Calculating m when reduced vector contains repeated values

[wleoncio]

Consider a case where the reduced vector contains duplicated values. For example, {12, 12, 16, 17, 13}. If we apply the ordering process (e.g. through the reducedVector() function in permChacko), we get:

Original vector: 12     12      16      17      13
Reduced vector : 12     12      15.333

With respective weights 1, 1, and 3 (since we had two reductions involving {17, 13} and then {16, 15}).

My question now is: what is the value of m, the number of distinct quantities? Should I take the word "distinct" literally (as permChacko 1.0.0 does) and say m = 2 or should m = 3 simply because that's the length of the reduced vector? The answer directly affects which elements appear in the test statistic.

I've tried to find an answer on both Chacko papers as well as on Brunk (1958), Section 6, which is a reference to the ordering process on Chacko (1963), all to no avail.

[wleoncio]

Replies from Graeme and Morten, respectively (TL;DR: m = 3)

My instinct is that we have three weights here so m = 3. However I am happy to hear different views.

My instincts agree, but there's two ways of looking at it. Either you have
12, 12, 15.333 with weights 1, 1, 3, and m = 3

or

12, 15.333 with weights 2, 3, and m = 2

These two will give identical values of the test statistic; however, the reference chi-squared distribution has m-1 degrees of freedom, so when the test statistic is compared with the reference distribution, we will get two different P-values. My instinct would be to interpret "distinct" as "non-decreasing", i.e. as complying with the assumption of the ordering, thus m=3.