📉 Quantile Level

Quantile Level#

The probability level (e.g. 0.9) targeted by a quantile forecast.

Important

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What it is#

The quantile level \(\tau \in (0,1)\) (sometimes written \(\alpha\)) is the probability that names which quantile a forecast targets — the value \(q_\tau\) below which the outcome is expected to fall a fraction \(\tau\) of the time:

\[\Pr\!\left(Y \le q_\tau\right) = \tau, \qquad \tau \in (0, 1).\]

Reading levels#

\(\tau = 0.5\) is the median; \(\tau = 0.1\) is the 10th percentile (a pessimistic lower value the outcome undercuts 10% of the time); \(\tau = 0.9\) the 90th (an optimistic upper value). A set of levels \(\{0.1, 0.5, 0.9, \dots\}\) traces out the whole predictive distribution.

Monotonicity#

In quantile regression the level is the tilting parameter of the pinball loss. Estimated quantiles should be monotonically increasing in \(\tau\) — when a lower level’s forecast exceeds a higher one’s, that is quantile crossing, an error to constrain away.

Theme: Risk & Probabilistic Forecasting · All terminology

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Mind map — connected ideas

Quantile Forecasts · Quantile Regression · Predicting Percentiles · Prediction Intervals (PI) · Probabilistic Forecasts · Cumulative Distribution Function (CDF)

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Continuous Probabilistic Forecasts · Continuous Ranked Probability Score (CRPS) · Deterministic forecasts · Full Distribution · Pinball Loss (a.k.a. Quantile Loss) · Point Forecasts · Predicting Percentiles · Prediction Intervals (PI) · Probabilistic Forecasts · Probabilistic Scoring · Probability Forecasts · Quantile Forecasts · Quantile Regression · Return Distribution