Module hankel 

McMillan degree by singular-value thresholding of the Hankel matrix of sample autocovariances (M3 route 2).

The rank of the autocovariance Hankel H[i][j] = γ(i+j+1) is the state dimension. A raw SVD cut fails because the threshold must carry the scale of the estimation error (the day-1 finding); following the calibration lesson of Zheng, Jedra & Proutière (2025, arXiv:2506.08720), the cut is

ξ_n = c · ŝ · sqrt(K · log n / n),

where ŝ² = 2 Σ_{|h| ≤ 2K} γ̂(h)² is a Bartlett-style plug-in bound on the per-entry error variance n·Var(γ̂(h)) ≈ Σ_k γ(k)² + γ(k+h)γ(k−h), and sqrt(K) is the random-matrix scaling of the operator norm of an entrywise-noise perturbation (the worst-case factor K proved hopelessly conservative in calibration: it underestimated everywhere). Recovery additionally needs separation: the smallest non-zero singular value of the true Hankel must exceed ξ_n — e.g. MA(2) with θ = (0.6, 0.3) has true Hankel singular values ≈ (0.88, 0.10), and 0.10 sits below any honest threshold at practical n; that is the problem being hard, not the threshold failing.

Structs

HankelRankConfig

Configuration of hankel_rank_order.

HankelRankResult

Output of hankel_rank_order.

HankelSvtConfig

Configuration of hankel_svt_order.

HankelSvtResult

Output of hankel_svt_order.

Enums

RankAlpha

How the sequential rank test sets its level at sample size n.

RankPvalue

How the weighted-χ²₁ null p-value is computed.

Functions

hankel_rank_order

Sequential rank test on the autocovariance Hankel: Robin & Smith’s (2000) characteristic-root statistic CRT_r = n Σ_{i>r} σ̂_i² (their Wald form, with identity weighting, so the roots are the squared singular values of the Hankel), with the null law Σ λ_i χ²₁ of their Theorem 3.2. The weights are the eigenvalues of (D ⊗ C)' Ω (D ⊗ C) with C, D the null-space singular vectors; the Hankel’s duplication structure reduces this to the small matrix (L' W)' (L' W) where Ξ = L L' is the Bartlett (Gaussian) covariance of the distinct sample autocovariances and W[ℓ, (a,b)] = Σ_{i+j+1=ℓ} C[i,a] D[j,b]. Stops at the first non-rejection (Camba-Méndez & Kapetanios’s sequence, started at r = 0); with RankAlpha::LogDecay the estimate is consistent (Hosoya 1989).

hankel_svt_order

Singular-value-thresholding estimator of the McMillan degree.