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arxiv: 2605.01349 · v1 · submitted 2026-05-02 · 📡 eess.SY · cs.SY

Sequentially decoupling estimators for Box-Jenkins model estimation

Biqiang Mu This is my paper

Pith reviewed 2026-05-09 18:28 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords Box-Jenkins modelssystem identificationsequentially decoupling estimatorGauss-Newton iterationprediction error methodARX modeloutput-error modelclosed-loop identification
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The pith

A sequentially decoupling estimator for Box-Jenkins models, refined by one Gauss-Newton step, is asymptotically equivalent to the prediction error method when the ARX order grows mildly with sample size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a two-stage estimation procedure for Box-Jenkins models that works with both open-loop and closed-loop data. The first stage builds a consistent sequentially decoupling estimator through three ordinary least-squares fits: a high-order autoregressive model with exogenous inputs, followed by two auxiliary output-error models that isolate the process dynamics and then the noise dynamics. A single Gauss-Newton iteration applied to this starting point then produces the final estimate. Under a mild growth condition on the initial autoregressive order, the refined estimate matches the statistical performance of the full prediction-error method without requiring its complete iterative optimization. Readers would care because the approach offers a practical route to efficient parameter estimates using only standard least-squares routines as building blocks.

Core claim

The paper shows that the sequentially decoupling estimator, constructed from three sequential least-squares steps consisting of a high-order ARX model followed by an auxiliary output-error model for the dynamics and another auxiliary output-error model for the noise, is consistent for Box-Jenkins models under standard regularity conditions. One Gauss-Newton iteration started from this estimator yields an estimator that is asymptotically equivalent to the prediction error method, provided the ARX model order satisfies a mild growth condition with the sample size.

What carries the argument

The sequentially decoupling estimator, built by chaining three least-squares steps (high-order ARX, auxiliary OE for dynamics, auxiliary OE for noise), which supplies a consistent initial value that one Gauss-Newton iteration converts into an asymptotically efficient result.

If this is right

  • The sequentially decoupling estimator itself is consistent for Box-Jenkins models under open-loop and closed-loop conditions.
  • The one-step Gauss-Newton refinement from the initial estimator achieves the same asymptotic efficiency as direct prediction-error minimization.
  • The procedure provides a simpler computational alternative to weighted null-space fitting methods.
  • All stages rely on standard least-squares solvers without needing specialized nonlinear optimization from the outset.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decoupling structure may offer robustness advantages in closed-loop data where full prediction-error methods encounter correlation between input and noise.
  • In practice the mild growth condition could be met by setting the ARX order to grow like log of sample size, which is easy to implement and check via Monte Carlo trials.
  • The same sequential pattern might apply to other linear model classes where dynamics and noise can be isolated through auxiliary output-error fits.

Load-bearing premise

The ARX model order must grow mildly with sample size so that the one-step Gauss-Newton refinement achieves asymptotic equivalence to the prediction error method.

What would settle it

A simulation study or asymptotic analysis in which the ARX order is held fixed while sample size increases, showing that the refined estimator's covariance exceeds the prediction error method's covariance.

Figures

Figures reproduced from arXiv: 2605.01349 by Biqiang Mu.

Figure 1
Figure 1. Figure 1: The average MSEs over 1000 Monte Carlo runs. view at source ↗
Figure 2
Figure 2. Figure 2: The boxplot of the fits given by the estimators under view at source ↗
Figure 3
Figure 3. Figure 3: The boxplot of the running times of the estimators w view at source ↗
Figure 4
Figure 4. Figure 4: The boxplot of the number of iterations for the esti view at source ↗
read the original abstract

In this paper, we propose a consistent and asymptotically efficient estimation method for Box-Jenkins (BJ) models that is applicable under both open-loop and closed-loop data conditions, serving as a possible alternative to the weighted null-space fitting approach. The method comprises two stages: an initial sequentially decoupling (SD) estimator, followed by Gauss-Newton (GN) refinement step. The SD estimator is constructed from three sequential least squares (LS) estimators: (i) estimation of a high-order autoregressive model with exogenous inputs (ARX) model; (ii) estimation of the BJ model's dynamic model via an auxiliary output-error (OE) model; and (iii) estimation of the noise model of the BJ model using another auxiliary OE model. We establish the consistency of the SD estimator under standard regularity conditions, leveraging the consistency of the underlying LS estimators for both the ARX and OE models. Moreover, we show that one-step GN iteration starting from the SD estimator yields an estimator that is asymptotically equivalent to the prediction error method, provided the ARX model order satisfies a mild growth condition. Simulation studies confirm the theoretical properties of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proposes a sequentially decoupling (SD) estimator for Box-Jenkins models consisting of three least-squares steps (high-order ARX, auxiliary OE for the dynamics, auxiliary OE for the noise model) followed by a single Gauss-Newton iteration. It claims that the SD estimator is consistent under standard regularity conditions by leveraging consistency of the underlying ARX and OE LS estimators, and that the one-step GN refinement is asymptotically equivalent to the prediction-error method (PEM) when the ARX order satisfies a mild growth condition with sample size. The method is positioned as applicable to both open- and closed-loop data and as an alternative to weighted null-space fitting.

Significance. If the asymptotic equivalence holds with the stated growth condition, the approach would supply a computationally attractive, consistent and asymptotically efficient estimator for BJ models that avoids the iterative optimization of full PEM while retaining its statistical properties; this would be particularly useful in closed-loop identification settings where direct PEM can be sensitive to initialization.

major comments (1)
  1. [Abstract / asymptotic-equivalence derivation] Abstract and the section deriving asymptotic equivalence: the claim that one GN step from the SD estimator is asymptotically equivalent to PEM requires the initial estimator to satisfy ||θ_SD − θ_0|| = o_p(N^{−1/4}) so that the linearization remainder vanishes. The manuscript only invokes consistency of the three LS stages plus a “mild growth condition” on the ARX order n(N); no explicit rate bound (e.g., n = o(N^{1/5}) or similar) is derived that guarantees the required o_p(N^{−1/4}) rate, especially under closed-loop feedback where the ARX bias-variance trade-off depends on the relative growth of n(N) and the feedback bandwidth.
minor comments (1)
  1. [Simulation studies] The simulation section should report the exact ARX orders used relative to sample size and the observed convergence rates of the SD estimator to confirm the growth condition is satisfied in the reported experiments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We appreciate the referee's detailed review and insightful comments on our manuscript. We address the major concern regarding the asymptotic equivalence derivation below.

read point-by-point responses

  1. Referee: [Abstract / asymptotic-equivalence derivation] Abstract and the section deriving asymptotic equivalence: the claim that one GN step from the SD estimator is asymptotically equivalent to PEM requires the initial estimator to satisfy ||θ_SD − θ_0|| = o_p(N^{−1/4}) so that the linearization remainder vanishes. The manuscript only invokes consistency of the three LS stages plus a “mild growth condition” on the ARX order n(N); no explicit rate bound (e.g., n = o(N^{1/5}) or similar) is derived that guarantees the required o_p(N^{−1/4}) rate, especially under closed-loop feedback where the ARX bias-variance trade-off depends on the relative growth of n(N) and the feedback bandwidth.

    Authors: We agree with the referee that establishing the o_p(N^{-1/4}) rate for the SD estimator is necessary to rigorously justify the asymptotic equivalence of the one-step GN refinement to the PEM. The manuscript relies on a mild growth condition for the ARX order to ensure consistency and the desired asymptotic properties, but we acknowledge that the explicit rate bound guaranteeing ||θ_SD − θ_0|| = o_p(N^{-1/4}) was not derived in detail, particularly for the closed-loop case. In the revised version, we will include a derivation of the convergence rate of the SD estimator under the specified growth condition on n(N). This will involve analyzing the bias-variance trade-off in the high-order ARX estimation step and showing that the condition ensures the required rate even in the presence of feedback, by relating n(N) to the feedback bandwidth. We believe this addition will strengthen the theoretical contribution without altering the main results. revision: yes

Circularity Check

0 steps flagged

No circularity; claims derived from external standard results on LS consistency

full rationale

The paper constructs the SD estimator explicitly from three sequential LS steps (high-order ARX, auxiliary OE dynamics, auxiliary OE noise) and invokes the known consistency of those underlying LS estimators under standard regularity conditions to establish consistency of the SD estimator. The one-step GN refinement is shown to be asymptotically equivalent to PEM by appealing to general one-step Newton theory once the initial estimator satisfies a rate condition, which the paper claims follows from a mild growth condition on ARX order. No step reduces by construction to its own fitted values, renames a known result, or relies on a load-bearing self-citation whose content is unverified; the derivation chain remains independent of the target quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the consistency properties of ordinary least-squares estimators for ARX and OE models together with a growth condition on the ARX order; no new entities are postulated and no parameters are fitted beyond those already present in the model class.

free parameters (1)
  • ARX model order
    Must grow with sample size under a mild condition; the precise rate is not fixed by the method but is required for the asymptotic equivalence result.
axioms (1)
  • domain assumption Standard regularity conditions for consistency of least-squares estimators for ARX and OE models
    Invoked to transfer consistency from the auxiliary models to the SD estimator.

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