Abstract
This chapter presents the notion of statistical models, a structure associated with a family of probability distributions, which can be given a geometric structure. This chapter deals with statistical models given parametrically. By specifying the parameters of a distribution, we determine a unique element of the family. When the family of distributions can be described smoothly by a set of parameters, this can be considered as a multidimensional surface. We are interested in the study of the properties that do not depend on the choice of model coordinates.
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Notes
- 1.
There are several ways of fitting data to a distribution. For instance, minimizing the Kullback–Leibler divergence or the Hellinger distance are just a couple of ways of doing this.
- 2.
Some authors consider a more general definition for parametric models, without requiring properties (i) and (ii). All examples provided in this book satisfy relations (i) and (ii); we required them here for smoothness reasons.
- 3.
The reason why f takes values in \(\mathcal{F}(\mathcal{X}, \mathbb{R})\) is because we consider f(p ξ ) as a real-valued function of x, \(x \in \mathcal{X}\).
- 4.
This follows the 1940s idea of C. R. Rao and H. Jeffreys to use the Fisher information to define a Riemannian metric.
Bibliography
B.S. Clarke, A.R. Barron, Information theoretic asumptotics of Bayes methods. IEEE Trans. Inform. Theory 36, 453–471 (1990)
J.M. Corcuera, F. Giummolé, A Characterization of Monotone and Regular Divergences (Annals of the Institute of Statistical Mathematics, 1998)
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Calin, O., Udrişte, C. (2014). Statistical Models. In: Geometric Modeling in Probability and Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-07779-6_1
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DOI: https://doi.org/10.1007/978-3-319-07779-6_1
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