The Annals of Statistics

Fisher's Information in Terms of the Hazard Rate

Bradley Efron and Iain M. Johnstone

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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.

Article information

Ann. Statist. Volume 18, Number 1 (1990), 38-62.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F12: Asymptotic properties of estimators
Secondary: 62G05: Estimation 62P10: Applications to biology and medical sciences

Score function length-preserving transformation information metric proportional hazards model counting process martingale Greenwood's formula binary trees tangent space $\alpha$-expectation statistical curvature


Efron, Bradley; Johnstone, Iain M. Fisher's Information in Terms of the Hazard Rate. Ann. Statist. 18 (1990), no. 1, 38--62. doi:10.1214/aos/1176347492.

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