Papers / Updated at: 2026-10-31

Park et al. (2021): Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting

Authors:

• Youngsuk Park
• Danielle Maddix
• François-Xavier Aubet
• Kelvin Kan
• Jan Gasthaus
• Yuyang Wang

Source: https://arxiv.org/abs/2111.06581

DOI: https://doi.org/10.48550/arXiv.2111.06581

Mentioned in:

• Baron et al. (2025): Efficiently Generating Correlated Sample Paths from Multi-step Time Series Foundation Models
• Park et al. (2021): Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting
• Shuffler

A paper with plenty of details. For example, instead of referring to GluonTS only, the authors define their evaluation metrics in appendix F. In appendix F.2, they define the Mean Weighted Quantile Loss as an approximation of the CRPS. This is the same approximation GIFT-eval uses when reporting the CRPS. Providing the definition of an evaluation metric isn’t a ton of work but makes everything so much clearer to the reader.

In section 3.2 “Inter/Extrapolation beyond Knots”, the authors describe how given only a limited set of quantiles of an unknown distribution, they inter- and extrapolate arbitrary quantiles from the known quantiles. This extrapolation of an unknown distribution is a fundamental component of their main contribution.

The same method is also picked up by Efficiently Generating Correlated Sample Paths from Multi-step Time Series Foundation Models (Baron et al., 2025). Baron et al. use the same method to evaluate arbitrary quantile levels from a forecast distribution of which again only a limited set of quantiles are known. This allows them to construct sample paths, first sampling quantile levels from a copula and then evaluating the forecast distribution at the sampled quantile levels, resulting in samples from the forecast distribution.

With Shuffler implementing the method defined by Baron et al. (2025), the extrapolation defined by Park et al. (2021) is a main component of Shuffler, too.

The following are some notes I took while reproducing the Park et al.’s approach of fitting an Exponential distribution to fit the right tail of the unknown distribution. Please forgive any abuse of notation.

Deriving Equation 6 of Section 3.2

Let’s start by repeating the definition of the Exponential distribution. If you parameterize the Exponential distribution via a rate parameter $\beta$¹, its cumulative distribution function is

and equal to 0 when $x < 0$. The inverse of the CDF is the quantile function, taking a quantile level $\alpha$ and returning quantile $q(\alpha)$:

The mean of the Exponential distribution is $1 / \beta$.

Adding a second parameter $\gamma$, one can right-shift the distribution so that it starts at $\gamma$ instead of at zero. Write:

This is the parameterization the authors use to extrapolate the right tail of the unknown distribution.

As far as I can tell, the Exponential distribution is a fitting choice for extrapolation because:

• Using a probability distribution to fit the tail ensures that the inter- and extrapolated distribution keeps the basic characteristics of a probability distribution with respect to its domain and codomain.
• The Exponential probability density function $f(x)$ decreases monotonously as $x$ increases, which implies that the derivative of its quantile function $q(\alpha)$ increases monotonously as $\alpha$ increases, making it a good functional shape to fit tails of other distributions.
• The Exponential distribution’s simple functional form lead to workable equations and closed form results. Its two parameters can be fitted given just two observations, as you’ll see in the following.

The authors aim to use the Exponential distribution to fit the right tail of an unknown distribution. What is known is a set of $K$ quantile knots, each knot being a tuple of a quantile level $\alpha_k$ and its corresponding (predicted) quantile $\hat{q}(\alpha_k)$. The set of known quantile levels is $\{\alpha_k\}_{k=1}^K$ with $\alpha_k<\alpha_{k+1}$. The two right-most quantile levels are $\alpha_{K-1}$ and $\alpha_{K}$ with corresponding quantiles $\hat{q}(\alpha_{K-1})$ and $\hat{q}(\alpha_{K})$.

The inter- and extrapolation of the CDF and quantile function (inverse CDF) through these quantiles needs to obey the standard characteristics of monotonicity and constraints of the respective function’s domain. As such it makes sense to fit a parametric distribution to the available quantiles such that the fitted distribution’s quantiles equal the known quantiles and the resulting fitted function obeys also all other characteristics.

Assuming that the interpolation for all $\alpha < \alpha_{K}$ is taken care of, the Exponential distribution you fit to the right tail only needs to pass through $\hat{q}(\alpha_K)$ at the $\alpha_K$ quantile level.

The authors additionally constrain the fit of the Exponential distribution by enforcing that it passes through $\hat{q}(\alpha_{K-1})$ at the $\alpha_{K-1}$ quantile level. Consequently, they are fitting a curve through two points.

The Exponential distribution that fulfills these two requirements has the two parameters $\beta_R$ and $\gamma_R$ such that the following two equations hold:

Solving the first equation for $\gamma$ provides:

Plugging this back in to the second equation, you can solve for $\beta$

which is the choice reported in equation (9) of appendix B.1 in the paper.

Plug these parameters back into the CDF and take the inverse to arrive at the quantile function. After rearranging the equation a bunch, $\beta$ reappears, leading to the reported result of:

Note that with $\alpha_K > \alpha_{K-1}$ and $\hat{q}(\alpha_K) > \hat{q}(\alpha_{K-1})$ it holds that $\beta > 0$. Since $\log(1-\alpha_{K-1}) < 0$, it then follows that $\gamma < \hat{q}(\alpha_{K-1})$. This means that the fitted Exponential distribution is shifted away from zero but starts to the left of $q(\alpha_{K-1})$, which makes sense given that it has to pass through that quantile knot.