My question follows on this discussion of medials and tantiles vs medians and quantiles from earlier this year:
When would we use tantiles and the medial, rather than quantiles and the median?
As described in the link, medials are a measure of location for cumulants, kind of like a weighted median, where a median is a measure of location for the unweighted, uncumulated distribution of values. The literature on cumulants is extensive and deep but, until I saw this link, my reading of it did not cover medials or tantiles. I can think of lots of circumstances where cumulants and tantiles are useful, e.g., in adoption and diffusion processes in technology or new product sales, in wealth-based partitioning in financial statistics, etc.
How might the analysis of cumulants differ? One obvious example would be to time series analysis. Cumulants would almost certainly be nonstationary, autocorrelated, etc. If one wishes to model the cumulant distribution then this is inappropriate input for a "Box-Jenkins" approach requiring residuals to be HAC. But there are circumstances where an analysis of cumulants is to be preferred, e.g., when the available time series is too short for standard approaches and one is interested only in the adoption rate over that short time span. Nonlinear approaches positing an underlying S-curve to the cumulant's growth over time such as Bass-Anderson diffusion models are relevant here.
Apologies for my vagueness but I'm struggling here...The question occurred to me whether or not tantile regression would require a qualitatively different functional form from quantile regression?