Fisher's Information in Terms of the Hazard Rate

If _θ(t)\ is a regular family of probability densities on the real line, with corresponding hazard rates _θ(t)\, then the Fisher information for θ can be expressed in terms of the hazard rate as follows: I_θ ≡ ∫ (ġ_θg_θ)² g_θ = ∫ (ḣ_θh_θ)² g_θ, θ ∈ R, where the dot denotes ∂/∂θ. This identity shows that the hazard rate transform of a probability density has an unexpected length-preserving property. We explore this property in continuous and discrete settings, some geometric consequences and curvature formulas, its connection with martingale theory and its relation to statistical issues in the theory of life-time distributions and censored data.