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In the literature there have been many suggestions on how to parametrize models. Some properties you can seek are (1) stability of variance of the MLE; (2) normal likelihood; (3) zero asymptotic skewness of the MLE; (4) asymptotic unbiasedness of the MLE. The parametrizations corresponding to these demands are found in the one-dimensional curved exponential family. They all belong to a general class of transformations, but they are in general not identical. The transformations in this class are characterized by a differential equation. The transformations are identical in the non-linear normal regression model.
Journal of the Royal Statistical Society. Series B (Methodological) © 1982 Royal Statistical Society