Quantile Level
1. What is a Quantile?
- A quantile is a cut-point that divides a probability distribution (or dataset) into intervals of equal probability.
- Example:
- 0.5 quantile = median (50% of values are below).
- 0.25 quantile = first quartile (Q1).
- 0.75 quantile = third quartile (Q3).
2. Quantile Level
- The quantile level refers to the probability threshold (α) at which you compute a quantile.
- Denoted as $q_\alpha$, where $0 < \alpha < 1$.
Formally: $q_\alpha = \inf \{ x : F(x) \geq \alpha \}$
where $F(x)$ = cumulative distribution function (CDF).
- Example:
- At quantile level 0.9, $q_{0.9}$ is the value such that 90% of the data lies below it.
- At quantile level 0.1, it’s the 10th percentile.
3. In Forecasting
- In quantile forecasting, you don’t predict just the mean, but specific quantiles of the future distribution.
- Example:
- 0.1 quantile forecast = lower bound (pessimistic scenario).
- 0.5 quantile forecast = median forecast.
- 0.9 quantile forecast = upper bound (optimistic scenario).
This is used to construct prediction intervals.
- A 90% prediction interval might use quantile levels 0.05 and 0.95.
4. In Quantile Regression
- Instead of modeling $E[y|x]$ (the mean), quantile regression estimates $q_\alpha(y|x)$.
- Example:
- α = 0.5 → median regression.
- α = 0.9 → regression for the 90th percentile.
- Useful when distribution is skewed or when you want to model extremes (like high sales, high demand).
5. Examples
Dataset: $[2, 3, 5, 7, 11, 13, 17, 19]$
- 0.25 quantile (25% level) = 4 (approx between 3 and 5).
- 0.50 quantile (50% level) = 9 (median, between 7 and 11).
- 0.75 quantile (75% level) = 15 (between 13 and 17).
Summary:
- Quantile level = probability threshold (α) that defines where a quantile lies.
- Example: quantile level 0.9 → value below which 90% of data falls.
- Widely used in forecasting (prediction intervals) and quantile regression to model uncertainty beyond just the mean.