Title: Warped metrics for location-scale models,
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.