On generalized f-statistics

Abstract

In statistics, there exists several means such as the arithmetic, geometric, harmonic, or power means. Using axiomatic deduction, they have been unified into the generalized f-mean. Similarly, there are several standard deviations and they have been unified into the generalized f-standard deviation, even though without derivation. However, standard score, as known in its arithmetic and geometric versions, has never been f-generalized. The goal of this article is to expose a full derivation of generalized f-statistics, adopting a parameter estimation point of view. Generalized f-mean and generalized f-variance are derived thanks to a maximum likelihood estimation. The non trivial relation between generalized f-standard deviation and generalized f-variance is also studied. Furthermore, generalized f-standard score is defined and extended to multivariate case. Finally, these generalized f-statistics are applied on simulated data for pre-processing and outlier detection.