# Why Student Distributions? Why Matern’s Covariance Model? A Symmetry-Based Explanation

## Abstract

In this paper, we show that empirical successes of Student distribution and of Matern’s covariance models can be indirectly explained by a natural requirement of scale invariance – that fundamental laws should not depend on the choice of physical units. Namely, while neither the Student distributions nor Matern’s covariance models are themselves scale-invariant, they are the only one which can be obtained by applying a scale-invariant combination function to scale-invariant functions.

## Notes

### Acknowledgments

This work was performed when Olga Kosheleva and Vladik Kreinovich were visiting researchers with the Geodetic Institute of the Leibniz University of Hannover, a visit supported by the German Science Foundation. This work was also supported in part by NSF grant HRD-1242122.

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