577e5f63-c3af-44b6-886b-f93186a8024820210702055119248wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS2224-26781109-277710.37394/23202http://wseas.org/wseas/cms.action?id=4067129202112920212010.37394/23202.2021.20https://wseas.org/wseas/cms.action?id=23288The Necessary And Sufficient Condition For Consistency Of The MLEQiqingYuDepartment of Mathematical Sciences State University of New York Binghamton, NY 13902 USASuppose that the observations are i.i.d. from a density f(.; θ), where θ is an identifiable parameter. One expects that the maximum likelihood estimator of θ is consistent. But its consistency proof is non-trivial and various sufficient conditions have been proposed (see, e.g., the classical statistics textbooks). All these sufficient conditions require f(x; θ) being somewhat upper semi-continuous (in θ), with various smoothness conditions or conditions needed for the dominated convergence theorem. We study the sufficient and necessary condition.72202172202112413214https://wseas.com/journals/systems/2021/a285102-011(2021).pdf10.37394/23202.2021.20.14https://wseas.com/journals/systems/2021/a285102-011(2021).pdf10.1016/s0378-3758(99)00218-9Berlinet, A., Liese, F. and Vajda, I., Necessary and sufficient conditions for consistency of Mestimates in regression models with general errors, Journal of Statistical Planning and Inference, Vol.89, No.1-2, 2000, pp. 243-267. Royden, H.L., Real analysis, Macmillan, NY, 1968. Bickel, P.J. and Doksum, K.A., Mathematical Statistics, Holden-Day, Oakland, 1997. Casella, G. and Berger, R. Statistical inference 2nd Ed., Duxbury, NY, 2001. Ferguson, T.S., A course in large sample theory, Chapman & Hall, NY, 1996. Kullback, S. and Leibler, R. A., On information and sufficiency. Ann. Math. Stat., Vol.22, 1951, pp.79–86. Lehmann, E.L. and Casella, G., Theory of Point estimation 2nd edition, Springer-Verlag, NY, 1998. Rossi, R. J., Mathematical Statistics : An Introduction to Likelihood Based Inference, pp. 227. John Wiley & Sons, NY, 2018. Rudin, W., Principles of mathematical analysis, McGraw-Hill, NY, 1976. Stuart, A. Ord, J.K., and Arnold, S., Advanced Theory of Statistics, Vol. 2A: Classical Inference and the Linear Model, 6th edition. Oxford University Press, London, 1999. Van der Vaart, A. W., Asymptotic Statistics, Cambridge University Press, 1998. 10.1007/s00362-017-0928-2Zhang, J., Consistency of MLE, LSE and Mestimation under mild conditions. Statistical Papers Vol.61, 2017, pp.189-199. 10.2174/2666148902010010021Yu, Q.Q., Consistency Of The Semi-parametric MLE Under The Cox Model With Right- Censored Data, The Open Mathematics, Statistics and Probability Journal Vol.10 2020, pp.21-27. Yu, Q.Q., Technical Report on “The necessary and sufficient condition for consistency of the MLE”, 2021.