# Literature audit C — jackknife / bias-correction classics and 2010–2026 extrapolation search

Method (verification task, 2026-10-28 18:47–19:03): Crossref-verified metadata for every item; full texts read for Dhaene–Jochmans (accepted manuscript), Hahn–Hughes–Kuersteiner–Newey 2024 (QE PDF), ZDW 2013, Hahn–Newey 2002 WP, Lin et al. 2024; secondary sources for formulas (Kosmidis 2014; Kosmidis–Kenne Pagui–Sartori 2018; Giles 2011). JSTOR / T&F / ScienceDirect / SIAM / Euclid blocked: several classics are "metadata verified, text NOT VERIFIED".

# Corrections to the brief

• Schucany, Gray & Owen (1971) JASA 66(335):524–533: DOI 10.1080/01621459.1971.10482296 (the brief's …297 is Light & Margolin).
• Sharot (1976) JASA 71(354):451–454: DOI 10.1080/01621459.1976.10480367. (Also Sharot 1976 Biometrika 63(2):315–321 "Sharpening the jackknife".)
• Cox & Snell (1968) JRSS B 30(2): 248–265 on OUP/Crossref (248–275 with discussion per JSTOR).

# Dhaene & Jochmans (2015), RES 82(3):991–1030, DOI 10.1093/restud/rdv007 (accepted manuscript read)

• Assumptions 2.1–2.3: α-mixing stationary panel; influence-function representation; bias expansion θ_T − θ0 = B₁/T + o(1/T). Eq. (2.4) half-panel jackknife θ̃_{1/2} = 2θ̂ − ½(θ̄_{S1} + θ̄_{S2}). Theorem 2.1: √(NT)(θ̃ − θ0) ⇒ N(0, Σ^{-1}) — "bias reduction can be achieved without variance inflation" (first-order).
• Higher order (§5.1, eqs. 5.1–5.3): with expansion (2.5) to order k, the h-th order jackknife θ̃_{1/G} = (1 + Σ_g a_{1/g})θ̂ − Σ_g a_{1/g} θ̄_{1/g}, weights a = (1 − ι'A^{-1}ι)^{-1}A^{-1}ι, [A]{r,s} = Σ{S∈S_{g_s}} (T/|S|)^{r−1}; for G = {2,3}: 3θ̂ − 3θ̄_{1/2} + θ̄_{1/3}, inconsistency o(T^{-2}). Supplement Theorem S.2.2: G = {2, 3, …, h+1} minimises the inflation of the non-eliminated bias terms (thirds, not quarters). First-order variance unchanged; "How to choose h optimally in practice is a difficult issue because the choice should also be guided by variance considerations" — higher-order variance left open. Overlapping subpanels inflate variance by d_g = ½g/(g−1) (Thm S.3.1).

# Hahn, Hughes, Kuersteiner & Newey (2024), Quantitative Economics 15(3):783–816, DOI 10.3982/QE2350 (full text read)

• Split-sample bias estimate b̂_SS = ½(θ̂(1) + θ̂(2)) − θ̂ is unbiased but inconsistent (O_p(1), U-statistic structure; Prop. 5, App. A.1). Panel Prop. 6: Var(θ̂_SS) ≈ Var(A₁) + 2ῦ/T vs Var(θ̂_J) ≈ Var(A₁) + ῦ/(T−1): split-sample corrections have twice the higher-order variance of leave-one-out. Prop. 7: √(nT)(b̂_SS/T − B) = O_p(T^{-1/2}) vs O_p(T^{-1}) for leave-one-out. Analytical, bootstrap and leave-one-out corrections share the same higher-order variance (Thms 1–2). Direct relevance: quantifies the finite-sample variance cost of split-sample (and quarter-sample) extrapolation.

# Classics (metadata verified; texts behind paywalls unless noted)

• Quenouille (1949) JRSS B 11(1):68–84, DOI 10.1111/j.2517-6161.1949.tb00023.x; Quenouille (1956) Biometrika 43(3/4):353–360, DOI 10.1093/biomet/43.3-4.353 (per Kosmidis 2014: nθ̂ − (n−1)θ̄ has bias −b₂/n² + O(n^{-3})).
• Tukey (1958) Ann. Math. Statist. 29(2):614 (abstract within "Abstracts of Papers" pp. 614–623, DOI 10.1214/aoms/1177706647).
• Schucany, Gray & Owen (1971): generalized jackknife (θ̂₁ − Rθ̂₂)/(1 − R); multi-term system NOT VERIFIED from the original.
• Efron (1982) SIAM CBMS-NSF 38, DOI 10.1137/1.9781611970319 (§2.3 jackknife as linear extrapolation in 1/n, per Kosmidis).
• Cox & Snell (1968): O(1/n) bias Bias(θ̂_s) = Σ k^{si}k^{jl}[½k_{ijl} + k_{ij,l}] + O(n^{-2}) (form per Giles 2011 eq. 10).
• Cordeiro & McCullagh (1991) JRSS B 53(3):629–643, DOI 10.1111/j.2517-6161.1991.tb01852.x (abstract read): first-order GLM biases via a supplementary weighted regression; form b_β = −i^{-1}X'Wξ, ξ_i = h_i d'_i/(2d_i w_i) per Kosmidis et al. 2018 eq. (4).
• Firth (1993) Biometrika 80(1):27–38, DOI 10.1093/biomet/80.1.27 (abstract read).
• Rilstone, Srivastava & Ullah (1996) J. Econometrics 75(2):369–395, DOI 10.1016/0304-4076(96)89457-7 (corrigendum 2005, 124(1):203–204); Bao & Ullah (2007) J. Econometrics 140(2):650–669, DOI 10.1016/j.jeconom.2006.07.007.
• Hahn & Newey (2004) Econometrica 72(4):1295–1319, DOI 10.1111/j.1468-0262.2004.00533.x (abstract + 2002 WP read): delete-one-period jackknife Tθ̂ − (T−1)Σθ̂_(t)/T, bias O(T^{-2}).
• Le Cam (1956) Proc. Third Berkeley Symp. 1:129–156, DOI 10.1525/9780520313880-014; Bickel (1975) JASA 70(350):428–434, DOI 10.1080/01621459.1975.10479884 (abstract); van der Vaart (1998) CUP, DOI 10.1017/CBO9780511802256, §5.7 "One-Step Estimators" p. 71 (TOC verified).
• Bhattacharya & Ghosh (1978) AoS 6(2):434–451, DOI 10.1214/aos/1176344134 (abstract); Ghosh (1994) NSF-CBMS 4, DOI 10.1214/cbms/1462297300; Pfanzagl (1985) LNS 31, DOI 10.1007/978-1-4615-6479-9.

# 2010–2026 search: nested-subsample Richardson extrapolation of MLEs + Newton in divide-and-conquer

No paper found that extrapolates over nested sub-sample sizes m, m/2, m/4 for MLEs/M-estimators and combines it with a Newton correction in a distributed setting. Nearest: ZDW (2013) SAVGM (one-level subsampled extrapolation, Theorem 4, eq. 8); Hahn et al. (2024) (variance of split-sample corrections); Bach (2021) SIMODS 3(4):1251–1277, DOI 10.1137/21M1397349 (Richardson in iterations/regularisation, not sample size); Lin, Su, Mou, Ding & Wainwright (2024) arXiv 2411.02909 (leave-one-out jackknife, high-dim Z-estimators); Wu, Zhu & Wang (2023) Statistica Sinica 33:2041–2064, DOI 10.5705/ss.202021.0257; Chudik, Pesaran & Yang (2018) JAE 33(6):816–836, DOI 10.1002/jae.2623; Fernández-Val & Weidner (2016) J. Econometrics 192(1):291–312, DOI 10.1016/j.jeconom.2015.12.014.