Existence of the Buckley-James estimate

Under the semi-parametric linear regression model Y=β′X+W with right-censored data, the Buckley-James estimator (BJE) of the parameter β (∈ℛp) is the standard extension of the least squares estimator (LSE). Since the original definition of the BJE may not exist, James and Smith (1984) proposed a modification which has been implemented in an R package. However, this algorithm may fail to yield an estimate. Thus, it has been a long open problem regarding the existence of BJE. In this article, we show that the BJE as defined in the literature always exists if p = 1. If p≥2, we show that further modifications are necessary to define BJE accurately. We establish that the modified BJE always exists, and it may happen that each point in an unbounded open set is a BJE. For instance, if the data are all censored or the covariate satisfies 𝐗≡𝐜, then every 𝐛∈ℛp is a BJE of β, which is undesirable. Therefore, we propose to further modify the definition of the BJE.