So, for most stations in the databank we have data for maximum (Tx) and minimum (Tn) that we then average to get Tm. Now, that is not the only transform possible - there is also DTR which is Tx-Tn. Although that is not part of the databank archive its a trivial transform. In looking at results running NCDC's pairwise algorithm distinct differences in breakpoint detection efficacy and adjustment distribution arise, which have caused great author team angst.
This morning I constructed a simple toy box where I just played what if. More precisely what if I allowed seeded breaks in Tx and Tn in the bound -5 to 5 and considered the break size effects in Tx, Tn, Tm and DTR:
red: Break largest in Tx
blue: Break largest in Tn
purple: break largest in DTR
green: break largest in Tm (yes, there is precisely no green)
Cases with breaks equal in size are no colour (infintesimally small lines along diagonal and vertices at Tx and Tn =0)
if we just randomly seeded Tx and Tn breaks in an entirely uncorrelated manner into the series then we would get 50% of breaks largest in DTR and 25% each in Tx and Tn. DTR should be broader in its overall distribution and Tm narrower with Tx and Tn intermediate.
if we put in correlated Tx and Tn breaks such that they were always same sign (but not magnitude) then they would always be largest in either Tx or Tn (or equal with Tm when Tx=Tn)
If we put in anti-correlated breaks then they would always be largest in DTR.
Perhaps most importantly, as alluded to above, breaks will only be equal largest for Tm in a very special set of cases where Tx break = Tn break. Breaks, on average will be smallest in Tm. If breakpoint detection and adjustment is a signal to noise problem its not sensible to look where the signal is smallest. This has potentially serious implications for our ability to detect and adjust for breakpoints if we limit ourselves to Tm and is why we should try to rescue Tx and Tn data for the large amount of early data for which we only have Tm in the archives.
Maybe in future we can consider this as an explicitly joint estimation problem of finding breaks in the two primary elements and two derived elements and then constructing physically consistent adjustment estimates from the element-wise CDFs. Okay, I'm losing you now I know so I'll shut up ... for now ...
Bonus version showing how much more frequently DTR is larger than Tm: