Negated tick loss for quantile forecasts

Computes the positively oriented (negated) tick/quantile loss (Koenker & Bassett, 1978).

Usage

tick_loss(q, y, alpha)

Arguments

q

Numeric vector. Quantile forecasts at level alpha.

y

Numeric vector. Realised outcomes.

alpha

Numeric in (0,1). Quantile level.

Value

Numeric vector of negated tick loss scores. Higher = better.

Details

The standard tick loss is $$\rho_\alpha(u) = u \left(\alpha - \mathbb{1}(u < 0)\right),$$ where \(u = y - q_\alpha\) is the forecast error. This is loss-oriented (lower = better), so the function negates it: $$S_T(q, y; \alpha) = -(y - q)\left(\alpha - \mathbb{1}(y < q)\right).$$

Tick loss is unbounded on general real-valued outcomes. Bounds derived from an empirical data range are ex-post and do not provide theorem-valid constants for finite-sample Hoeffding/Bernstein confidence sequences or e-processes.

Sign convention: the negation means hat_delta_t > 0 when forecaster p has smaller tick loss, hence a better quantile forecast, than forecaster q.

Examples

q <- c(1.0, 1.5, 2.0)
y <- c(1.2, 1.4, 2.3)
tick_loss(q, y, alpha = 0.5)
#> [1] -0.10 -0.05 -0.15