Module threshold 

Vanishing thresholds τ_n and growing memory p_n for Fine–Hwang estimators.

Thresholds are expressed on the studentized scale: lag k survives iff |stat_k / se_k| > τ_n. On the raw scale of Fine & Hwang this is η_n = τ_n · se_k ≈ τ_n / sqrt(n).

Which theory justifies which threshold. Hwang’s (1978, Thm 5.3) sufficient condition p_n / (n η_n²) → 0 translates to τ_n² / p_n → ∞, i.e. the threshold must grow faster than sqrt(p_n). None of the log n-type thresholds below satisfies it with p_n = n^0.4 (the ratio p_n/τ_n² grows from 0.9 at n=100 to 5.8 at n=10⁵). Hwang’s condition comes from a Chebyshev bound and is far from necessary; the sqrt(c log n) and Gumbel-type thresholds are instead justified by maximal inequalities / extreme-value limits for the vector of long-AR estimates — see Jirak (2012, Ann. Statist.), who proves a Gumbel limit for max_{i ≤ d_n} |θ̂_i − θ_i| with d_n = O(n^δ) and derives consistent band-based order estimators from it.

Enums

Memory

How the growing memory p_n is chosen; the default was fixed by benchmark. Memory::resolve turns it into a concrete window for a sample size.

Threshold

Threshold rule on the studentized scale.

Functions

polylog_memory

Poly-logarithmic memory p_n = ceil(c · log n), clamped to [2, n/4]. This is the window size covered by the uniform law-of-the-iterated-logarithm bounds for sample autocovariances (Hannan & Deistler 1988, Thm 5.3.2; Bauer 2001, Lemma 1), hence a candidate default pending theory work. c = 2 gives 9 at n=100, 14 at n=1000, 19 at n=10⁴.

power_memory

Power-law growing memory p_n = floor(n^0.4), clamped to [2, n/4] — the Fine–Hwang-style window. Satisfies p_n²/n → 0 (Hwang) but not p_n³/n → 0 (Berk 1974 / Lewis–Reinsel 1985 uniform results). It was the package default until the benchmark showed the poly-log window dominating it (notably on seasonal AR, where n^0.4 < 12 until n ≈ 800); the default is now polylog_memory(n, 2.0) via crate::Statistic::default_memory. Kept as an option and for the benchmark.