On the Existence and Uniqueness of the Maximum Likelihood Estimate of a Vector-Valued Parameter in Fixed-Size Samples

The maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the Hessian matrix is negative definite whenever the gradient vector vanishes. The condition of constancy on the boundary cannot be completely removed, cf. Tarone and Gruenhage (1975). The theory is illustrated with several examples.

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