Subset selection with additional order information

Traditional subset selection procedures were developed without assuming any order information about the response variable. However, in some applications there is additional, even though incomplete, order information about the treatment effects at increasing treatment levels. One important example is the up-then-down umbrella ordering with an unknown peak. This type of additional order information is utilized explicitly in this paper to construct subset selection procedures for several settings studied in the literature where only order restricted tests are known to exist. This paper also proposes a straightforward algorithm to compute the isotonic regression with respect to umbrella orderings, which can be used to carry out the proposed procedures. Examples are given to illustrate the procedures and algorithm.