Computes the inverse CDF: Q(u | a, b) = F⁻¹(u).
Arguments
- u
Numeric vector of probability levels in the unit interval.
Values u ≤ 0 return 0 and u ≥ 1 return 1.
- a
Numeric; shape parameter of the Gamma prior on α (a > 0).
- b
Numeric; rate parameter of the Gamma prior on α (b > 0).
Value
Numeric vector of quantile values Q(u | a, b).
Details
The quantile function has the closed form:
$$Q_{w_1}(u | a, b) = 1 - \exp\left(b \left[1 - (1-u)^{-1/a}\right]\right)$$
The implementation computes (1-u)^(-1/a) in log space for stability
when u is close to 1.
Numerical Note: For small values of a (a < 1) and u close to 1,
the quantile approaches 1 very rapidly and may round to 1.0 in double
precision.
References
Lee, J. (2025). RN-06: Dual-Anchor Design II, Corollary 2.
Examples
# Median of w₁
quantile_w1(0.5, a = 2, b = 1) # ~0.339
#> [1] 0.3391402
# 90th percentile
quantile_w1(0.9, a = 2, b = 1) # ~0.732
#> [1] 0.8849373
