Consistency of the Buckley–James estimator

A major open problem in survival analysis since the 1970s is the consistency of the Buckley–James estimator (BJE). The BJE is the extension of the least squares estimator (LSE) of β∈Rp for the linear regression model Y=α+β′X+ϵ with right-censored data. The LSE is consistent if E(ϵ)=0, ‖X‖<∞, ϵ and X are independent, and if β is identifiable. Under these assumptions with censoring, in order to establish the consistency of the BJE, its definition had been modified 3 times so far, as there exist counterexamples based on each of the previous 2 definitions. In this paper, we find another example of inconsistent BJE based on the 3rd definition. We propose a new definition of the BJE and study its consistency.