The Annals of Statistics

The Asymptotic Joint Distribution of Regression and Survival Parameter Estimates in the Cox Regression Model

Kent R. Bailey

Full-text: Open access

Abstract

In this paper it is shown that the Cox likelihood (Cox, 1972) may be treated as a standard likelihood, in the sense that its maximizer $\hat{\beta}$ is asymptotically normally distributed with asymptotic covariance matrix equal to $-\{E \partial^2 \log L (\beta)/\partial\beta\partial\beta'\}^{-1}$. In the process, an asymptotic representation of the score function is obtained in terms of functions of the independent observations. This representation may have some uses in itself such as: (1) providing a kind of residual for each observation, censored or uncensored, thereby indicating the relative influence of the observations, and (2) providing some information about the applicability of the asymptotics in a particular small sample. The asymptotic joint distribution of $\hat{\beta}$ and of the cumulative hazard function estimator $\hat{Lambda}_0(t)$ is also derived via a representation of the latter involving an independent increments process. Bailey (1982) shows that the "joint likelihood function" of the regression parameters $\beta$ and of the cumulative hazard jump parameters $\{\Lambda_i\}$ can be used in a natural way to obtain consistent estimates of these joint asymptotic covariances in the case of no ties. This justifies, to some extent, use of the general ML method for joint estimation of $\beta$ and $\Lambda_0(t)$.

Article information

Source
Ann. Statist., Volume 11, Number 1 (1983), 39-48.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346054

Digital Object Identifier
doi:10.1214/aos/1176346054

Mathematical Reviews number (MathSciNet)
MR684861

Zentralblatt MATH identifier
0509.62015

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62F10: Point estimation

Keywords
Asymptotic representation Cox regression Hajek projection independent increments maximum likelihood proportional hazards

Citation

Bailey, Kent R. The Asymptotic Joint Distribution of Regression and Survival Parameter Estimates in the Cox Regression Model. Ann. Statist. 11 (1983), no. 1, 39--48. doi:10.1214/aos/1176346054. https://projecteuclid.org/euclid.aos/1176346054


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