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REVIEW 3 major objections 4 minor 26 references

A Quantized Order Estimator

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that a penalty-based least-squares criterion can recover the true orders of a stochastic ARX system from quantized output alone, almost surely as sample size grows.

desk verdict Novel problem, plausible conditional proof, but the advertised concrete penalty choice is self-referential and the simulation violates the theorem's own conditions. read the letter →

arxiv 2506.00984 v1 pith:KIECWNGS submitted 2025-06-01 math.ST stat.TH

Lida Jing This is my paper
classification math.STstat.TH MSC 62F1262M1093E12
keywords quantizedoutputARXsystemsorderestimationleastsquarescontrolinformationcriterionalmostsureconsistencyquantizationsteppersistentexcitation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the unknown orders of a stochastic ARX (autoregressive with exogenous input) system can be recovered consistently from uniformly quantized output alone. The proposed estimator runs least squares at every candidate order pair in a finite grid and minimizes a criterion that adds a linear penalty proportional to the order to the sum of squared prediction errors. Under stability, bounded-input, and persistent-excitation assumptions, the paper proves that the minimizing orders converge almost surely to the true orders, provided the quantization step is small and the penalty rate per sample lies in a specified interval. The contribution is to extend order estimation, which is classically done on precise measurements, to the quantized-data setting where parameter estimates do not converge to the true values; the catch, acknowledged by the paper, is that the allowed penalty interval is expressed through the true order and unknown system constants, so the result is a consistency theorem rather than a directly implementable tuning rule.

What carries the argument

The load-bearing object is the quantized regression vector $\psi_i(p,q)=[s_i,\dots,s_{i-p+1},u_i,\dots,u_{i-q+1}]^\top$ built from quantized outputs $s_i$, together with the criterion $L_n(p,q)=\sum_{i=0}^{n-1}(s_{i+1}-\theta_n^\top(p,q)\psi_i(p,q))^2+l_n(p+q)$. The proof decomposes $\sigma_n(p,q)-\sigma_n(p_0,q^*)$ into a bias term that is positive and linear in $n$ for wrong orders and a quantization-error term of order $\varepsilon n$ for the true order; the penalty rate $l_n$ is the knife-edge that lets the bias dominate quantization noise.

What would settle it

Take the paper's simulation setup (ARX with $p_0=2$, $q_0=1$, $\varepsilon=0.001$, $p^*=3$, $q^*=3$) and choose a penalty $l_n$ such that the interval in (26)-(27) is empty, for instance by making the upper bound negative through a larger $\varepsilon$; if $\hat p_n$ still converges to 2, the interval condition is not the operative restriction, while failure would confirm that the bound is doing real work.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1 and Theorem 3.2: with $L_n(p,q^*)=\sigma_n(p,q^*)+l_n(p+q^*)$ and $V_n(p^*,q)=\sigma_n(p^*,q)+v_n(p^*+q)$, where $\sigma_n$ is the least-squares prediction-error sum, the estimates $\hat p_n$ and $\hat q_n$ converge almost surely to the true orders $p_0$ and $q_0$. The proof shows that for over-ordered models the quantized-noise contribution to $\sigma_n$ is at most $5(1+p^*c)\varepsilon n$, while for under-ordered models the missing-regressor bias contributes at least $a_{p_0}^2 c_1 n$; a penalty rate growing linearly in $n$ and bounded between these two scales selects the truth asymptotically.

Load-bearing premise

The consistency proof requires the penalty rate per sample, $l_n/n$, to lie between a lower bound set by the quantization step and an upper bound whose value depends on the true order $p_0$ and on unknown constants $a_{p_0}, c_1, \gamma, c_3$; no data-based rule is given to place $l_n$ there, so the theorem guarantees consistency only for a penalty interval one cannot compute before identifying the system.

Editorial extensions

If this is right

  • If the assumptions hold, the estimated AR order $\hat p_n$ converges almost surely to the true $p_0$, so the selected order can be trusted once enough quantized samples are collected.
  • A non-empty penalty interval forces the quantization step $\varepsilon$ to be small relative to the smallest-regressor eigenvalue, quantifying the resolution the quantizer must provide for consistency.
  • Larger search bounds $p^*, q^*$ preserve consistency but slow the convergence rate, as the numerical trajectories show.
  • The criterion can be evaluated on every candidate pair $(p,q)$ in a finite grid, making the method straightforward to implement once the penalty rate is chosen.
  • Order consistency is obtained without knowing the parameters, so the estimated order can be fed into an existing quantized parameter estimator as the correct model structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the penalty interval in (26)-(27) depends on the true order and unknown constants, turning the theorem into a practical rule requires an adaptive choice of the penalty rate; notably, the paper's own numerical values for $l_n$ fall below the theorem's lower bound, suggesting the bound is conservative rather than sharp.
  • A natural extension would be a data-driven penalty schedule, for instance one scaling like $\log n$ with constants selected by validation, that achieves the same consistency without prior knowledge of $p_0$.
  • The proof structure should transfer to other quantization schemes whose error is uniformly bounded by $O(\varepsilon)$ and to noise sequences satisfying the same moment assumptions, broadening the class of observation models for which order-consistent estimation is possible.
  • A falsifiable consequence of the consistency claim is that the order estimates should still converge for a range of penalty constants around the numerical choices used in the example; failure in some part of that range would locate the true restriction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies order estimation of ARX systems from uniformly quantized outputs. It defines a CIC-type criterion L_n(p,q) based on least-squares estimates from quantized regressors and proves, under Assumptions 3.1-3.5 and provided the penalty rate l_n lies in the interval (26)-(27), that the estimated AR order converges almost surely to p0; Theorem 3.2 gives an analogous result for the exogenous order q0. A numerical example with two quantization steps is presented. The abstract claims a concrete method for choosing quantization parameters to ensure consistency.

Significance. If Theorems 3.1-3.2 were accompanied by a data-based construction of the penalty rate, the paper would be a useful extension of quantized system identification to order estimation. The conditional consistency result is a nontrivial technical step, and the paper is clearly written where it does not depend on unverified constants. However, the advertised practical contribution is not delivered: the conditions (26)-(27) and (53)-(54) depend on the true orders and system constants that the estimator is supposed to discover, and the numerical example runs outside the theorem's hypotheses. The paper therefore does not provide an implementable consistency guarantee.

major comments (3)
  1. [Section 3.2, Theorem 3.1 (Eqs. (26)-(27))] The abstract promises "a concrete method is given for choosing quantization parameters to ensure that the system order estimates are consistent." In the body, the only guidance is Remark 3.2, which asserts that the intervals in (26)-(27) and (53)-(54) are nonempty for suitably chosen ε, α1, α2, β1, β2. No such choice is specified, and the upper bound (27) contains a_{p0}^2, c1, γ, c3 and the true order p0 itself. Remark 3.3 explicitly admits that these quantities are not accessible. Thus the load-bearing hypothesis of the theorem—that l_n lies in the stated interval—cannot be verified from data, and the central advertised contribution is missing. The same problem applies to v_n in Theorem 3.2.
  2. [Section 4, numerical example] The simulation chooses l_n=0.006n for ε=0.001, p*=3 and l_n=0.012n for ε=0.002, p*=6. Under Assumption 3.3, c≥1 because |b1|=1, so condition (26) requires l_n ≥ 5(1+p*c)ε n, i.e., at least 0.02n for the first setting and 0.07n for the second. The chosen values are lower by a factor of roughly 3 to 6, so the simulation is not carried out under the hypotheses of Theorem 3.1. The trajectories in Figs. 1-8 therefore do not demonstrate that the consistency conditions can be satisfied, and the paper gives no explanation for these l_n values in terms of (26)-(27).
  3. [Remark 3.2 and Theorem 3.1] Even the non-emptiness of the interval (26)-(27) is not established. Remark 3.2 merely states that suitable parameters make the intervals nonempty, but it does not quantify how small ε must be relative to a_{p0}^2 c1, γ, c3, and c1 and c3 are themselves not verified in the simulation. A conditional consistency theorem with an unverifiable and possibly empty parameter range does not support the paper's stated claim of a concrete method.
minor comments (4)
  1. [Equation (40)] The term a_{p0}^2 λ_min(p0,q*)(n-1) is used for a matrix whose sum runs from i=0 to n-1, while P_{n+1} in (7) has sum to n; the indexing of λ_min should be made consistent with Assumption 3.5 and Eq. (7).
  2. [Equations (13), (20), and (33)] The remainder terms o(n) are used together with "a.s."; the statements should specify whether the o(n) is uniform in the random realization and whether the almost-sure bound is meant to hold along every sample path.
  3. [Section 4] The numerical example uses uniform inputs on [-3,3] for p estimation and [-1,1] for q estimation, but the role of δ from Assumption 3.1 is not explained, and the choice of different δ values in the two simulations is not reconciled with the constants in the theorem.
  4. [References] Several references contain typographical artifacts such as "S¨ oderstr¨ om" and "Birkhauser"; these should be formatted consistently.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised concrete choice of penalty and quantization parameters is self-referential: Theorem 3.1's sufficient conditions (26)-(27) require the true order p0 and unknown constants a_{p0}, c1, γ, c3, which Remark 3.3 admits are inaccessible.

  1. self definitional [Section 3.2, Theorem 3.1, Eq. (27) and Remark 3.3; also Abstract]
    "a concrete method is given for choosing quantization parameters to ensure that the system order estimates are consistent. ... Selecting ap0, bq0, γ and γ′ in (26)-(27), (53)-(54) depends on the exact model and order of the system, and we do not have access to them."

    The only rule offered for choosing l_n (and the quantization step) is the interval (26)-(27). Its upper bound (27) contains a_{p0}^2, c1, γ, c3, and p0 itself, i.e., the true order and system constants that the estimator is supposed to find. Thus the 'concrete method' cannot be applied without already knowing the target; the advertised consistency guarantee is conditional on a self-referential selection rule. No data-based construction is supplied, and Remark 3.3 explicitly concedes the quantities are inaccessible.

full rationale

The conditional implication in Theorem 3.1 is not itself circular: the proof does not substitute the conclusion into its hypotheses, and the order estimator is not a mere renaming of the parameter estimator of Jing (2022). The reliance on Theorem 1 of Jing (2022) for the key error bound is load-bearing but has independent content (parameter estimation, not order estimation), so it is not scored as a definitional reduction. The circularity is in the paper's advertised practical recipe: the only stated way to choose l_n so that (26)-(27) hold requires the true order p0 and unknown constants a_{p0}, c1, γ, c3, which Remark 3.3 admits are inaccessible. The numerical example does not repair this gap: setting l_n=0.006n or 0.012n falls below the lower bound 5(1+p*c)εn in every configuration shown (e.g., p*=3, ε=0.001 gives at least 0.02n), so the simulation runs outside the theorem's own hypotheses. This makes the central 'concrete method' claim partially circular, though the underlying conditional consistency statement retains independent mathematical content.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central proof rests on seven domain assumptions and on a same-author theorem about quantized least squares; the penalty rates used in the simulation are free parameters. No new physical entities are introduced.

free parameters (3)
  • l_n (penalty rate in L_n) = 0.006n and 0.012n in Section 4
    Chosen by hand in the simulation; the theorem only guarantees consistency if l_n lies in an interval depending on unknown system constants (26)-(27), which is not checked.
  • v_n (penalty rate in V_n) = 0.006n and 0.012n in Section 4
    Chosen by hand in the simulation with the same non-verifiable interval condition (53)-(54).
  • quantization step epsilon = 0.001 and 0.002 in Section 4
    Design parameter; the theorem requires epsilon small enough to satisfy Assumption 3.3 and to make the penalty intervals nonempty, but no computable threshold is given.
assumptions (7)
  • domain assumption Assumption 3.1: inputs {u_i} are i.i.d. uniform on [-delta, delta].
    Used throughout for regressor growth and persistent excitation; not derived from data.
  • domain assumption Assumption 3.2: A(z) is stable.
    Standard in system identification; needed for output boundedness in Lemma 3.1.
  • domain assumption Assumption 3.3: |a_i|, |b_j| <= c and epsilon < 1/(2(1+p0*c)).
    Controls the quantization noise bound in equation (6) and appears in every main bound.
  • domain assumption Assumption 3.4: (p0,q0) lies in a known finite set with upper bounds p*, q*.
    Required for the finite search over candidate orders.
  • domain assumption Assumption 3.5: lambda_min^(p,q*)(n) >= c1(n+1) for all 0 <= p <= p*.
    Persistent excitation for the AR-order search; used in Lemmas 3.2-3.4 and Theorem 3.1.
  • domain assumption Assumption 3.6: lambda_min^(p*,q)(n) >= c2(n+1) for all 0 <= q <= q*.
    Persistent excitation for the exogenous-order search; used in Theorem 3.2.
  • domain assumption Theorem 1 of (Jing, 2022) is true and directly applicable.
    Quoted in Lemmas 3.2 and 3.4 and Appendix B to bound the least-squares estimation error; the bound is not proved or independently verified here, and the cited result is by the same author.

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Cite this review

Pith. "Pith review of A Quantized Order Estimator." pith.science (2026). https://pith.science/paper/KIECWNGS

@misc{pith2026250600984,
  author       = {Pith},
  title        = {Pith review of: A Quantized Order Estimator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIECWNGS}},
  note         = {Machine review of arXiv:2506.00984}
}
read the original abstract

This paper considers the order estimation problem of stochastic autoregressive exogenous input (ARX) systems by using quantized data. Based on the least squares algorithm and inspired by the control systems information criterion (CIC), a new kind of criterion aimed at addressing the inaccuracy of quantized data is proposed for ARX systems with quantized data. When the upper bounds of the system orders are known and the persistent excitation condition is satisfied, the system order estimates are shown to be consistent for small quantization step. Furthermore, a concrete method is given for choosing quantization parameters to ensure that the system order estimates are consistent. A numerical example is given to demonstrate the effectiveness of the theoretical results of the paper.

Figures

Figures reproduced from arXiv: 2506.00984 by the authors.

Figure 1
Figure 1. The trajectories of ˆpn with ε = 0.001, p ∗ = 3, q ∗ = 3 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The trajectories of ˆpn with ε = 0.001, p ∗ = 6, q ∗ = 6 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The trajectories of ˆpn with ε = 0.002, p ∗ = 3, q ∗ = 3 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The trajectories of ˆpn with ε = 0.002, p ∗ = 6, q ∗ = 6 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The trajectories of ˆqn with ε = 0.001, p ∗ = 3, q ∗ = 3 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: The trajectories of ˆqn with ε = 0.001, p ∗ = 6, q ∗ = 6 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: The trajectories of ˆqn with ε = 0.002, p ∗ = 3, q ∗ = 3 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n 0 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The trajectories of ˆqn with ε = 0.002, p ∗ = 6, q ∗ = 6 squares method is the key to the algorithm of this pa￾per. It is shown that the estimated order is consistent. For further research, a method is required for the ver￾ification of the assumptions and conditions in…

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