Royal Statistical Society, Ser. B, vol. 24, pp. 406–424,1962.

D. Kundu and M. Z. Raqab. "Discriminating between the generalized Rayleigh and Log-normal distribution". Statistics, Vol. 41(6), pp. 505-515, 2007.

D. Kundu and A. Manglick. Discriminating between the Weibull and Log-normal distributions. Navel Research Logistics, Vol. 51 (6), pp.893-905, 2004.

D. Kundu, R. D. Gupta and A.Manglick." Discriminating between Log-normal and generalized exponential distributions. Journal of Statistical planning and inference 127: 213-227, 2005.

F. Aucoin and F. Ashkar." Discriminating between the logistic and the normal distributions based on likelihood ratio". http://interstat.statjournals.net/YEAR/2010/articles/1008003.pdf, 2010.

G. Corsini, F. Gini, M. V. Greco and L. Verrazzani. "Cramer-Rao bounds and estimation of the parameters of the Gumbel distribution". Aerospace and Electronic Syst., Vol. 31 (3), 1202-1204,1995.

G.S. Mudholkar & A.D. Hutson. "LQ-moments: Analogs of L-moments". Journal of Statistical Planning and Inference, vol. 71, pp. 191-208,1998.

H.L. Harter & A.H. Moore. "Maximum-likelihood estimation, from censored samples, of the parameters of a logistic distribution". J. Amer. Statist. Assoc., vol. 62, pp. 675–684, 1967.

H. N Phien." A review of methods of parameter estimation for the extreme value type-1 distribution". Journal of Hydrology, vol. 90 (3–4), pp. 251–268,1987.

J. S. Kim, and B. J. Yum. "Selection between Weibull and lognormal distributions: A comparative simulation study". Computational Statistics and Data Analysis, vol. 53, pp. 477-485, 2008.

J. Maciunas, N. C. Matalas and J. R. Wallis. Probability weighted moments compared with some traditional techniques in estimating Gumbel Parameters and quantiles. Water Resources Research, Vol. 15(5), pp. 1055–1064, 1979.

M. Engelhardt." Simple Linear Estimation of the Parameters of the Logistic Distribution from a Complete or Censored Sample". Journal of the American Statistical Association, Vol. 70(352), pp. 899- 902, 1975.

M. Fiorentino and S. Gabriele." A correction for the bias of maximum-likelihood estimators of Gumbel parameters. Journal of Hydrology. Vol. 73 (1-2),pp. 39–49, 1984.

M. N. Leese. "Use of censored data in the estimation of Gumbel distribution parameters for annual maximum flood series". Water Resources Research. Vol. 9 (6), pp. 1534–1542,1973.

M. A. M. Mousa, Z. F. Jaheen and A.A Ahmad." Bayesian Estimation, Prediction and Characterization for the Gumbel Model Based on Records". A Journal of Theoretical and Applied Statistics, vol. 36(1), pp. 65-74, 2002.

N. Balakrishnan. Handbook of the logistic distribution. New York, Marcel Dekker, Inc, 1992.

N. Balakrishnan, S.S. Gupta & S. Panchapakesan. 1995. "Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples". Statistics, Vol. 27(1), pp. 127 – 142, 1995.

R. D. Gupta and D. Kundu." Generalized Exponential Distributions". Australian and New Zealand Journal of Statistics, vol. 41, pp. 173-188, 1999.

R. D. Gupta and D. Kundu. "Exponentiated Exponential Distribution; An alternative to gamma or Weibull distribution". Biometrical Journal,vol. 43, pp. 117-130,2001b.

R. D. Gupta and D. Kundu. "Generalized Exponential Distributions; Different Methods of Estimation". Journal of Statistical Computation and Simulation, vol. 69, pp.315-338, 2001a.

R. D. Gupta and D. Kundu. "Generalized exponential distributions". Statistical Inferences. Journal of Statistical Theory and Applications, vol. 1, pp. 101-118,2002.

R. D. Gupta and D. Kundu." Closeness of gamma and generalized exponential distributions". Commun. Stat.-Theory Methods, vol. 32, pp. 705–721,2003a.

R. D. Gupta and D. Kundu." Discriminating between Weibull and generalized exponential distributions". Comput. Stat. Data Anal., vol. 43, pp. 179–196, 2003b.

R. D. Gupta and D. Kundu. "Discriminating between gamma and generalized exponential distributions". Journal of Statistical Computations and Simulations, vol. 74, pp. 107–121,2004.

R.L. Prentice." Discrimination among Some Parametric Models". Biometrika, vol. 62 (3), pp. 607-614,1975.

W. G. Strupczewski, H. T. Mitosek, K. Kochanek, V. P. Singh and S. Weglarczyk. "Probability of correct selection from lognormal and convective diffusion models based on the likelihood ratio". Stochastic Environmental Research and Risk Assessment,Vol. 20(3), pp. 152-163,2006.

W. Abu-Dayyeh, S. Al-Subh and H. Muttlak. "Logistic parameters estimation using simple random sampling and ranked set sampling data". Applied Mathematics and Computation, vol. 150(2), pp. 543-554, 2004.


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