# Warped metrics for location-scale models

## TL;DR

This paper demonstrates that warped Riemannian metrics naturally arise in location-scale models, providing explicit curvature and geodesic solutions, which facilitate the development of efficient statistical algorithms for classification and estimation.

## Contribution

It establishes that the Rao-Fisher metric of location-scale models is a warped metric, derives its curvature and geodesic solutions, and shows their application in statistical algorithm development.

## Key findings

- Rao-Fisher metric of location-scale models is a warped metric under invariance conditions.
- Explicit formulas for sectional curvature of these metrics.
- Geodesic equations have exact analytic solutions.

## Abstract

This paper argues that a class of Riemannian metrics, called warped metrics, plays a fundamental role in statistical problems involving location-scale models. The paper reports three new results : i) the Rao-Fisher metric of any location-scale model is a warped metric, provided that this model satisfies a natural invariance condition, ii) the analytic expression of the sectional curvature of this metric, iii) the exact analytic solution of the geodesic equation of this metric. The paper applies these new results to several examples of interest, where it shows that warped metrics turn location-scale models into complete Riemannian manifolds of negative sectional curvature. This is a very suitable situation for developing algorithms which solve problems of classification and on-line estimation. Thus, by revealing the connection between warped metrics and location-scale models, the present paper paves the way to the introduction of new efficient statistical algorithms.