Summary
The existence of an estimator constrained to lie in a certain type of bounded set is established for a fairly wide class of probability density functions. The necessary and sufficient conditions thus obtained provide a convenient means of finding such an estimator by mathematical programming methods. This result is a generalization of Cramer’s demonstration of the existence of an unconstrained maximum likelihood estimator and of Aitchison and Silvey’s demonstration of the existence of a maximum likelihood estimator constrained to satisfy certain equations.
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References
J. Aitchison and S. D. Silvey, “Maximum likelihood estimation of parameters subject to restraints,”Ann. Math. Statist., 29 (1958), 813–828.
H. W. Kuhn and A. W. Tucker, “Non-linear programming,”Second Berkeley Symposium on Math. Stats, and Prob., Univ. of Calif. Press, (1951), 481–492.
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Hanson, M.A. Inequality constrained maximum likelihood estimation. Ann Inst Stat Math 17, 311–321 (1965). https://doi.org/10.1007/BF02868175
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DOI: https://doi.org/10.1007/BF02868175