Summary
Let P={P θ: θ∈Θ} be an exponential family of probability distributions with θ the canonical parameter and consider the one to one mapping ϕ: P θ→θ. It is shown that, under mild regularity assumptions, ϕ and ϕ −1 are continuous with respect to the Lévy metric in P and Euclidean metric in θ.
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Reference
Nölle, G.: Zur Theorie der bedingten Tests. Inaugural-Dissertation an der Universität Münster (1966).
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Barndorff-Nielsen, O. Lévy homeomorphic parametrization and exponential families. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 56–58 (1969). https://doi.org/10.1007/BF00538522
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DOI: https://doi.org/10.1007/BF00538522