# Inconsistency of the Breslow estimator

The Breslow estimate is commonly used in the Cox proportional hazards model. However this paper by Deborah Burr

Burr, D. (1994). On Inconsistency of Breslow's Estimator as an Estimator of the Hazard Rate in the Cox Model. Biometrics, 50(4), 1142-1145.

claims that the estimator is inconsistent (I do not have this paper so I have read the first page only).

Does the conclusion of the paper mean that we should really stop to use this estimate ?

• From page 2: The estimator in (1.2) was intended as an intermediate step in the construction of the estimator of $\Lambda$, not as a bona fide estimator of the hazard rate function. However, it is tempting to view it as an estimator of $\lambda$ itself; in fact, $\hat\lambda$ is frequently referred to as "the Breslow estimate of $\lambda$."  Their result is: For t such that $0 < S(t) < 1$, as $n \rightarrow \infty$, $${\hat \lambda(t) \over \lambda(t)} \mathop{\rightarrow}^d (E_1 + E_2)^{-1}$$ where $E_1$ and $E_2$ are independent unit exponentials. – Elvis Jan 8 '12 at 14:36
• Is the Breslow estimate of the hazard rate really commonly used? You're not confusing it with the Breslow method for handling ties?? – onestop Jan 8 '12 at 15:55

The estimator should not be viewed as such, for it is inconsistent as an estimator of $\lambda$ in the Cox model (although erroneous use of (1.2) has occurred in the literature). This inconsistency of $\hat\lambda$ is well known, but the result has not been written down explicitly. The purpose of this note is to do that.