On parameters for inner products under imprecision and their applications to certain geometric and statistical problems

Abstract

This work aims to study the imprecision on the inner product of a real vector space. It starts defining the notion of imprecise inner product. This notion can also model the uncertainty about the variability of a multivariate random variable. The primary goal is to introduce parameters in order to provide information about the shape and the size of this kind of inner products. Imprecision is inherent to any quantity which depends on the inner product (or the covariance matrix). Useful techniques are developed by means of the introduced parameters to obtain fast and easy approximations for that kind of quantities. Throughout several examples, the convenience of the techniques is proven. These techniques can be applied to a wider class of statistical problems dealing with imprecision.

Keywords:

• Inner product
• imprecision
• precision parameters
• generalized information theory
• approximations

Acknowledgments

The author would like to thank the referees for their valuable comments which helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

By project PGC2018-098623-B-I00.

Notes on contributors

Juan Jesús Salamanca

Juan Jesús Salamanca is currently lecturer in Statistics at the Department of Statistics and O.R., University of Oviedo, Spain. He did his M.Sc. and Ph.D. from the University of Granada, Spain, in 2012 and 2015, respectively. His research interest lies in Imprecise Probability; more precisely, in random sets and other imprecise random elements.