Consistency of the Semi-parametric MLE under the Cox Model with Right-Censored Data

Qiqing Yu

doi:10.2174/2666148902010010021

Abstract

Objective: We studied the consistency of the semi-parametric maximum likelihood estimator (SMLE) under the Cox regression model with right-censored (RC) data. Methods: Consistency proofs of the MLE are often based on the Shannon-Kolmogorov inequality, which requires finite E(lnL), where L is the likelihood function. Results: The results of this study show that one property of the semi-parametric MLE (SMLE) is established. Conclusion: Under the Cox model with RC data, E(lnL) may not exist. We used the Kullback-Leibler information inequality in our proof.

Highlights

We studied the consistency of the semi-parametric maximum likelihood estimator (SMLE) under the Cox model with right-censored (RC) data
Suppose that C is a random variable with the df fC (t) and the survival function SC (t), X takes at least p +1 values, say 0, x1, ..., xp, where x1, ..., xp are linearly independent, (Y,X) and C are independent
The computation issue of the SMLE under the Cox model has been studied, but its consistency has not been established under the model [3]. Their simulation results suggest that the SMLE is more efficient than the partial likelihood estimator under the Cox model

Summary

Introduction

Conditional on X = x, Y satisfies the Cox model if its hazard function satisfies h(y|x) = ho (y) eβ'x, y < τY, (1.1) S(t |x) ≠ (S(t |0))exp(x' β) under the discrete Cox model ( [3]). The SMLE of S(t|x) is denoted by Ŝ (t|x), which is a function of (Ŝo, β).

Results

Conclusion