  We revisit the problem of selecting an item from $n$ choices that appear
before us in random sequential order so as to minimize the expected rank of the
item selected. In particular, we examine the stopping rule where we reject the
first $k$ items and then select the first subsequent item that ranks lower than
the $l$-th lowest-ranked item among the first $k$. We prove that the optimal
rule has $k \sim n/{\mathrm e}$, as in the classical secretary problem where
our sole objective is to select the item of lowest rank; however, with the
optimally chosen $l$, here we can get the expected rank of the item selected to
be less than any positive power of $n$ (as $n$ approaches infinity). We also
introduce a common generalization where our goal is to minimize the expected
rank of the item selected, but this rank must be within the lowest $d$.
