The Annals of Statistics

Fisher's Information in Terms of the Hazard Rate

Bradley Efron and Iain M. Johnstone

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

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

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

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


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