## The Annals of Statistics

### Fisher's Information in Terms of the Hazard Rate

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347492

Digital Object Identifier
doi:10.1214/aos/1176347492

Mathematical Reviews number (MathSciNet)
MR1041385

Zentralblatt MATH identifier
0722.62022

JSTOR

#### Citation

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. https://projecteuclid.org/euclid.aos/1176347492