Fisher's Information in Terms of the Hazard Rate

Efron, Bradley ; Johnstone, Iain M.

Ann. Statist., Tome 18 (1990) no. 1, p. 38-62 / Harvested from Project Euclid

Résumé

If $\{g_\theta(t)\}$ is a regular family of probability densities on the real line, with corresponding hazard rates $\{h_\theta(t)\}$, then the Fisher information for $\theta$ can be expressed in terms of the hazard rate as follows: $\mathscr{I}_\theta \equiv \int \big(\frac{\dot{g}_\theta}{g_\theta}\big)^2 g_\theta = \int \big(\frac{\dot{h}_\theta}{h_\theta}\big)^2 g_\theta, \theta \in \mathbb{R},$ where the dot denotes $\partial/\partial\theta$. 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.

Détail

Publié le : 1990-03-14
Classification:  Score function,  length-preserving transformation,  information metric,  proportional hazards model,  counting process,  martingale,  Greenwood's formula,  binary trees,  tangent space,  $\alpha$-expectation,  statistical curvature,  62F12,  62G05,  62P10