A unified approach to the Pythagorean identity and projection theorem for a class of divergences based on M-estimations

Abstract

Divergences that satisfy Pythagorean-type identity are found to be quite useful in statistical inference. Projection theorem of these divergences explores the associated inference problems through closed-form estimators and a fixed number of sufficient statistics. This work attempts to unify the Pythagorean property and projection theorem of such divergences, including the well-known KL, Basu et al., and Jones et al. divergences, minimization of which result in an M-estimation. This unification characterizes a general form of power-law distributions that includes the well-known Student-t and Student-r families. Rao-Blackwell-type best estimators for these families are derived using sufficient statistics based on the proposed unified class of divergences. Finally, it comments on the efficacy of these estimators through numerical studies.

Keywords:

• Divergence function
• M-estimation
• Pythagorean identity
• power-law distributions
• sufficient statistics

2020 Mathematics Subject Classifications:

• 62F10
• 62F35
• 94A15

Disclosure statement

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