Summary
This paper reviews recent developments in statistical catastrophe theory. A connection is established between a class of stochastic catastrophe models (the ‘cuspoid’ catastrophes, with Weiner input) and a class of regular exponential families, which are the stationary probability densities of the stochastic catastrophe models. These are called the exponential catastrophe densities. Parameter estimation is examined from the point of view of three methods: maximum likelihood, moments, and approximation theory. Special attention is given to the cusp densities, and a comparative example is presented. Then an inferential theory is presented, based on the likelihood ratio test. This test can be used on a hierarchy of catastrophe densities. At the base of the heirarchy are the familiar normal, gamma, and beta densities, while at the top are complex multimodal forms. The theory as presented has none of the topolological flavor of catastrophe theory, but the principle of invariance up to diffeomorphism is discussed in relation to the inferential theory.
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© 1981 D. Reidel Publishing Company
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Cobb, L. (1981). The Multimodal Exponential Families of Statistical Catastrophe Theory. In: Taillie, C., Patil, G.P., Baldessari, B.A. (eds) Statistical Distributions in Scientific Work. NATO Advanced Study Institutes Series, vol 79. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8549-0_4
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DOI: https://doi.org/10.1007/978-94-009-8549-0_4
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