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Exact properties of the maximum likelihood estimator in exponential regression models: a differential geometric approach

Using recently developed methods for obtaining exact distribution results for implicitly defined estimators, we study the exact properties of the maximum likelihood estimator in exponential regression models. The main technical problem is the evaluation of a surface integral over an n-k)-dimensional hyperplane embedded in the n-dimensional sample space. Details of the calculation are given for the cases k = 1 and k = 2 and some general properties of the densities for arbitrary k are indicated.