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March, 1990 Fisher's Information in Terms of the Hazard Rate
Bradley Efron, Iain M. Johnstone
Ann. Statist. 18(1): 38-62 (March, 1990). DOI: 10.1214/aos/1176347492


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.


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Bradley Efron. Iain M. Johnstone. "Fisher's Information in Terms of the Hazard Rate." Ann. Statist. 18 (1) 38 - 62, March, 1990.


Published: March, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0722.62022
MathSciNet: MR1041385
Digital Object Identifier: 10.1214/aos/1176347492

Primary: 62F12
Secondary: 62G05, 62P10

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 1 • March, 1990
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