Calculations and Discussion of the Examples in "U-Statistics and Imperfect Ranking in Ranked Set Sampling"

this paper, some discussion of issues in modeling judgement ranking seems to be needed. The main point to be made is that modeling the judgement ranking process by conditioning on the unmeasured data is not only correct, but is a natural and perhaps the only reasonable way to proceed. We believe that some reflection will convince the reader that this view is correct. A useful thought experiment is to imagine judgement ranking of a pair of books selected at random from the shelf (without checking the actual number of pages of course). If the two books selected happen to be very di#erent in length, then they are easy to rank correctly. However if the numbers of pages of the two books are very nearly the same, it is very di#cult to guess which is actually larger. The crucial point is that it is the actual (ordered) lengths of the books, X (1) , X (2) , which largely determine the di#culty of ranking, and hence the conditional probability that we get it right. Modeling judgement ranking using only the expected values of X (1) , X (2) , for example, does not capture this aspect of judgement ranking at all, since seen through the expected values all samples look the same. Expected values can at most tell us something about the average di#culty of ranking over all possible samples, and are thus not a suitable basis for modeling the ranking process itself. The remainder of this note simply provides detailed calculations for the examples that were used in the paper to illustrate some errors in the literature. This is all elementary, and there's no doubt that much of it could be done more elegantly. The objective is simply to demonstrate the calculations in su#cient detail that they can be seen to be correct with minimal e#ort on the part of the reader. 2 The First Example Supp..