In References for methods for calculating the confidence interval for Theil-Sen Estimator, we note that Wilcox (1998, 2009?) proposed the use of bootstrap as a method for finding confidence intervals for Theil regression of heteroscedastic conditions. There is a practical problem doing this and that is the treatment of ties. That is, since bootstrap is sampling with replacement, it is probable to eventually sample the same point twice leading to an indeterminate slope pair. Thus, we need to either discard indeterminate results or use something else, for example a variant of leave n/2 out.
Q1: How do we resolve the indeterminate bootstrap problem for Theil regression?
Possible answers: Heretofore unspecified correction for ties. Incomplete Jackknife of n/2 samples (e.g., I have 412 samples, leave out 206 cannot be completely re-sampled as the combination of 412 samples taken 206 at a time is a geological time to completion answer competitive with the age of the universe on my pc. Moreover, leave one out would not be very useful for establishing confidence intervals).
BTW, leave half out appears to work +/- circa 7% on Monte-Carlo simulation of my data. However, that is a far cry from proving that the leave half out approach is valid.
Any clarification would be welcome.