pub enum Threshold {
BicLike {
c: f64,
},
Lil {
c: f64,
},
MaxOfK {
c: f64,
},
Fixed(f64),
Gumbel {
alpha: f64,
},
}Expand description
Threshold rule on the studentized scale.
Variants§
BicLike
sqrt(c · log n). With c = 1 this equals the per-increment threshold implicit in BIC
(adjacent-order BIC difference is negative iff t_k² > log n to first order) — this is
not the same as BIC’s global argmin. Requires c > 0.
Lil
sqrt(2 c · log log n): Hannan–Quinn / law-of-the-iterated-logarithm rate. Requires c > 0;
consistency arguments need c > 1.
MaxOfK
sqrt(2 · log p_n) + c: Bonferroni/extreme-value correction for the number of lags
inspected — the natural companion of Jirak’s (2012) Gumbel limit. Requires c ≥ 0.
Fixed(f64)
Fixed value on the studentized scale, e.g. 1.96 reproduces the Box–Jenkins band
(not consistent; for benchmarking). Requires a finite value ≥ 0.
Gumbel
Level-calibrated threshold: the 1−alpha quantile of the Gumbel limit of
max_{k ≤ p_n} |Z_k|, Z_k approximately standard normal, i.e.
b_p + a_p · (−log(−log(1−alpha))) with a_p = 1/sqrt(2 log p) and
b_p = sqrt(2 log p) − (log log p + log π) / (2 sqrt(2 log p)) (absolute values ⇒ 2p tails).
With a fixed alpha this controls the family-wise overestimation probability at ≈ alpha
(not consistent); with alpha = alpha_n → 0 slowly it is the Jirak-type consistent rule.
Jirak (2012) gives the exact normalising constants for Yule–Walker estimates
(see Statistic::JirakBand); this generic form uses the universal Gumbel constants.
Requires 0 < alpha < 1.