What is tantile regression?

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?

• google scholar does not give any hits on tantile regression ... Commented Nov 26, 2017 at 19:57
• @kjetilbhalvorsen I know. That seems to imply that it hasn't been invented yet. Commented Nov 27, 2017 at 0:09
• "Box-Jenkins" approach requiring residuals to be HAC? I guess you mean white noise instead. Commented Dec 29, 2020 at 10:14
• I am having a conceptual disconnect with the concept, because regressions (including quantile) summarize the variable relative to individuals to predict or describe processes in groups of individuals. But the tantile defines the mid-point of the resource, regardless of the distribution within the population. I cannot see how you could use it to create a meaningful estimator of an individual. In order to change a prediction , the entire cumulative system volume would need to change for a variable. Or am I understanding this wrong? Commented Oct 8, 2022 at 16:23
• @sconfluentus Thanks for your comment. Your question is at the heart of the original query which was and remains an exploratory one. If there is a misunderstanding it may be in confounding individual vs aggregate model description and/or prediction. Cumulants, by definition, are aggregate metrics, as are averages, medians, quantiles, and so on. Regression models which predict aggregate tendencies are neither interpretable nor meaningful at the individual level, except in the case of repeated measures for each individual in the data, enabling random coefficient-type individual estimation Commented Oct 11, 2022 at 16:39