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1. What the student is asking (in plain language)

The student has read about medials and tantiles – weighted analogues of the median and quantiles that are defined on a cumulative (or “cumulant”) distribution rather than on the raw data.
They wonder:

  • What exactly is tantile regression?
  • Is it just another name for ordinary quantile regression, or does it require a fundamentally different model‑building approach?

In other words: If we want to model a conditional “tantile” of a response variable given covariates, do we have to change the functional form that we use in quantile regression?

2. Step‑by‑step answer

Below we develop the concept from the ground up, compare it to ordinary quantile regression, and show that the only real difference is the choice of loss/weights – the functional form (the set of regressors) stays the same.

2.1 Recall: median, quantile and quantile regression

Concept Definition Estimator (sample)    
Median 0.5‑quantile of the (un‑weighted) distribution of a random variable (Y). (\displaystyle \hat m = \arg\min_{b}\sum_{i=1}^{n} Y_i-b ).
Quantile (q_\tau) ((0<\tau<1)) Value (c) such that (P(Y\le c)=\tau). (\displaystyle \hat q_\tau = \arg\min_{b}\sum_{i=1}^{n}\rho_\tau(Y_i-b)) where (\rho_\tau(u)=u(\tau-{\bf 1}{u<0})).    
Quantile regression Models the conditional quantile (Q_Y(\tau\mid X)=X\beta(\tau)). (\displaystyle \hat\beta(\tau)=\arg\min_{\beta}\sum_{i=1}^{n}\rho_\tau\bigl(Y_i-X_i^\top\beta\bigr)).    

The check‑loss (\rho_\tau(\cdot)) penalises under‑prediction and over‑prediction asymmetrically, giving exactly the (\tau)‑quantile as the minimiser.

2.2 Cumulative (or “cumulant”) distribution

Suppose each observation carries a non‑negative weight (w_i) (e.g., the number of units sold up to time (t_i), the size of a household, etc.).

Define the cumulative weight up to observation (i) as

[ W_i=\sum_{j=1}^{i} w_j ,\qquad\text{with }W_0=0. ]

If we order the data by the outcome (Y) (or by a time index, whichever makes sense), the cumulative distribution function (CDF) at the (i)-th ordered point is

[ F_i =\frac{W_i}{W_n}\quad\in[0,1]. ]

Thus each observation now has an associated cumulative probability (F_i) that reflects the weighted share of the total mass that lies at or below that point.

2.3 Definition of a tantile

A tantile is simply a quantile of the cumulative distribution.

For a given level (\tau\in(0,1)):

[ \text{tantile } T_\tau = \inf{y: F(y)\ge \tau}, ]

where (F(y)) is the weighted CDF defined above.
If all weights are equal ((w_i=1)), then (F_i=i/n) and a tantile collapses to the ordinary quantile.

The term “tantile” comes from “t‑weighted quantile” (t = t for t‑weight).

2.4 From tantiles to a regression problem

We now want to model the conditional tantile of a response (Y) given covariates (X).
Let

[ T_\tau(Y\mid X)=X\beta(\tau) . ]

Because a tantile is a quantile of a transformed variable, we can rewrite the problem in a way that makes the connection to ordinary quantile regression explicit.

2.4.1 Transform the response

Define the cumulative‑weight transformed response

[ \tilde Y_i \;=\; \frac{W_i}{W_n}\;=\;F_i . ]

Note that (\tilde Y_i) is already a number in ([0,1]) and its distribution is un‑weighted: every observation now contributes equally to the empirical CDF of (\tilde Y).

2.4.2 Loss function for tantile regression

The tantile (\tau) is the ordinary (\tau)-quantile of (\tilde Y). Therefore the natural loss to minimise is the standard check loss applied to (\tilde Y):

[ \boxed{\; \hat\beta(\tau)=\arg\min_{\beta}\; \sum_{i=1}^{n} \rho_\tau\bigl(\tilde Y_i- X_i^\top\beta\bigr) \;} \tag{1} ]

Equation (1) is exactly the ordinary quantile‑regression objective, except that the dependent variable has been replaced by the cumulative‑weight proportion.

2.5 Is the functional form different?

The functional form—the way we relate covariates to the location parameter (linear, splines, additive, etc.)—does not change.

What changes is:

Aspect Quantile regression Tantile regression
Dependent variable Raw outcome (Y) Cumulative weight proportion (\tilde Y = W_i/W_n)
Loss weighting Implicit (all observations equal) Implicit via the transformation; alternatively one may keep the original (Y) and weight each observation by (w_i) in the check loss: (\sum_i w_i\,\rho_\tau(Y_i - X_i^\top\beta)).
Interpretation “(\tau)-th conditional quantile of (Y)” “(\tau)-th conditional tantile (weighted quantile) of the cumulative distribution of (Y)”.

Thus, tantile regression can be implemented by simply feeding weighted observations into any standard quantile‑regression routine (most software packages allow observation weights). No new optimisation algorithm or different regression basis is required.

2.6 Practical implementation steps

  1. Choose the weighting scheme that reflects the cumulative quantity of interest (e.g., sales volume, household size). Compute a weight (w_i\ge0) for each observation.
  2. Compute cumulative proportions (or, equivalently, keep the original (Y) and supply the weights).
    • Option A (transform):
      [ \tilde Y_i = \frac{\sum_{j\le i} w_j}{\sum_{j=1}^{n} w_j}, ] where the data are sorted by the original (Y).
    • Option B (weighted loss):
      [ \min_\beta \sum_{i=1}^{n} w_i\,\rho_\tau!\bigl(Y_i-X_i^\top\beta\bigr). ]
  3. Run ordinary quantile regression on (\tilde Y) (or weighted (Y)). The estimated coefficients are the tantile‑regression coefficients.
  4. Interpret the fitted line (X\beta(\tau)) as the conditional (\tau)-tantile of the cumulative distribution. For example, with (\tau=0.5) you obtain the medial (weighted median) curve.

2.7 When is tantile regression preferable?

Situation Why tantile (weighted) matters
Adoption / diffusion studies (e.g., cumulative product sales) Early adopters contribute little to total sales; later adopters have large weights, so a weighted location measure captures the “center” of the adoption curve better than an un‑weighted median.
Wealth‑based partitions Rich households have a disproportionate share of total wealth; a weighted quantile (tantile) gives a more meaningful “median wealth holder” (the medial) than the ordinary median person.
Short time‑series where only the cumulative count is reliable Modeling the cumulative series directly (with tantiles) avoids differencing that would amplify noise.
Survey data with design weights The design weight is exactly the (w_i); tantile regression yields design‑consistent location estimates.

In all these cases the shape of the regression model (linear, spline, etc.) can be chosen exactly as in ordinary quantile regression; the only extra ingredient is the weight.

3. Final answer

Tantile regression is quantile regression applied to a weighted (cumulative) version of the data.
Formally, for a chosen level (\tau),

[ \boxed{\displaystyle \hat\beta(\tau)=\arg\min_{\beta}\; \sum_{i=1}^{n} w_i\, \rho_\tau\bigl(Y_i- X_i^\top\beta\bigr) } ]

or, equivalently, after transforming the response to its cumulative‑weight proportion (\tilde Y_i),

[ \boxed{\displaystyle \hat\beta(\tau)=\arg\min_{\beta}\; \sum_{i=1}^{n}\rho_\tau\bigl(\tilde Y_i- X_i^\top\beta\bigr) }. ]

The functional form (the set of regressors, linear vs. spline, additive, etc.) is identical to that used in ordinary quantile regression; only the weights (or the transformed dependent variable) differ. Consequently, any software that implements weighted quantile regression can be used to fit tantile regression.

4. Common mistakes

Mistake Why it’s wrong How to avoid it
Treating tantiles as a new kind of loss (inventing a “tantile‑check” function) The check loss is already the correct loss; the only change is the observation weight. Use the standard (\rho_\tau) and supply the appropriate weights (or transform (Y)).
Weighting the covariates instead of the observations Weights reflect the cumulative contribution of each observation to the total mass, not a scaling of the predictors. Compute (w_i) based on the outcome (e.g., sales) and pass them as observation‑level weights.
Forgetting to sort the data before computing cumulative weights Cumulative proportion must be monotone in the outcome; unsorted data give a meaningless (\tilde Y). Sort by the raw outcome (or by time, if that is the natural ordering) before forming (W_i).
Interpreting the tantile coefficient as a median of the raw outcome The tantile is a weighted median; its location can be far from the un‑weighted median. Report that the coefficient refers to the (\tau)-tantile (weighted quantile) and, when useful, compare it to the ordinary quantile.
Assuming the same asymptotic variance as ordinary quantile regression Weights affect the variance; naïve standard errors are biased. Use the weighted‑bootstrap or the sandwich variance estimator that incorporates the observation weights

Original question: What is tantile regression? on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.