## Info

accurate. Among the situations considered, there are many instances where the Theil-Sen estimator provides a striking advantage, and there are none where the reverse is true, the lowest value for R being 0.76. It should be remarked that direct comparisons in terms of power are hampered by the fact that for many of the situations considered in Tables 3.1 and 3.2, conventional hypothesis testing methods based on least squares perform very poorly. Perhaps there are situations where the very inadequacies of conventional techniques result in relatively high power. That is, probability coverage might be extremely poor, but in a manner that increases power. Experience suggests, however, that it is common to find situations where the hypothesis of a zero slope is rejected when using Theil-Sen, but not when using least squares.

A technical issue when using the Theil-Sen estimator is that when there is heteroscedasticity, an explicit expression for its standard error is not available. However, a percentile bootstrap method has been found to provide fairly accurate probability coverage and good control over the probability of a Type I error for a very wide range of situations, including situations where the conventional method based on least squares is highly inaccurate. But rather than use the modified percentile bootstrap method previously described, now it suffices to use the standard percentile bootstrap method instead.

In particular, again let (X1, Y1),. ..,(Xn, Yn) be n randomly sampled pairs of points and generate a bootstrap sample by resampling with replacement n pairs of points from (Xi, Yi),...,(Xn, Yn). Let b\ be the Theil-Sen estimate of the slope based on this bootstrap sample. Next, repeat this process B times yielding b*n,...,b{B. The standard percentile bootstrap method uses the middle 95% of these B bootstrap estimates as a .95 confidence interval for slope. That is, put the B bootstrap samples in ascending order, label the results b1(l) <■■ ■ < b1(B), in which case a 1 — a confidence interval for the population slope is (b1(L+1), b1(U)), where L = aB /2, rounded to the nearest integer, and U = B — L. (B = 600 seems to suffice, in terms of accurate probability coverage, when using Theil-Sen.) Obviously this approach requires a computer, but even with a moderately large sample size, execution time is fairly low.

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