CDF of First Stick-Breaking Weight w₁

Computes P(w₁ ≤ x | a, b) using the closed-form expression derived by marginalizing over α ~ Gamma(a, b).

Usage

cdf_w1(x, a, b)

Arguments

x: Numeric vector. Values outside the unit interval are allowed but are mapped to the boundary values of the CDF (0 for x ≤ 0, 1 for x ≥ 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 CDF values F(x | a, b) with same length as x.

Details

The unconditional CDF is given by: $$F_{w_1}(x | a, b) = 1 - \left(\frac{b}{b - \log(1-x)}\right)^a$$

The implementation uses log1p and expm1 for numerical stability, particularly when the CDF is close to 0 (small x).

Interpretation

The weight w₁ is in GEM (size-biased) order, not ranked by size. It represents the asymptotic cluster share of a randomly chosen unit, not the largest cluster proportion. See RN-06 §1 for details.

References

Lee, J. (2025). RN-06: Dual-Anchor Design II.

Vicentini, C. and Jermyn, I. H. (2025). Prior selection for the precision parameter of Dirichlet Process Mixtures. arXiv:2502.00864.

Examples

# P(w₁ ≤ 0.3) under standard prior
cdf_w1(0.3, a = 2, b = 1)
#> [1] 0.4566891

# Vectorized computation
cdf_w1(c(0.1, 0.3, 0.5, 0.7), a = 1.6, b = 1.22)
#> [1] 0.1241267 0.3365805 0.5131689 0.6666263