We are indebted to David Hoaglin for pointing out an unfortunate error in two formulas in the article. He also pointed out some other errors and raised some additional issues, and we briefly respond here to those we consider most significant. We sincerely thank him for all of his comments.
The term “min” appears in two formulas where we should have written “max.” In particular, in formula (3) on pp. 253–254 the first four instances of “min” should be replaced by “max. (“min” also appears inside the braces in the last line of that formula, and is correct there.) Corresponding to this change, Step 1(f) of the algorithm later on p. 254 should read “Find the maximum probability Cm = 2max i{min (Ami, 1 − Ami)}.” The textual description of the algorithm elsewhere in the article is correct, and the procedure (as corrected) is consistent with the programmatic ideas in Buja and Rolke (2006), as claimed in the article. The R function described and cited at the end of Section 2.1 runs properly since it was created using formula (3) and the algorithm in their correct form.
Step 1(c) of the algorithm should say, “Apply the probability integral transformation, with Ymi = Φ(Zim).” This is because the are in the (standardized) scale of the data, not the probability scale. In Step 1(e) of the algorithm, B−1 should be B.
In Figure 3, the right-hand plot was created using the algorithm from p. 257 of the article. This algorithm involves a slightly different centering and estimate of scale for the Y-values than the naïve estimates used in the left-hand plot. In particular, because of the scaling of the axis the largest two Y-values do not appear in the right-hand plot, but they are used in all computations leading to the plot. We apologize for the graphical confusion, and note that it does not occur in the paired plots in Figures 1 and 2.
We drew the descriptive term “corner distributions” that appears on p. 257 from Randal and Thomson (Citation2003), not from Morgenthaler and Tukey (1991) as stated in the article. We are indebted to Hoaglin for pointing out that much more definitive, earlier references that describe those distributions are Goodall (Citation1983) and Iglewicz (Citation1983). The formulas in the footnote on p. 257 for the “wild” (which should really be termed “one-wild”) and “slash” densities are unfortunately garbled and should be ignored. These densities, as used in Tables 1 and 2, are as described in both these references: the “wild” sample with n = 100 has 99 observations from a standard normal and one from a normal with standard deviation = 10; the “slash” distribution is that of a standard normal divided by an independent uniform (0,1) variable.
Finally, we would like to mention the interesting, recent article by Dümbgen and Wellner (Citation2014), which came to our attention after publication of our article. Their article describes an algorithm that is different from that in our article, but could be used graphically for the same purpose.
ADDITIONAL REFERENCES
- Dümbgen, L., and Wellner, J.A. (2014), ``Confidence Bands for a Distribution Function: A New Look at the Law of the Iterated Logarithm'' (arxiv:1402.2918). See also the brief version in Adaptive Statistical Inference (2014), eds. M. Low, A. Munk and A. Tsybakov. Available at http://www.mfo.de/occasion/1411/www_view.; workshop 1411.
- Goodall, C. (1983), “M-Estimators of Location: An Outline of the Theory,” in Understanding Robust and Exploratory Data Analysis, eds. D.C. Hoaglin, F. Mosteller, and J.W. Tukey, New York: Wiley, pp. 339–403.
- Iglewicz, B. (1983), “Robust Scale Estimators and Confidence Intervals for Location,” in Understanding Robust and Exploratory Data Analysis, eds. D.C. Hoaglin, F. Mosteller, and J.W. Tukey, New York: Wiley, pp. 404–431.
- Randal, J.A., and Thomson, P.J. (2003), ``Maximum Likelihood Estimation for Tukey's Three Corners,'' available at http://www.victoria.ac.nz/staff/john_randal/Papers/corners.pdf