Doubly truncated data are found in astronomy, econometrics and survival analysis literature. They arise when each observation is confined to an interval, i.e., 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 data set, Efron and Petrosian (1999) proposed some nonparametric methods for doubly truncated data. Motivated by their approach, as well as by the work of Bhattacharya et al. (1983) 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 are 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 (1999) and an AIDS incubation data are analyzed by the new method. Simulation results show that the proposed method works well.
Keywords: Confidence interval; Empirical process; L1 method; Linear programming; Rank estimation; Resampling; U-process; Wilcoxon-Mann-Whitney Statistic.