¹ Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ08544, USA
² Department of Statistics, Shanghai University of Finance and Economics, Shanghai 200433, China 
³ Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 
⁴ Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7420, USA

Multivariate failure time data arise frequently in survival analysis. A commonly used technique is the working independence estimator for marginal hazard models. Two natural questions are how to improve the efficiency of the working independence estimator and how to identify the situations under which such an estimator has high statistical efficiency. In this paper, three weighted estimators are proposed based on three different optimal criteria in terms of the asymptotic covariance of weighted estimators. Simplified close-form solutions are found, which always outperform the working independence estimator. We also prove that the working independence estimator has high statistical efficiency, when asymptotic covariance of derivatives of partial log-likelihood functions is nearly exchangeable or diagonal. Simulations are conducted to compare the performance of the weighted estimator and working independence estimator. A data set from Busselton population health surveys is analyzed using the proposed estimators.

This work was supported by National Natural Science Foundation of China (Grant No. 10628104), Fan was also supported by National Institutes of Health (Grant No. R01-GM072611) and Natural Science Foundation (Grant No. DMS-0714554), Zhou was supported by National Natural Science Funds for Distinguisheel Young Scholar (Grant No. 70825004), National Natural Science Foundation of China (Grant Nos. 10731010, 10628104), the National Basic Research Program (Grant No. 2007CB814902), Creative Research Groups of China (Grant No. 10721101) and Leading Academic Disipline Program, the $10^{th}$ five year plan of 211 Project for Shanghai University of Finance and Economics (the $3^{rd}$ phase), Cai was supported by National Institutes of Health (Grant No. R01-HL57444)