We introduce and analyze a class of estimators for distribution parameters based on the relationship between the distribution function and the empirical distribution function. This class includes the nonlinear least squares estimator and the weighted nonlinear least squares estimator which has been used in parameter estimation for lifetime data (see e.g. [6, 8]) as well as the generalized nonlinear least squares estimator proposed in [3]. Sufficient conditions for consistency and asymptotic normality are given. Capability and limitations are illustrated by simulations.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1172,
author = {Mirta Ben\v si\'c},
title = {Properties of the generalized nonlinear least squares method applied for fitting distribution to data},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {35},
year = {2015},
pages = {75-94},
zbl = {1333.62064},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1172}
}
Mirta Benšić. Properties of the generalized nonlinear least squares method applied for fitting distribution to data. Discussiones Mathematicae Probability and Statistics, Tome 35 (2015) pp. 75-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1172/
[000] [1] P.A. Al-Baidhani and C.D. Sinclair, Comparison of methods of estimation of parameters of the Weibull distribution, Communications in Statistics - Simulation and Computation 16 (1987), 373-384. | Zbl 0621.62032
[001] [2] T.W. Anderson and D.A. Darling, Asymptotic theory of certain 'goodness of fit' criteria based on stochastic processes, Ann. Math. Statist. 23 (1952), 193-212. | Zbl 0048.11301
[002] [3] M. Benšić, Fitting distribution to data by a generalized nonlinear least squares method, Communications in Statistics - Simulation and Computation (2012), in press.
[003] [4] P. Billingsly, Convergence of Probability Measures (John Wiley & Sons, New York, 1968). | Zbl 0172.21201
[004] [5] J. Castillo and J. Daoudi, Estimation of the generalized Pareto distribution, Statistics and Probability Letters 79 (2009), 684-688. | Zbl 1156.62313
[005] [6] D. Jukić, M. Benšiś and R. Scitovski, On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution, Computational Statistics & Data Analysis 52 (2008), 4502-4511. | Zbl 05565032
[006] [7] D. Harris and L. Matyas, Introduction to the generalized method of moment estimation, Matyas, L. (Ed.), Generalized method of moment estimation (Cambridge University Press, Cambridge, 1999), 3-30.
[007] [8] D. Kundu and M.Z. Raqab, Generalized Rayleigh distribution: different methods of estimations, Computational Statistics & Data Analysis 49 (2005), 187-200. | Zbl 05374154
[008] [9] D. Kundu and M.Z. Raqab, Burr Type X distribution: Revisited, J. Prob. Stat. Sci. 4 (2006), 179-193.
[009] [10] J.F. Lawless, Statistical Models and Methods for Lifetime Data (Wiley, New York, 1982). | Zbl 0541.62081
[010] [11] A. Luceño, Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimator, Comp. Stat. & Data Anal. 51 (2006), 904-917. | Zbl 1157.62399
[011] [12] D.N.P. Murthy, M. Bulmer and J.A. Eccleston, Weibull model selection for reliability modelling, Reliability Engineering and System Safety 86 (2004), 257-267.
[012] [13] D. Pollard, The minimum distance method of testing, Metrika 27 (1980), 43-70. | Zbl 0425.62029
[013] [14] H. Rinne, The Weibull Distribution. A Handbook (Chapman & Hall/CRC, Boca Raton, 2009).
[014] [15] G.A.F. Seber and C.J. Wild, Nonlinear Regression (John Wiley & Sons, New York, 1989). | Zbl 0721.62062
[015] [16] R.L. Smith and J.C. Naylor, A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Biometrika 73 (1987), 67-90.
[016] [17] J.G. Surles and W.J. Padgett, Inference for reliability and stress-strength for a scaled Burr Type X distribution, Lifetime Data Analysis 7 (2001), 187-200. | Zbl 0984.62082
[017] [18] F.J. Torres, Estimation of parameters of the shifted Gompertz distribution using least squares, maximum likelihood and moments methods, J. Comp. and Appl. Math. 255 (2014), 867-877. | Zbl 1291.62057
[018] [19] J. Wolfowitz, Estimation by minimum distance method, Ann. Inst. Statisti. Math. 5 (1953), 9-23. | Zbl 0051.37004
[019] [20] J. Wolfowitz, The minimum distance method, Ann. Math. Statist. 28 (1957), 75-88. | Zbl 0086.35403