Regression Analysis of Doubly Truncated Data

Abstract

Doubly truncated data are found in astronomy, econometrics, and survival analysis literature. They arise when each observation is confined to an interval, that is, only those which fall within their respective intervals are observed along with the intervals. Unlike the one-sided truncation that can be handled by counting process-based approach, doubly truncated data are much more difficult to handle. In their analysis of an astronomical dataset, Efron and Petrosian proposed some nonparametric methods for doubly truncated data. Motivated by their approach, as well as by the work of Bhattacharya et al. for right truncated data, we propose a general method for estimating the regression parameter when the dependent variable is subject to the double truncation. It extends the Mann–Whitney-type rank estimator and can be computed easily by existing software packages. Weighted rank estimation is also considered for improving estimation efficiency. We show that the resulting estimators are consistent and asymptotically normal. Resampling schemes are proposed with large sample justification for approximating the limiting distributions. The quasar data in Efron and Petrosian and an AIDS incubation data are analyzed by the new method. Simulation results show that the proposed method works well.

Keywords:

• Empirical process
• Rank estimation
• Resampling
• Wilcoxon–Mann–Whitney statistic
• U-process

Acknowledgments

The authors thank the editor, associate editor, and two referees for their insightful comments and suggestions, which have led to many improvements. The authors also thank Professor Bradley Efron for providing the quasar data.

Additional information

Funding

Zhiliang Ying’s research was supported in part by NIH grant R37GM047845 and NSF grants DMS-1308566, SES-1323977. Wen Yu’s research was supported in part by the Fudan Young Scholar Research Enhancement Program (20520133107) and the National Natural Science Foundation of China (11671097). Ming Zheng’s research was supported in part by the National Natural Science Foundation of China (11771095).