Abstract
In the theory of complex systems, long tailed probability distributions are often discussed. For such a probability distribution, a deformed expectation with respect to an escort distribution is more useful than the standard expectation. In this paper, by generalizing such escort distributions, a sequence of escort distributions is introduced. As a consequence, it is shown that deformed expectations with respect to sequential escort distributions effectively work for anomalous statistics. In particular, it is shown that a Fisher metric on a q-exponential family can be obtained from the escort expectation with respect to the second escort distribution, and a cubic form (or an Amari–Chentsov tensor field, equivalently) is obtained from the escort expectation with respect to the third escort distribution.
Highlights
Long tailed probability distributions and their related probability distributions are important objects in anomalous statistical physics
Since an escort distribution gives a suitable weight for tail probability, the escort expectation which is the expectation with respect to an escort distribution is more useful than the standard one
A deformed exponential family is an important statistical model in anomalous statistics. Such a statistical model is described by such a deformed exponential function
Summary
Long tailed probability distributions and their related probability distributions are important objects in anomalous statistical physics (cf. [1,2,3]). A deformed exponential family is an important statistical model in anomalous statistics Such a statistical model is described by such a deformed exponential function. The author showed that a deformed score function is unbiased with respect to the escort expectation [8,9]. This implies that a deformed score function is regarded as an estimating function on a deformed exponential family. Properties of escort expectations are closely related to geometric structures on a deformed exponential family. See [12,13], for example
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