Abstract
In this article, maximum likelihood estimator(s) (MLE(s)) of the scale and shape parameters \(\alpha \) and \(\beta \) from log-logistic distribution will be respectively considered in cases when one parameter is known and when both are unknown under simple random sampling (SRS) and ranked set sampling (RSS). In addition, the MLE of one parameter, when another parameter is known using a RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. These MLEs will be compared in terms of asymptotic efficiencies. These MLEs based on RSS can be real competitors against those based on SRS. All efficiencies of these MLEs are simulated under imperfect ranking.
Similar content being viewed by others
References
Abu-Dayyeh W, Assrhani A, Ibrahim K (2013) Estimation of the shape and scale parameters of pareto distribution using ranked set sampling. Stat Pap 54(1):207–225
Ahmad MI, Sinclair CD, Werritty A (1988) Log-logistic flood frequency analysis. J Hydrol 98(3):205–224
Al-Saleh MF, Al-Hadhrami SA (2003) Estimation of the mean of the exponential distribution using moving extremes ranked set sampling. Stat Pap 44(3):367–382
Ashkar F, Mahdi S (2003) Comparison of two fitting methods for the Log-logistic distribution. Water Resour Res 39:12–17
Balakrishnan N, Malik HJ, Puthenpura S (1987) Best linear unbiased estimation of location and scale parameters of the log-logistic distribution. Commun Stat Theory Methods 16:3477–3495
Bennett S (1983) Log-logistic regression models for survival data. J R Stat Soc 32(2):165–171
Chen W, Xie M, Wu M (2013) Parametric estimation for the scale parameter for scale distributions using moving extremes ranked set sampling. Stat Probab Lett 83(9):2060–2066
Chen W, Xie M, Wu M (2016) Modified maximum likelihood estimator of scale parameter using moving extremes ranked set sampling. Commun Stat Simul Comput 45(6):2232–2240
Chen W, Tian Y, Xie M (2017) Maximum likelihood estimator of the parameter for a continuous one-parameter exponential family under the optimal ranked set sampling. J Syst Sci Complex 30(6):1350–1363
Chen Z (2006) Estimating the shape parameter of the log-logistic distribution. Int J Reliab Qual Saf Eng 13(3):257–266
Dell TR, Clutter JL (1972) Ranked set sampling theory with order statistics background. Biometrics 28(2):545–555
Fisk PR (1961) The graduation of income distributions. Econometrica 29(2):171–185
Geskus RB (2001) Methods for estimating the AIDS incubation time distribution when data of seroconversion is censored. Stat Med 20:795–812
Gupta RC, Akman O, Lvin S (1999) A study of log-logistic model in survival analysis. Biom J 41(4):431–443
Kus CS, Kaya MF (2006) Estimation of parameters of the log-logistic distribution based on progressive censoring using the EM algorithm. Hacettepe J Math Stat 35(2):203–211
Lesitha G, Thomas PY (2013) Estimation of the scale parameter of a log-logistic distribution. Metrika 76(3):427–448
McIntyre GA (1952) A method of unbiased selective sampling, using ranked sets. Aust J Agric Res 3:385–390
Omar A, Ibrahim K (2013) Estimation of the shape and scale parameters of the pareto distribution using extreme ranked set sampling. Pak J Stat 29(1):33–47
Reath J, Dong J, Wang M (2018) Improved parameter estimation of the log-logistic distribution with applications. Comput Stat 33(1):339–356
Robson A, Reed D (1999) Statistical procedures for flood frequency estimation. Flood estimation handbook. Institute of Hydrology, Wallingford, p 3
Shoukri MM, Mian IUM, Tracy D (1988) Sampling properties of estimators of log-logistic distribution with application to Canadian precipitation data. Can J Stat 16(3):223–236
Stokes L (1995) Parametric ranked set sampling. Ann Inst Stat Math 47(3):465–482
Tiku ML, Suresh RP (1992) A new method of estimation for location and scale parameters. J Stat Plan Inference 30:281–292
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, X., Chen, W. & Qian, W. Maximum likelihood estimators of the parameters of the log-logistic distribution. Stat Papers 61, 1875–1892 (2020). https://doi.org/10.1007/s00362-018-1011-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-018-1011-3