  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 more widely studied one-sided
truncation that can be handled effectively by the 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, including a generalization of Kendall's tau test, for
doubly truncated data. Motivated by their approach, as well as by the work of
Bhattacharya et al. (1983) for right truncated data, we proposed 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. We show
that the resulting estimator is consistent and asymptotically normal. A
resampling scheme is proposed with large sample justification for approximating
the limiting distribution. The quasar data in Efron and Petrosian (1999) are
re-analyzed by the new method. Simulation results show that the proposed method
works well. Extension to weighted rank estimation are also given.
