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XMSE-Aware Adaptive Empirical Bayes Estimation

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XMSE-Aware Adaptive Empirical Bayes Estimation

arXiv:2606.26975v1 [stat.ML] 25 Jun 2026

Minghao Chen and Jiale Zheng

Abstract—Empirical Bayes (EB) estimators can match the first-order asymptotic risk of maximum likelihood (ML) while behaving very differently at second order: recent excess mean squared error (XMSE) analysis shows that kernel-based EB estimation may be worse than ML when the kernel is poorly aligned with the true parameter. This paper turns that diagnostic into a design principle. We propose an XMSE-aware mixed estimator that interpolates between ML and EB shrinkage. Its fixed-weight XMSE is a scalar quadratic, yielding a closed-form oracle mixing weight that is no worse than both ML and the base EB estimator at the XMSE scale. A plug-in implementation based on finite-sample XMSE approximations is proved consistent, with a second-order oracle regret rate for an interior oracle weight. We further establish a transfer of the regret bound to the fixedweight risk curve evaluated at the selected weight, a thresholded boundary rule, and extensions to compact kernel families and to finite and growing kernel dictionaries with high-probability oracle bounds. Finite impulse response simulations with SUREtuned, hard-selection, and trace-corrected baselines, together with the public Silverbox and Cascaded Tanks benchmarks, show that the proposed estimator retains most of the benefit of regularization when it is helpful and retreats toward ML under kernel misspecification, with an identified finite-sample calibration failure mode analyzed on the benchmarks. Index Terms—Empirical Bayes, excess mean squared error, regularized system identification, hyperparameter estimation, kernel selection.

I. I NTRODUCTION Empirical Bayes (EB) and kernel-based regularized estimators are widely used in linear regression and system identification [1]–[3]. Their first-order asymptotic MSE often coincides with that of the maximum likelihood (ML) estimator, so first-order theory cannot explain when regularization helps or hurts. The XMSE framework addresses this limitation by studying the second-order difference h i b = lim N 2 MSE(θ) b − MSE(θbML ) . XMSE(θ) N →∞

Existing XMSE analysis is primarily diagnostic: it decomposes the EB risk gap and identifies kernel-parameter misalignment as a source of positive XMSE. This paper turns the diagnostic into a design principle. The main contributions are as follows. 1) We introduce an XMSE-aware mixed estimator that interpolates between ML and a base EB estimator. 2) We derive the fixed-weight XMSE formula and show that its oracle mixing weight has a closed form, including explicit improvement regimes that explain when the oracle rule selects ML, a strict mixture, or EB. M. Chen is with Tencent Technology, Shenzhen, China (e-mail: [email protected]). J. Zheng is with Huawei Noah’s Ark Lab, Shenzhen, China (e-mail: [email protected]).

3) We prove that the oracle mixed estimator is no worse than both ML and the base EB estimator at the XMSE scale. 4) We develop a plug-in implementation based on finitesample XMSE component estimates and establish consistency, a boundary-robust plug-in oracle inequality, and a second-order oracle regret bound for the selected weight at an interior oracle optimum. We further prove that the inherited expansion holds uniformly in the mixing weight, so the regret bound transfers from the XMSE criterion to the deterministic fixed-weight risk curve evaluated at the selected weight; the corresponding expansion for the randomly weighted estimator itself is posed as an open problem. We also give a thresholded zero-bias boundary rule. For the scaled-kernel case, we provide primitive sufficient conditions for consistency of the plug-in XMSE components, strengthen these conditions to compact scaled-kernel families by uniform convergence, and extend the same logic to compact multi-parameter kernel families under uniform criterion convergence. 5) We show that the same plug-in criterion yields a finiteset oracle inequality and, under a positive oracle gap, a consistent selector over a finite candidate set of kernels. The same argument extends to growing candidate dictionaries under a uniform plug-in approximation condition, with a high-probability oracle bound and a sub-Gaussian rate specialization. 6) We demonstrate, in FIR system-identification simulations that include SURE-tuned, hard-selection, tracecorrected, and marginal-likelihood baselines, that the proposed estimator protects against kernel mismatch while retaining most of the benefit of EB regularization under better-aligned choices. On the public Silverbox and Cascaded Tanks benchmarks we report both the intended retreat behavior and an identified finite-sample calibration failure mode of the plug-in components. II. R ELATED W ORK The empirical Bayes view of shrinkage estimation goes back to Robbins [4] and to the classical James–Stein and empirical Bayes literature on risk reduction [1], [5], [6]. Modern parametric empirical Bayes methods use data-dependent hyperparameters to tune prior or penalty structure [7], [8]. SURE-type risk estimation traces to Stein’s unbiased risk estimate [9], while GCV and kernel smoothing have a long connection to spline and Gaussian-process methods [10], [11]. In system identification, regularized FIR estimation builds on the classical prediction-error framework [12], and kernelbased regularization has become a standard way to encode

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smoothness and decay of impulse responses [3], [13]–[24]. Hyperparameters are often selected by marginal likelihood, SURE, or GCV, with scalable implementation and robustness issues studied in [25], [26]; their asymptotic properties have been studied in [27]–[31], and a unified family of hyperparameter estimators linking EB and SURE has been proposed in [32]. The present paper is closest to the recent XMSE analysis of empirical Bayes estimators [34], which shows that first-order asymptotic risk is insufficient for distinguishing EB and ML estimators and that kernel misalignment can make EB worse than ML. A related line gives deterministic conditions under which kernel-based regularization cannot improve on the leastsquares (ML) estimate [33]; rather than certifying such cases in advance, our rule detects harmful regularization from data at the second-order risk scale and retreats toward ML. Our contribution is complementary: instead of using XMSE only as an explanatory quantity, we use the same second-order risk scale to design a mixed estimator, derive its oracle rule, and analyze the plug-in implementation. The fixed-weight quadratic criterion, closed-form projected oracle weight, plug-in oracle inequality, XMSE-transfer result, thresholded boundary rule, and finite/growing dictionary selectors are therefore decision rules and guarantees built on top of the reference decomposition, not direct restatements of the diagnostic XMSE formula. Another recent XMSE-based direction constructs generalized Bayes and closed-form biased estimators that match the XMSE of an EB regularized estimator while avoiding hyperparameter estimation [35]. That line replaces the EB implementation by computationally simpler estimators with comparable secondorder behavior. In contrast, the present paper keeps the base EB estimator available and uses XMSE to decide how much to mix it with ML, including plug-in oracle-regret, XMSEtransfer, boundary, and kernel-selection guarantees. III. P ROBLEM S ETUP Consider the finite impulse response model Y = Φθ0 + E,

E ∼ N (0, σ 2 IN ),

where Φ ∈ RN ×n is deterministic with full column rank and 1 ⊤ Φ Φ → Σ ≻ 0. N Let θbML = (Φ⊤ Φ)−1 Φ⊤ Y be the ML estimator. Throughout the paper, the risk of an estimator θb is the parameter-space mean squared error b = E ∥θb − θ0 ∥2 , MSE(θ) 2 where the expectation is over the noise E, and Π[0,1] (x) = min{1, max{0, x}} denotes the projection onto [0, 1]. In this Gaussian model the ML estimator is exactly unbiased, E[θbML ] = θ0 , with covariance σ 2 (Φ⊤ Φ)−1 . Let θbEB = θbEB (b η (θbML )) be an EB estimator whose hyperparameter estimate ηb is a function of θbML , as in the reference XMSE analysis [34].

Assumption 1 below states, in self-contained moment form, the minimal part of the reference XMSE expansion XMSE(θbEB ) = BEB + VEB + HEB needed for the mixed estimator, where BEB , VEB , and HEB are, respectively, the second-order squared-bias, variance, and hyperparameter-estimation components. IV. XMSE-AWARE M IXED E STIMATOR For a mixing weight ω ∈ [0, 1], define θbmix (ω) = θbML + ω(θbEB − θbML ). The endpoints recover ML and EB: θbmix (0) = θbML ,

θbmix (1) = θbEB .

V. M AIN R ESULTS The results below are organized by the strength of the assumption needed. The fixed-weight and oracle results use only the inherited XMSE expansion for the base EB estimator. The plug-in weight results add consistency of the estimated XMSE components. The compact scaled-kernel corollary gives a uniform continuous-mapping route for practical hyperparameter searches, while the compact multi-parameter result replaces closed-form scale consistency with uniform criterion and component convergence over a compact kernel-parameter set. The XMSE-transfer theorem evaluates the deterministic fixed-weight risk curve at the selected weight; its key step, uniform convergence of that curve over the weight interval, is proved from the inherited expansion rather than assumed. The kernel-selection results replace scalar component consistency by uniform approximation of the minimized criterion over the candidate dictionary. This separation is important: the paper does not claim a general nonsmooth random-weight XMSE expansion—the risk of the estimator with the data-dependent weight inside the expectation is explicitly left open—and the boundary and dictionary results are stated under the weaker conditions needed for their respective conclusions. For orientation, the theory proceeds in six layers. The fixedweight result uses only the inherited EB XMSE expansion and gives the exact quadratic criterion for θbmix (ω). The oracle result minimizes this deterministic criterion over [0, 1] and gives dominance over both endpoints. The plug-in results add consistent component estimates and give consistency, an oracle inequality, and an interior second-order regret rate. The compact scaled-kernel corollary supplies a uniform route for continuous searches over scaled kernel families, and the compact multi-parameter result supplies a uniform argmin route for more general continuous kernel families. The XMSEtransfer result shows that the fixed-weight risk curve converges to the quadratic criterion uniformly in the weight, so the criterion regret bound carries over to that curve at the selected weight. Finally, the kernel-dictionary results replace scalar component consistency by uniform plug-in approximation over the candidate set.

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Assumption 1 (Inherited XMSE expansion): Let dN = θbEB − θbML and split it along the hyperparameter map, as in the reference decomposition [34], into

Moreover, XMSE(θbmix (ω ⋆ )) ≤ min{0, XMSE(θbEB )}. Proof. By Theorem 1, the oracle problem is the scalar convex minimization

dN = d1,N + d2,N , with the fixed-hyperparameter part and the hyperparameterestimation part d1,N = θbEB (b η (θ0 )) − θbML , d2,N = θbEB (b η (θbML )) − θbEB (b η (θ0 )). All moments below exist for every N , and the following limits hold: (i) (bias) N E[dN ] → bEB , and BEB = ∥bEB ∥22 ; (ii) (variance component) 2N 2 E{(θbML − θ0 )⊤ d1,N } → VEB ; (iii) (hyperparameter-estimation component) 2N 2 E{(θbML − θ0 )⊤ d2,N } → HEB ; (iv) (higher-order term) N 2 E ∥dN − E[dN ]∥22 → 0. Remark 1 (Relation to the reference decomposition): Conditions (i)–(iv) are a self-contained moment restatement of the limits established in [34] for kernel-based EB estimators with hyperparameter estimators that are functions of θbML : (i) is the second-order bias limit, (ii) and (iii) are the limits of the variance and hyperparameter-estimation cross terms in the reference decomposition, and (iv) is the vanishing N 2 limit of the centered higher-order term. Since E[θbML ] = θ0 and E∥dN ∥22 = ∥E[dN ]∥22 + E∥dN − E[dN ]∥22 , conditions (i)–(iv) immediately reproduce the reference expansion XMSE(θbEB ) = BEB + VEB + HEB . For orientation on the orders involved: in the scaled-kernel case d1,N = Op (N −1 ) with mean of exact order N −1 , while its centered part and d2,N ’s centered part are Op (N −3/2 ), so (iv) reflects the fact that the EB correction concentrates around its mean faster than the N −1 bias scale. Theorem 1 (Fixed-weight XMSE): Under Assumption 1, for any fixed ω ∈ [0, 1], XMSE(θbmix (ω)) = ω 2 BEB + ωVEB + ωHEB . Proof. Since θbmix (ω) − θbML = ωdN with deterministic ω, expanding the square gives the exact identity MSE(θbmix (ω)) − MSE(θbML ) = 2ω E{(θbML − θ0 )⊤ dN } + ω 2 E ∥dN ∥2 . 2

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Multiply by N and take limits term by term. By (ii) and (iii) of Assumption 1, the cross term converges to ω(VEB + HEB ). For the quadratic term, write E∥dN ∥22 = ∥E[dN ]∥22 + E∥dN − E[dN ]∥22 ; by (i) the first part contributes ω 2 BEB , and by (iv) the second part vanishes in the N 2 limit. □ Theorem 2 (Oracle XMSE weight): Let CEB = VEB + HEB . An XMSE-minimizing fixed mixing weight over [0, 1] is    CEB   Π − , BEB > 0,    [0,1] 2BEB ω⋆ =  BEB = 0, CEB < 0,  1,  0, BEB = 0, CEB ≥ 0.

min q(ω), ω∈[0,1]

q(ω) = BEB ω 2 + CEB ω.

If BEB > 0, the unconstrained minimizer is −CEB /(2BEB ) and projection gives the constrained minimizer. If BEB = 0, then q is linear; the minimizer is 1 when CEB < 0 and 0 otherwise. Since the feasible set contains ω = 0 and ω = 1, q(ω ⋆ ) ≤ q(0) = 0,

q(ω ⋆ ) ≤ q(1) = XMSE(θbEB ),

which proves the dominance claim. □ Corollary 1 (Oracle improvement regimes): Assume BEB > 0 and write CEB = VEB + HEB . The minimized oracle XMSE criterion is   0, CEB ≥ 0,     C2 −2BEB ≤ CEB < 0, q(ω ⋆ ) = − EB ,  4BEB     BEB + CEB , CEB < −2BEB . Consequently the oracle rule returns ML when the linear XMSE component is nonnegative, uses a strict mixture when the component is negative but not large enough to offset the full EB bias penalty, and returns EB when the linear benefit dominates that penalty. Proof. For BEB > 0, the unconstrained minimizer of q(ω) = BEB ω 2 + CEB ω is −CEB /(2BEB ). If CEB ≥ 0, its projection is 0 and q(ω ⋆ ) = 0. If −2BEB ≤ CEB < 0, the unconstrained minimizer lies in [0, 1], and substitution gives 2 /(4BEB ). If CEB < −2BEB , the projection is q(ω ⋆ ) = −CEB 1 and q(ω ⋆ ) = BEB + CEB . □ VI. S CALED K ERNEL S PECIAL C ASE For implementation and experiments we consider the scaledkernel family P (η) = ηK,

K ≻ 0,

η > 0,

where K is fixed. The fixed kernels used in this paper are, for k, l = 1, . . . , n and a decay parameter γ ∈ (0, 1), KRI [k, l] = 1{k = l}, KDI [k, l] = γ k−1 1{k = l}, KTC [k, l] = γ max(k,l) , γ 3 max(k,l) γ k+l+max(k,l) − , 2 6 i.e., ridge (RI), diagonal-decay (DI), tuned/correlated (TC), and stable spline (SS), matching the kernel definitions in [13], [14], [34]. In particular, KSS divides the whole first term by two, not the exponent; at n = 20 and γ = 0.95 its condition number is 7.55 × 106 , which reproduces the ill-conditioning reported for the SS kernel in [34]. Let K −1 be denoted by Q and define S1 = Σ−1 , S2 = Σ−2 . KSS [k, l] =

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For the scaled EB hyperparameter estimator, the limiting scale is θ⊤ Qθ0 ηb⋆ = 0 , αn where α > 0 is a fixed normalization constant of the scale rule, set to α = 1 in all experiments. For the SURE/GCV-type hyperparameter estimator, the limiting scale is ηy⋆ =

θ0⊤ QS1 Qθ0 . αTr(S1 Q)

For either choice of η ⋆ , the fixed-hyperparameter bias and variance components of the regularized estimator are B=

σ4 ⊤ θ QS2 Qθ0 , (η ⋆ )2 0

and V =−

2σ 4 Tr(S1 QS1 ). η⋆

The hyperparameter-estimation component depends on the hyperparameter estimator. For scaled EB, 4

Hb =

4σ θ⊤ QS2 Qθ0 . αn(ηb⋆ )2 0

For SURE/GCV-type scale estimation, Hy =

4σ 4 θ⊤ QS1 QS2 Qθ0 . αTr(S1 Q)(ηy⋆ )2 0

The last quadratic form need not be nonnegative when Q and Σ do not commute, which matches the phenomenon observed in the reference XMSE analysis. In the notation of Assumption 1, these B, V , and H are the scaled-kernel instances of the second-order components BEB , VEB , and HEB ; the plug-in p b− consistency statements below (B → BEB , etc.) are to be read under this identification. Remark 2 (Implementation of the hyperparameter estimators): The experiments implement the scaled EB and SUREtype hyperparameter estimates as the explicit one-step plug-in rules b −1 QθbML θb⊤ QθbML θb⊤ QΣ ηbb = ML , ηby = ML , b −1 Q) αn αTr(Σ b = Φ⊤ Φ/N , i.e., the limiting scale maps η ⋆ and ηy⋆ with Σ b evaluated at θbML . These are not the finite-sample minimizers of the EB and SURE criteria of [34]; the two implementations differ at finite N . They nevertheless have the same XMSE: by the reference analysis, the XMSE of θbEB (b η (θbML )) depends on the hyperparameter map only through its probability limit η ⋆ (θ0 ) and the derivative of the estimator map at θ0 , and for the scaled-kernel family the criterion minimizers and the onestep rules above share both quantities (cf. Corollaries 2.3–2.6 of [34], where the SURE and GCV minimizers are also shown to share the same limit and XMSE). Finite-sample GCV, by contrast, is implemented as an actual criterion minimization, by golden-section search on log η, so that the experiments contain at least one criterion-minimizing implementation; its limiting scale is again ηy⋆ . This asymmetry—closed-form onestep rules for EB/SURE, numerical criterion minimization for GCV—is deliberate: it keeps the EB/SURE rows exactly

reproducible in closed form while letting the GCV rows probe the finite-sample behavior of a minimized criterion. The plugb ηb) in in XMSE components replace (θ0 , Σ, η ⋆ ) by (θbML , Σ, the displayed formulas. Theorem 3 (Primitive plug-in consistency for scaled kernels): Fix n and a positive definite kernel K with Q = K −1 . p b = Φ⊤ Φ/N → Σ ≻ 0 and θbML − Assume Σ → θ0 . Assume also p that σ 2 is known, or is replaced by an estimator σ b2 − → σ2 . For the scaled EB rule, suppose θ0⊤ Qθ0 > 0 and define the plug-in scale by θb⊤ QθbML . ηbb = ML αn For the SURE/GCV-type rule, suppose θ0⊤ QΣ−1 Qθ0 > 0 and define b −1 QθbML θb⊤ QΣ . ηby = ML b −1 Q) αTr(Σ Then the plug-in quantities obtained from the scaled-kernel formulas satisfy p b− B → B,

p Vb − → V,

p b − H →H

for the corresponding hyperparameter rule. The same conclusion holds uniformly over any finite candidate set of positive definite kernels whose limiting scale denominators are nonzero. Proof. The ML consistency follows from the fixed-n deterministic-design regression assumptions because θbML − θ0 = (Φ⊤ Φ)−1 Φ⊤ E = Op (N −1/2 ). Matrix inversion is b −1 → Σ−1 and continuous on the positive definite cone, so Σ −2 −2 b Σ → Σ . The quadratic forms, traces, and scale maps disb σ played above are continuous functions of (θbML , Σ, b2 ) on the event where the scale denominators are bounded away from zero. The nonzero limiting denominator assumptions make this event have probability tending to one. The continuous mapping theorem therefore gives consistency of the plug-in scale and of each displayed plug-in component B, V , and H. For a finite candidate set, pointwise convergence for each kernel implies convergence of the maximum error over candidates. □ Corollary 2 (Uniform consistency for compact scaled-kernel families): Let Ξ be compact and let K(ξ) ≻ 0 be continuous on Ξ, with Q(ξ) = K(ξ)−1 . Assume the same fixed-order p b → Σ ≻ 0 and θbML − design conditions as above, with Σ → θ0 . For the scaled EB rule assume inf θ0⊤ Q(ξ)θ0 > 0,

ξ∈Ξ

and for the SURE/GCV-type rule assume inf θ0⊤ Q(ξ)Σ−1 Q(ξ)θ0 > 0.

ξ∈Ξ

Then the scaled-kernel plug-in scales and the corresponding XMSE components converge uniformly over Ξ. In particular, any compact hyperparameter search whose empirical criterion converges uniformly satisfies the component convergence condition in Theorem 4 below. Proof. Continuity and compactness imply that the eigenvalues of K(ξ) are bounded away from zero and infinity on Ξ, so Q(ξ) is continuous and uniformly bounded. The displayed nondegeneracy assumptions, together with positivity

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of Tr(Σ−1 Q(ξ)), keep all scale denominators bounded away from zero. Hence the scale maps, traces, and quadratic forms defining B(ξ), V (ξ), and H(ξ) are uniformly continuous on b σ a compact neighborhood of (θ0 , Σ, σ 2 , ξ). Since (θbML , Σ, b2 ) 2 converges in probability to (θ0 , Σ, σ ), the uniform continuous mapping argument gives the claimed sup-norm convergence. □ Theorem 4 (Compact multi-parameter plug-in consistency): Let Ξ be compact. Let P (ξ) be a continuous positive definite kernel family on Ξ. Suppose the population criterion m(ξ; θ0 , Σ, σ 2 ) and the component maps B(ξ), V (ξ), and H(ξ) are continuous on Ξ, and that m has a unique minimizer ξ ⋆ in a neighborhood where the required scale denominators are nonzero. Let m b N (ξ) be the same criterion with (θ0 , Σ, σ 2 ) replaced by consistent estimates, and let ξbN be an op (1) approximate minimizer of m b N over Ξ. If p

sup |m b N (ξ) − m(ξ; θ0 , Σ, σ 2 )| − → 0, ξ∈Ξ

p

then ξbN − → ξ ⋆ . If, moreover, the plug-in component maps converge uniformly, bN (ξ) − B(ξ)| + sup |VbN (ξ) − V (ξ)| sup |B ξ∈Ξ

ξ∈Ξ p b N (ξ) − H(ξ)| − + sup |H → 0, ξ∈Ξ

then p p bN (ξbN ) − B → B(ξ ⋆ ), VbN (ξbN ) − → V (ξ ⋆ ), p b N (ξbN ) − H → H(ξ ⋆ ).

Proof. Uniform convergence of m b N and op (1) empirical optimality give, for any fixed ε > 0, that the population criterion at ξbN is within op (1) of its minimum. Since Ξ is compact, m is continuous, and the minimizer is unique, the separation inf

ξ∈Ξ:∥ξ−ξ ⋆ ∥≥ε

{m(ξ; θ0 , Σ, σ 2 ) − m(ξ ⋆ ; θ0 , Σ, σ 2 )} > 0 p

holds for every ε > 0. Hence ξbN − → ξ ⋆ . The uniform component convergence and continuity of the population component maps then imply bN (ξbN ) − B(ξ ⋆ )| ≤ sup |B bN (ξ) − B(ξ)| |B ξ∈Ξ

+ |B(ξbN ) − B(ξ ⋆ )|, and the same argument applies to V and H.

□

VII. P LUG - IN A LGORITHM The oracle weight depends on unknown quantities. The implementable estimator uses finite-sample analogues of the XMSE components. b = Φ⊤ Φ/N . 1) Compute θbML = (Φ⊤ Φ)−1 Φ⊤ Y and Σ 2) Select a base hyperparameter estimate ηb using scaled EB, SURE, or finite-sample GCV. 3) Form the base regularized estimate θbEB = [Φ⊤ Φ + σ 2 P (b η )−1 ]−1 Φ⊤ Y.

b Vb , and H b by 4) Compute plug-in XMSE components B, ⋆ b b replacing (θ0 , Σ, η ) with (θML , Σ, ηb) in the scaledkernel formulas. 5) Set ! b Vb + H , ω b = Π[0,1] − b + ρN 2B with a small stabilizing ρN > 0. 6) Return θbmix = θbML + ω b (θbEB − θbML ). If several kernels are available, the same computation can be repeated for a finite candidate set K = {K1 , . . . , KM }. For each candidate, compute the plug-in minimized criterion bj ω b j )b qbj = B bj2 + (Vbj + H ωj , and select the kernel with the smallest qbj . This is an XMSEbased kernel-selection rule. Near the zero-bias boundary, a thresholded version can b ≤ τN , set ω replace the projected ratio in Step 5: if B b = b < 0}; otherwise use the projected ratio. Theorem 12 1{Vb + H below gives the corresponding boundary justification. Remark 3 (Why the weight is restricted to [0, 1]): The unconstrained minimizer −CEB /(2BEB ) of the quadratic criterion can leave [0, 1]: when CEB > 0 it is negative, so an unconstrained rule would extrapolate away from EB and attain 2 /(4BEB ), whereas the strictly negative criterion value −CEB the projected rule returns ML with criterion value 0; symmetrically, when CEB < −2BEB the unconstrained minimizer exceeds 1 and projection forfeits (CEB + 2BEB )2 /(4BEB ) relative to over-shrinking beyond EB. We nevertheless restrict the weight to convex combinations, for three reasons. First, the quadratic criterion is a second-order asymptotic surrogate whose accuracy we can only argue for on the segment between the two anchor estimators; an extrapolated weight amplifies exactly the higher-order terms that the expansion drops. Second, on [0, 1] the rule degrades gracefully under componentestimation error, since ∥θbmix (ω)− θbML ∥2 ≤ ∥θbEB − θbML ∥2 and the convex criterion satisfies q(ω) ≤ max{0, XMSE(θbEB )} for all ω ∈ [0, 1], while extrapolation has no such bound and turns weight estimation error into unbounded risk inflation. Third, the projected rule preserves the safeguard semantics of the method: ω b answers “how much of the EB correction should be accepted,” and a harmful correction is rejected rather than reversed. The experiments report how often the unconstrained plug-in ratio actually leaves [0, 1], so the cost of the projection is visible rather than hidden. Remark 4 (Numerical choices in the experiments): The theory lets ρN ↓ 0 at any rate; the experiments use the fixed value ρ = 10−10 , which acts purely as a division guard and b so is many orders of magnitude below every observed 2B, replacing it by a decreasing sequence would not change any reported digit. The thresholded boundary rule with τN exists for the zero-bias boundary case BEB = 0; in all reported b > 0 held in every realization, so the threshold experiments B branch was never active, and the reported results use the projected-ratio rule throughout.

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Remark 5 (Finite-sample bias of the plug-in components and a trace-corrected variant): The plug-in components evaluate quadratic forms at θbML . Since θbML ∼ N (θ0 , W ) with W = σ 2 (Φ⊤ Φ)−1 ,

b b b Consequently, if |B−B EB |+|V −VEB |+|H −HEB | = Op (rN ) for some deterministic rN ↓ 0, then

⊤ E[θbML AθbML ] = θ0⊤ Aθ0 + Tr(AW ),

Proof. The optimality of ω b for qb gives qb(b ω ) ≤ qb(ω ⋆ ). Therefore

so each plug-in quadratic form carries a systematic O(σ 2 /N ) trace term—upward for positive semidefinite A—that propb and H b and is one identifiable source of agates into ηb, B, finite-sample miscalibration of the selected weight. The tracecorrected variant therefore replaces every plug-in quadratic form, including the one inside the plug-in scale ηb, by its un⊤ c ) with W c=σ biased version θbML AθbML − Tr(AW b2 (Φ⊤ Φ)−1 , floored at zero whenever the population quantity is nonnegb can then reach zero, in which case ative. The corrected B the projected ratio with ρN degenerates to the 0/1 decision of the thresholded rule, which is the intended boundary behavior. The base estimator θbEB is left unchanged, so the corrected variant isolates the calibration of the mixing weight. This correction follows the same finite-sample motivation as the approximate-XMSE refinement in Section IV of [34]; the experiments below report it as a separate column, and the public benchmarks use it to test whether the identified SURErow failure mode is a weight-calibration artifact. Theorem 5 (Plug-in consistency away from the zero-bias b Vb , and H b be finite-sample plug-in estimaboundary): Let B, tors of BEB , VEB , and HEB , respectively, and assume p

p

b− B → BEB ,

p

Vb − → VEB ,

b − H → HEB .

Let ρN > 0 be deterministic with ρN ↓ 0 and define ! b Vb + H . ω bXMSE = Π[0,1] − b + ρN 2B If BEB > 0, then

p

ω bXMSE − → ω⋆ . b = Vb +H. b By assumption, Proof. Let CEB = VEB +HEB and C p b C− → CEB . Since BEB > 0 and ρN → 0, p

b + ρN − 2B → 2BEB > 0. Therefore −

b C b + ρN 2B

p

− →−

CEB . 2BEB

The projection Π[0,1] is continuous, so the continuous mapping theorem gives the claim. □ Theorem 6 (Plug-in oracle inequality): Let q(ω) = BEB ω 2 + (VEB + HEB )ω and b 2 + (Vb + H)ω. b qb(ω) = Bω Let ω ⋆ ∈ arg minω∈[0,1] q(ω) and let ω b ∈ arg minω∈[0,1] qb(ω), with an arbitrary deterministic tie-breaking rule. Define eN = sup |b q (ω) − q(ω)|. ω∈[0,1]

Then, deterministically, q(b ω ) − q(ω ⋆ ) ≤ 2eN .

q(b ω ) − min q(ω) = Op (rN ). ω∈[0,1]

q(b ω ) − q(ω ⋆ ) = {q(b ω ) − qb(b ω )} + {b q (b ω ) − qb(ω ⋆ )} + {b q (ω ⋆ ) − q(ω ⋆ )} ≤ |q(b ω ) − qb(b ω )| + |b q (ω ⋆ ) − q(ω ⋆ )| ≤ 2eN . For ω ∈ [0, 1], b − BEB | + |Vb − VEB | + |H b − HEB |, |b q (ω) − q(ω)| ≤ |B so the stochastic statement follows. □ Theorem 7 (Plug-in oracle regret rate): Under the assumptions of Theorem 5, define q(ω) = BEB ω 2 + (VEB + HEB )ω. Assume in addition that BEB > 0, ω ⋆ ∈ (0, 1), and for some deterministic sequence rN ↓ 0, b − BEB | + |Vb − VEB | + |H b − HEB | + ρN = Op (rN ). |B Then ω bXMSE − ω ⋆ = Op (rN ) and 2 q(b ωXMSE ) − q(ω ⋆ ) = Op (rN ).

b = Vb + H. b Since ω ⋆ ∈ Proof. Let CEB = VEB + HEB and C (0, 1), the projection is inactive with probability tending to one. On this event, ω bXMSE − ω ⋆ = −

b C

+

CEB 2BEB

b + ρN 2B b + ρN ) − 2BEB C b CEB (2B = . b + ρN ) 2BEB (2B

The denominator is bounded away from zero with probability tending to one, and the numerator is Op (rN ). Hence ω bXMSE − ω ⋆ = Op (rN ). Because ω ⋆ is the unconstrained minimizer in the interior, q ′ (ω ⋆ ) = 0 and 2 q(b ωXMSE ) − q(ω ⋆ ) = BEB (b ωXMSE − ω ⋆ )2 = Op (rN ).

□ For each fixed weight ω ∈ [0, 1], define the deterministic fixed-weight risk curve ∆N (ω) = N 2 {MSE(θbmix (ω)) − MSE(θbML )}, where the expectations are taken over all randomness with the weight held fixed at ω. Thus ω 7→ ∆N (ω) is a deterministic function, and Theorem 1 states its pointwise convergence to q(ω). The next result upgrades this to uniform convergence on [0, 1], using no assumption beyond Assumption 1. Theorem 8 (Uniform fixed-weight convergence): Under Assumption 1, εN := sup |∆N (ω) − q(ω)| −→ 0. ω∈[0,1]

7

p

Proof. For every N the curve is exactly a quadratic polynomial in ω, ∆N (ω) = aN ω + bN ω 2 ,

Since M is finite and qbj − → qj⋆ for each j, we have

with coefficients aN = 2N 2 E{(θbML − θ0 )⊤ dN } and bN = N 2 E ∥dN ∥22 . Assumption 1 gives aN → VEB + HEB = CEB and bN = N 2 ∥E[dN ]∥22 +N 2 E∥dN −E[dN ]∥22 → BEB . Hence, for all ω ∈ [0, 1],

On the event where this maximum error is smaller than g/3,

|∆N (ω) − q(ω)| ≤ |aN − CEB | + |bN − BEB |, and the right-hand side is a deterministic null sequence. □ Theorem 9 (Oracle transfer along the fixed-weight risk curve): Let the assumptions of Theorem 7 hold, and let εN be as in Theorem 8. Then the fixed-weight risk curve evaluated at the selected weight satisfies

p

max |b qj − qj⋆ | − → 0.

1≤j≤M

qbj ⋆ < qj⋆⋆ + g/3 < qj⋆ − g/3 < qbj for every j ̸= j ⋆ . Thus b j = j ⋆ on an event whose probability tends to one. □ Corollary 3 (Finite-set oracle inequality): In the setting of Theorem 10, let j ⋆ ∈ arg min qj⋆ 1≤j≤M

be any oracle minimizer and define eN = max |b qj − qj⋆ |.

2 ∆N (b ωXMSE ) − q(ω ⋆ ) = Op (rN ) + O(εN )

and 2 ) + O(εN ). ∆N (b ωXMSE ) − ∆N (ω ⋆ ) = Op (rN

Proof. Since ω bXMSE ∈ [0, 1] always, |∆N (b ωXMSE ) − q(b ωXMSE )| ≤ εN surely, by Theorem 8. Theorem 7 gives 2 ), and adding the two bounds q(b ωXMSE ) − q(ω ⋆ ) = Op (rN proves the first claim. The second claim follows by also applying |∆N (ω ⋆ ) − q(ω ⋆ )| ≤ εN . □ Remark 6 (What the transfer result does and does not control): ∆N (b ωXMSE ) is the deterministic risk curve evaluated at the realized weight: it is the second-order risk the mixed estimator would have if the selected weight were frozen and reused on independent data. It is not the XMSE of the data-dependent estimator θbmix (b ωXMSE ), in which the random weight sits inside the expectation. The difference consists of cross-moments between ω bXMSE and (θbML − θ0 , θbEB − θbML ), which arise because the weight and the EB correction are computed from the same data. Controlling these cross-moments requires a joint expansion of the weight map and the EB correction; for the scaled-kernel one-step rules of Remark 2, ω bXMSE is a smooth function of θbML away from the projection boundary, so a delta-method analysis appears feasible, but it is beyond the scope of this paper and is recorded as an open problem in the Limitations section. Theorem 10 (Finite-candidate kernel selection): Consider a finite set of candidate kernels K = {K1 , . . . , KM }. Let qj⋆ denote the oracle minimized XMSE criterion for candidate Kj , and let qbj be its plug-in counterpart. Assume p

qbj − → qj⋆ ,

j = 1, . . . , M.

If the oracle minimizer is unique, i.e., there exists j ⋆ such that

1≤j≤M

Then the selected candidate satisfies the deterministic inequality qbj⋆ − qj⋆⋆ ≤ 2eN . Consequently, if eN = Op (rN ) for some deterministic rN ↓ 0, then qbj⋆ − min qj⋆ = Op (rN ). 1≤j≤M

Proof. By definition of b j, qbbj ≤ qbj ⋆ . Adding and subtracting the plug-in criteria gives qbj⋆ − qj⋆⋆ = (qbj⋆ − qbbj ) + (b qbj − qbj ⋆ ) + (b qj ⋆ − qj⋆⋆ ) ≤ |qbj⋆ − qbbj | + |b qj ⋆ − qj⋆⋆ | ≤ 2eN . The stochastic statement follows immediately from the assumed rate for eN . □ Theorem 11 (Growing candidate dictionary): For each sample size, let KN = {KN,1 , . . . , KN,MN } ⋆ be a deterministic candidate dictionary. Let qN,j denote the oracle minimized XMSE criterion for candidate KN,j and let qbN,j be its plug-in counterpart. Define

b jN = arg

min

1≤j≤MN

qbN,j ,

⋆ jN ∈ arg

min

1≤j≤MN

⋆ qN,j ,

and eN =

⋆ max |b qN,j − qN,j |.

1≤j≤MN

Then the selected candidate satisfies

j̸=j

⋆ ⋆ qN, − qN,j ⋆ ≤ 2eN . b j N

b j = arg min qbj

If eN = op (1), the selected candidate is asymptotically oracle optimal over the growing dictionary. If, in addition, the oracle minimizer is unique with gap

qj⋆⋆ < min⋆ qj⋆ , then the plug-in selector 1≤j≤M

satisfies Pr(b j = j ⋆ ) → 1.

N

⋆ ⋆ gN = min⋆ (qN,j − qN,j ⋆ ) > 0 N j̸=jN

and eN = op (gN ), then

Proof. Let g = min⋆ (qj⋆ − qj⋆⋆ ) > 0. j̸=j

⋆ Pr(b jN = jN ) → 1.

8

Proof. The oracle inequality is the same add-and-subtract argument used for the finite-set oracle inequality: ⋆ ⋆ ⋆ qN,bjN − qbN,jN⋆ ) qN, − qN,j − qbN,bjN ) + (b ⋆ = (q b j N,b j N N

N

⋆ + (b qN,jN⋆ − qN,j ⋆ ) N ⋆ ⋆ qN,jN⋆ − qN,j ≤ |qN, − qbN,bjN | + |b ⋆ | ≤ 2eN . b j N N

Thus eN = op (1) implies asymptotic oracle optimality. On the ⋆ event eN < gN /3, for every j ̸= jN , ⋆ ⋆ qbN,jN⋆ < qN,j bN,j . ⋆ + gN /3 ≤ qN,j − gN /3 < q N ⋆ Hence b jN = jN on this event, and eN = op (gN ) makes its probability tend to one. □ Corollary 4 (High-probability dictionary oracle bound): In the setting of Theorem 11, suppose that for deterministic sequences aN ↓ 0 and βN ↓ 0, ⋆ Pr{|b qN,j − qN,j | > aN } ≤ βN ,

j = 1, . . . , MN .

Then, with probability at least 1 − MN βN , ⋆ qN, − b j N

min

1≤j≤MN

⋆ qN,j ≤ 2aN .

Consequently, if aN → 0 and MN βN → 0, the selected dictionary element is asymptotically oracle optimal. If the oracle gap gN is positive and aN < gN /3, then the same ⋆ . event implies exact oracle selection, b jN = jN Proof. By the union bound, Pr(eN > aN ) ≤

MN X

⋆ Pr{|b qN,j − qN,j | > aN } ≤ MN βN .

j=1

On the complementary event, Theorem 11 gives the oracle inequality with eN ≤ aN . If aN < gN /3, the gap argument ⋆ . □ in the proof of Theorem 11 gives b jN = jN Corollary 5 (Sub-Gaussian dictionary rate): In the setting of Theorem 11, suppose that for some scale sN ↓ 0 and all t > 0, j = 1, . . . , MN ,  2 t ⋆ Pr{|b qN,j − qN,j | > t} ≤ 2 exp − 2 . sN Then, for any δ ∈ (0, 1), with probability at least 1 − δ, s   2MN ⋆ ⋆ . qN,bj − min qN,j ≤ 2sN log N 1≤j≤MN δ Consequently, if s2N log MN → 0, the selected dictionary element is asymptotically oracle optimal. Proof. Apply Corollary 4 with s   2MN aN = sN log and βN = δ/MN . δ The stated tail bound gives the required per-candidate probability bound, and the oracle inequality follows. For fixed δ, the right-hand side is o(1) whenever s2N log MN → 0. □ Theorem 12 (Thresholded consistency at the zero-bias b = Vb + H. b Suppose boundary): Let CEB = VEB + HEB and C p b BEB = 0, CEB ̸= 0, C − → CEB , and for some deterministic τN ↓ 0, b ≤ τN ) → 1. Pr(B

Define the thresholded plug-in weight  b b ≤ τN ,  B  1{C < 0}, ! b ω e= C b > τN .  , B  Π[0,1] − 2B b + ρN Let

( ω0⋆ =

1, 0,

CEB < 0, CEB > 0,

be the minimizer of the linear oracle objective q0 (ω) = CEB ω over [0, 1]. Then p ω e− → ω0⋆ . b ≤ τN }. By assumption, Pr(AN ) → 1. Proof. Let AN = {B p b b is eventually Since C − → CEB and CEB ̸= 0, the sign of C correct with probability tending to one: b < 0} = 1{CEB < 0}} → 1. Pr{1{C b < 0}. Therefore On AN , the thresholded rule equals 1{C   b < 0} = 1{CEB < 0}} → 1, Pr(e ω = ω0⋆ ) ≥ Pr AN ∩ {1{C which implies convergence in probability. □ Remark 7 (Scope of the boundary result): The thresholded theorem gives consistency of the selected boundary decision. It does not claim a full XMSE expansion for the estimator with a random, nonsmooth projected or thresholded weight. Such an expansion would require tracking additional terms from the weight map near the projection and threshold boundaries. VIII. N UMERICAL E XPERIMENTS The experiments are organized to mirror the theory. The diagonal and tail-mismatch simulations are controlled fixedorder FIR settings where the XMSE expansion and plugin component estimates are intended to be informative. The sample-size and SNR sweeps check the asymptotic direction suggested by the plug-in consistency and regret results. The finite-candidate experiment tests the risk-oriented kernelselection criterion behind the finite-set oracle inequality. The Silverbox and Cascaded Tanks studies, by contrast, are outof-model public-data checks: they examine whether the same diagnostic weight can retreat from harmful shrinkage on nonlinear systems and where its finite-sample calibration limits appear, not whether the XMSE theorems hold outside their assumptions. A. Common protocol All synthetic experiments use the FIR model of the problem setup with known noise variance σ 2 = 1; only the two public benchmarks estimate the noise variance, as disclosed there. For each system, the input is drawn i.i.d. standard normal and the true parameter is constructed as follows and then rescaled so that the sample signal-to-noise ratio SNR = (Φθ0 )2 /σ 2 matches its target. Aligned and misaligned parameters are p p θ0ali [k] = K[k, k] zk , θ0mis [k] = zk / K[k, k],

9

with z ∼ N (0, In ), so that the parameter energy profile follows or opposes the kernel diagonal. The tail-mismatch parameters are θ0tail [k] = rk (−1)k+1 |zk |,

rk = 0.1 +

1.9 (k − 1) , n−1

an increasing ramp with alternating signs, which concentrates energy in the late impulse-response coefficients and is therefore intentionally misaligned with the decaying TC/SS kernels. The hyperparameter estimators are implemented as described in Remark 2, the projected ratio uses the numerical choices of Remark 4, and α = 1 throughout. Two error metrics are reported. For the synthetic experiments, the parameter-space fit of an estimate θb of θ0 is ! n 1X ∥θb − θ0 ∥2 b θ0 [k], , θ̄0 = FITθ (θ) = 100 1 − n ∥θ0 − θ̄0 1∥2 k=1

averaged over Monte Carlo repetitions. For the public benchmarks, the output-prediction fit on a test record y with prediction yb is   ∥y − yb∥2 FITy (b y ) = 100 1 − . ∥y − ȳ1∥2 The two definitions agree in form but live in different spaces; tables state which one is used. B. Diagonal-kernel calibration We first run a calibration experiment with n = 20, N = 50, SNR 10, 50 systems, and 200 Monte Carlo repetitions per system, using the RI kernel with neutral parameters (θ0 = z) and the diagonal-decay kernel KDI with aligned and misaligned parameters. Table I reports the results. In these diagonal settings the regularized estimator helps on average in both alignments, and the plug-in mixed estimator retains most of the benefit while giving up a small part of it (e.g., 0.730 versus 0.720 for the aligned diagonal kernel against 0.798 for ML). The misaligned construction reduces but does not eliminate the benefit of shrinkage at this sample size, so this experiment functions as a calibration check of the plug-in weight in benign conditions; the genuinely harmfulregularization regime is produced by the tail-mismatch construction studied next. C. Tail mismatch The main experiment considers TC and SS kernels with γ = 0.95 and the tail-mismatch parameter class of Section VIII-A, whose energy is concentrated in later impulse-response coefficients and which is therefore intentionally misaligned with decaying kernels. We use 100 systems and 500 Monte Carlo repetitions per system; the TC and SS rows use the same seed, and since the tail construction does not depend on the kernel, the two kernels see identical systems and noise realizations, so cross-kernel comparisons are paired. Table II reports mean MSEs. The column “base” denotes the regularized estimator using scaled EB, SURE-type, or finite-sample GCV scale selection.

The scaled EB rows show the clearest protection effect. With the TC kernel the base estimator is 8% worse than ML on average (1.001 versus 0.928) while the plug-in mixed estimator is 2% better (0.910); with the ill-conditioned SS kernel the base estimator is 15% worse (1.071) and the plug-in mixed estimator retreats essentially to ML (0.928, mean weight 0.155). SURE-type and GCV scale selection are intrinsically more conservative in these settings, with base estimators close to or slightly better than ML, and the mixed estimator matches or slightly improves them. In terms of mean MSE the plug-in mixed estimator is at or below ML in all six settings (Table IV); in terms of the per-system winning fraction against ML, mixing improves on the base estimator in four settings, ties in one (TC-SURE, 68%), and is lower in one (SS-SURE, 49% versus 42%), the setting where the base estimator is already statistically indistinguishable from ML. Both fractions are reported in Table II so this statement can be checked directly. The FIT values in Table V tell the same story in the relative metric. Table VI shows that the plug-in rule assigns smaller weights to the EB estimator under the more vulnerable EB hyperparameter choice (median 0.37 for TC, 0.11 for SS), while SURE and GCV receive larger weights because their base estimates are already more robust. Table VII connects the empirical risk gaps to the XMSE criterion: for the base estimators, the sign of the oracle XMSE agrees with the sign of the observed sample MSE gap in 88–97% of systems. After mixing, the average sample gap is negative in all six settings, although the plug-in XMSE sign is naturally noisier because it also contains the random weight-selection step. Table III adds the natural competitors. The SURE-tuned weight, which minimizes an unbiased estimate of the finitesample risk difference for the frozen hyperparameter, performs essentially identically to the XMSE plug-in weight under mismatch (within 0.003 in every row) while choosing systematically larger weights; its advantage appears in favorable settings (Section VIII-F). The hard {0, 1} rule is slightly worse than the soft mixture in the four rows where the two differ at the displayed precision and ties it in the remaining two (SS-SURE and SS-GCV), which quantifies the small gain of weight interpolation over pure ML/EB selection. The trace-corrected variant coincides with the plug-in rule to the displayed precision in every row except TC-GCV, where the two differ by 0.001, because with known σ 2 , n = 20, and N = 50 the trace terms are small relative to the tail-mismatch quadratic forms. Concerning the projection of the weight (Remark 3), the unconstrained ratio fell below 0 in 8–31% and exceeded 1 in 0–34% of realizations across the six settings, so the projection is active for a nontrivial minority of realizations. D. Sample-size and SNR sensitivity To check whether the plug-in mixed estimator approaches the oracle mixed estimator as the sample size increases, we run a sample-size sweep for the TC tail-mismatch setting with scaled EB hyperparameter selection, using 80 systems and 300 Monte Carlo repetitions per system. Table VIII shows that the plug-in mixed estimator moves closer to the oracle mixed estimator as N increases, matching it to three digits

10

Setting RI-EB-neutral DIAG DECAY-EB-aligned DIAG DECAY-EB-misaligned

ML 0.898 0.798 0.892

base 0.818 0.720 0.826

oracle mix 0.811 0.716 0.815

plug-in mix 0.830 0.730 0.840

ω̂ 0.824 0.798 0.696

base < ML 96% 94% 94%

mix < ML 96% 98% 98%

TABLE I D IAGONAL - KERNEL CALIBRATION EXPERIMENT, N = 50, n = 20, SNR 10, KNOWN σ 2 . M EAN MSE ACROSS 50 SYSTEMS ; THE LAST TWO COLUMNS GIVE THE FRACTION OF SYSTEMS WHERE THE BASE AND PLUG - IN MIXED ESTIMATORS HAVE LOWER MSE THAN ML.

Setting TC-EB TC-SURE TC-GCV SS-EB SS-SURE SS-GCV

ML 0.928 0.928 0.928 0.928 0.928 0.928

base 1.001 0.912 0.914 1.071 0.929 0.930

oracle mix 0.902 0.908 0.908 0.924 0.926 0.926

plug-in mix 0.910 0.911 0.912 0.928 0.927 0.927

ω̂ 0.437 0.602 0.585 0.155 0.419 0.423

base < ML 43% 68% 62% 10% 49% 43%

mix < ML 67% 68% 73% 42% 42% 57%

TABLE II M EAN MSE UNDER TC/SS TAIL MISMATCH , N = 50, n = 20, SNR 10, KNOWN σ 2 . T HE LAST TWO COLUMNS GIVE THE FRACTION OF SYSTEMS WHERE THE BASE AND PLUG - IN MIXED ESTIMATORS HAVE LOWER MSE THAN ML. TC AND SS ROWS SHARE THE SAME SYSTEMS AND NOISE REALIZATIONS .

Setting TC-EB TC-SURE TC-GCV SS-EB SS-SURE SS-GCV

ML 0.928 0.928 0.928 0.928 0.928 0.928

base 1.001 0.912 0.914 1.071 0.929 0.930

plug-in mix 0.910 0.911 0.912 0.928 0.927 0.927

SURE-ω mix 0.911 0.911 0.911 0.931 0.929 0.929

hard {0, 1} 0.914 0.912 0.913 0.931 0.927 0.927

corrected mix 0.910 0.911 0.911 0.928 0.927 0.927

TABLE III BASELINE COMPARISON UNDER TC/SS TAIL MISMATCH ( MEAN MSE). SURE-ω TUNES THE MIXING WEIGHT BY MINIMIZING THE bmix (ω)) − MSE(θbML ); HARD {0, 1} SELECTS ML OR EB BY THE SIGN OF THE PLUG - IN XMSE; FIXED - HYPERPARAMETER SURE OF MSE(θ CORRECTED MIX USES THE TRACE - CORRECTED COMPONENTS OF R EMARK 5.

Setting TC-EB TC-SURE TC-GCV SS-EB SS-SURE SS-GCV

base–ML 0.0726 (0.0216) -0.0167 (0.0040) -0.0142 (0.0041) 0.1425 (0.0232) 0.0007 (0.0009) 0.0019 (0.0011)

mix–ML -0.0187 (0.0039) -0.0175 (0.0036) -0.0167 (0.0034) -0.0006 (0.0005) -0.0015 (0.0005) -0.0014 (0.0004)

mix–base -0.0913 (0.0202) -0.0008 (0.0009) -0.0026 (0.0012) -0.1431 (0.0231) -0.0022 (0.0007) -0.0033 (0.0009)

mix < base 61% 38% 53% 90% 53% 62%

TABLE IV M EAN MSE GAPS WITH STANDARD ERRORS IN PARENTHESES UNDER TC/SS TAIL MISMATCH . N EGATIVE VALUES INDICATE THAT THE ESTIMATOR NAMED BEFORE THE DASH HAS LOWER MSE. T HE PER - SYSTEM WINNING FRACTIONS AGAINST ML APPEAR IN TABLE II.

from N = 100 on. At N = 30 the noise level is high enough that shrinkage helps on average; the plug-in mixed estimator improves over both ML and the base estimator, but remains visibly away from the oracle (4.79 versus 4.31), which reflects the noisier finite-sample XMSE approximation at small N . We also sweep SNR at N = 50, again using 80 systems and 300 Monte Carlo repetitions per system. Table IX shows that the plug-in mixed estimator improves over ML across all tested SNR levels. At very low SNR the base EB estimator is already helpful, while at moderate and high SNR the XMSEaware mixture provides protection against the tail-mismatch penalty. Figure 1 visualizes the same sensitivity experiments. Both panels plot the mean MSE of each estimator relative to ML, so the ML reference is the horizontal line at one and the curves remain separated at every N and SNR; absolute MSE values are in Tables VIII–IX. The sample-size panel shows the plugin mixed estimator tracking the oracle mixed estimator as N grows, while the SNR panel shows that the mixed estimator stays at or below the ML line across the tested signal-to-noise

ratios. E. Finite-candidate kernel selection We next evaluate the finite-candidate kernel-selection rule using the candidate set {RI, TC, SS} over 100 systems and 300 Monte Carlo repetitions per system. The protocol is the one a user faces: for each individual noise realization, the rule computes the single-realization plug-in criterion qbj for every candidate and selects the minimizing kernel, and the selected mixed estimator is scored on that same realization. This matches Theorem 10, whose plug-in criterion is a singlerealization quantity; averaging qbj over the Monte Carlo repetitions before selecting would use information a single user does not have. The selection is risk-oriented: it minimizes the plug-in XMSE criterion and is not intended to recover the datagenerating kernel. As a standard-practice baseline, we also report per-realization kernel selection by maximized marginal likelihood (evidence), where for each candidate the scale η is chosen by maximizing the Gaussian evidence and the winning kernel’s regularized estimator is scored (column “evid. MSE”).

11

Setting TC-EB TC-SURE TC-GCV SS-EB SS-SURE SS-GCV

ML 75.95 75.95 75.95 75.95 75.95 75.95

base oracle mix plug-in mix 75.52 76.34 76.24 76.23 76.26 76.22 76.20 76.26 76.21 74.68 76.02 75.96 75.96 75.99 75.97 75.94 75.99 75.97 TABLE V AVERAGE PARAMETER - SPACE FITθ UNDER TC/SS TAIL MISMATCH , N = 50, n = 20, SNR 10.

Setting ω̂ ML MSE base MSE plug-in MSE plug-in FIT TC-EB 0.366 [0.111, 0.804] 0.84 [0.71, 1.05] 0.87 [0.76, 1.12] 0.82 [0.71, 1.03] 77.43 [72.82, 79.69] TC-SURE 0.658 [0.199, 0.993] 0.84 [0.71, 1.05] 0.82 [0.71, 1.02] 0.82 [0.71, 1.03] 77.46 [72.80, 79.69] TC-GCV 0.639 [0.206, 0.937] 0.84 [0.71, 1.05] 0.82 [0.71, 1.03] 0.82 [0.71, 1.03] 77.45 [72.78, 79.69] SS-EB 0.112 [0.013, 0.228] 0.84 [0.71, 1.05] 0.94 [0.80, 1.22] 0.84 [0.71, 1.05] 77.29 [72.47, 79.65] SS-SURE 0.391 [0.000, 0.758] 0.84 [0.71, 1.05] 0.84 [0.72, 1.05] 0.84 [0.71, 1.05] 77.29 [72.52, 79.65] SS-GCV 0.408 [0.057, 0.707] 0.84 [0.71, 1.05] 0.84 [0.72, 1.06] 0.84 [0.71, 1.05] 77.29 [72.51, 79.65]

TABLE VI M EDIAN AND INTERQUARTILE RANGES ACROSS SYSTEMS UNDER TC/SS TAIL MISMATCH .

Setting TC-EB TC-SURE TC-GCV SS-EB SS-SURE SS-GCV

base ∆MSE 0.0726 -0.0167 -0.0142 0.1425 0.0007 0.0019

oracle XMSE/N 2 0.1278 -0.0174 -0.0174 0.2294 0.0018 0.0018

sign 95% 91% 97% 96% 88% 90%

mix ∆MSE -0.0187 -0.0175 -0.0167 -0.0006 -0.0015 -0.0014

plug-in XMSE/N 2 -0.0275 -0.0212 -0.0200 -0.0041 -0.0024 -0.0023

sign 70% 78% 73% 59% 68% 57%

TABLE VII S AMPLE MSE GAPS AND XMSE- BASED DIAGNOSTICS UNDER TC/SS TAIL MISMATCH . T HE BASE DIAGNOSTIC IS THE oracle XMSE EVALUATED AT θ0 , bML INCLUDING THE DATA - DEPENDENT WEIGHT; THE SIGN COLUMNS GIVE THE WHEREAS THE MIX DIAGNOSTIC IS THE plug-in XMSE EVALUATED AT θ FRACTION OF SYSTEMS WHERE THE DIAGNOSTIC SIGN MATCHES THE SAMPLE MSE- GAP SIGN . T HESE DIAGNOSTICS IMPLEMENT, AT THE LEVEL OF OUR MIXED RULE , THE FINITE - SAMPLE XMSE APPROXIMATION IDEA OF S ECTION IV OF [34].

N 30 50 70 100 150

oracle mix plug-in mix ω̂ mix < ML 4.310 4.792 0.418 70% 0.900 0.908 0.368 69% 0.479 0.480 0.344 71% 0.281 0.281 0.376 84% 0.161 0.161 0.344 89% TABLE VIII S AMPLE - SIZE SENSITIVITY UNDER TC TAIL MISMATCH , SCALED EB, SNR 10. ML 5.128 0.918 0.485 0.282 0.162

base 5.044 1.010 0.509 0.286 0.163

Table X reports the realization-level selection frequencies, the fraction of realizations where the selected kernel matches the per-system best fixed candidate (“match best”), the realization-level MSE of the selected estimator, the best fixed-candidate MSE, and the regret, i.e., the mean excess of the selected MSE over the best fixed candidate. The regret column is the empirical analogue of Corollary 3: the theory controls the oracle criterion regret of the selected candidate by the uniform plug-in criterion error, while the table reports the same risk-oriented comparison on realized Monte Carlo test errors. Thus a small regret is the target predicted by the oracle inequality even when the selected kernel is not the datagenerating kernel. In the tail-mismatch setting, the XMSE rule chooses RI in 99% of realizations, effectively avoiding the decaying TC/SS kernels that are poorly aligned with the true parameter shape; the selected MSE (0.839) is within 0.002 of the best fixed candidate and improves over both ML (0.890) and the evidence baseline (0.869), whose selected EB estimator inherits the harmful shrinkage. In the aligned setting, the realization-level selection is noisier (it distributes 68/15/17% across RI/TC/SS and matches the best fixed candidate in 74% of realizations),

but the regret remains small (0.013), which is the risk-oriented target. In this favorable setting the evidence baseline is slightly better than the XMSE rule (0.746 versus 0.768): when the kernel family is well aligned, standard evidence maximization combined with full EB shrinkage is hard to beat, and the value of the XMSE rule lies in the mismatched regime. F. Semi-real fixed impulse-response benchmark As a complementary check, we use a fixed stable oscillatory impulse response rather than drawing a new parameter vector for each system. The benchmark uses n = 30, N = 80, SNR 10, 80 input realizations, and 300 Monte Carlo repetitions per input; TC and SS rows share the same seed and therefore the same inputs and noise (paired). The impulse response is normalized and then rescaled to the desired SNR for each input realization. Table XI reports the results for TC and SS kernels with EB, SURE-type, and finite-sample GCV scale selection, and Table XII the corresponding baseline comparison. In this benchmark the decaying kernels are well matched to the stable impulse response, so the base regularized estimator is very strong (mean MSE 0.31–0.59 against 0.77 for ML). The plug-in mixed estimator is never worse than ML, but

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oracle mix plug-in mix ω̂ mix < ML 0.729 0.766 0.394 98% 0.792 0.811 0.426 86% 0.816 0.824 0.438 72% 0.845 0.848 0.413 62% 0.850 0.851 0.458 69% TABLE IX SNR SENSITIVITY UNDER TC TAIL MISMATCH , SCALED EB, N = 50.

SNR 1 3 10 30 100

ML 0.854 0.857 0.842 0.854 0.853

base 0.762 0.881 0.879 0.875 0.859

Fig. 1. Sample-size and SNR sensitivity under TC tail mismatch with scaled EB hyperparameter selection. Both panels show mean MSE relative to ML.

Setting RI TC SS match best selected MSE best MSE regret evid. MSE ML MSE tc aligned 68% 15% 17% 74% 0.768 0.755 0.013 0.746 0.824 99% 1% 0% 90% 0.839 0.837 0.002 0.869 0.890 tc tail TABLE X F INITE - CANDIDATE KERNEL SELECTION WITH PER - REALIZATION SELECTION . RI/TC/SS COLUMNS ARE REALIZATION - LEVEL SELECTION FREQUENCIES OF THE PLUG - IN XMSE RULE ; “ EVID . MSE” IS THE PER - REALIZATION MARGINAL - LIKELIHOOD BASELINE , WHICH SELECTS RI IN 99–100% OF REALIZATIONS IN BOTH SETTINGS .

Setting TC-EB-semireal TC-SURE-semireal TC-GCV-semireal SS-EB-semireal SS-SURE-semireal SS-GCV-semireal

ML 0.766 0.766 0.766 0.766 0.766 0.766

base 0.443 0.474 0.400 0.451 0.586 0.313

oracle mix 0.443 0.488 0.409 0.451 0.601 0.334

plug-in mix 0.546 0.565 0.674 0.699 0.677 0.765

ω̂ 0.711 0.631 0.263 0.169 0.427 0.004

base < ML 100% 100% 100% 100% 100% 100%

mix < ML 100% 100% 100% 100% 100% 99%

TABLE XI S EMI - REAL FIXED IMPULSE - RESPONSE BENCHMARK , N = 80, n = 30, SNR 10, KNOWN σ 2 .

it is overly conservative here: it retains only part of the achievable gain, most visibly in the GCV and SS rows (e.g., SS-GCV keeps almost none of it, with mean weight 0.004). The mechanism is the finite-sample inflation of the plug-in squared-bias component discussed in Remark 5: quadratic −1 forms in θbML involving the ill-conditioned Q = KSS are dominated by their trace bias, which makes shrinkage look more harmful than it is. Consistent with this diagnosis, the trace-corrected variant recovers a large part of the gap (TCGCV 0.674 → 0.487, SS-GCV 0.765 → 0.438), and the SURE-tuned weight, which estimates the finite-sample risk difference directly, recovers nearly all of it (within 0.020 of the

base in every row). The unconstrained oracle ratio exceeds one in these settings (mean values between 1.1 and 27), i.e., overshrinkage beyond EB would be profitable, so the projected oracle weight sits at or near the upper boundary. Two readings of Table XI deserve comment. First, the oracle mixed estimator is slightly worse than the base estimator in the SURE/GCV rows (e.g., 0.488 versus 0.474 for TC-SURE). This is not a contradiction of Theorem 2: the oracle weight minimizes the limiting XMSE criterion, and its dominance over the base estimator holds at the XMSE scale asymptotically, not at every finite N ; in strongly favorable settings the finite-sample optimal weight can even exceed one, which

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Setting TC-EB-semireal TC-SURE-semireal TC-GCV-semireal SS-EB-semireal SS-SURE-semireal SS-GCV-semireal

ML 0.766 0.766 0.766 0.766 0.766 0.766

base 0.443 0.474 0.400 0.451 0.586 0.313

plug-in mix 0.546 0.565 0.674 0.699 0.677 0.765

SURE-ω mix 0.448 0.474 0.414 0.463 0.586 0.333

hard {0, 1} 0.569 0.563 0.719 0.755 0.693 0.765

corrected mix 0.495 0.544 0.487 0.567 0.631 0.438

TABLE XII BASELINE COMPARISON ON THE SEMI - REAL BENCHMARK ( MEAN MSE); COLUMNS AS IN TABLE III.

the criterion, restricted to [0, 1], cannot express. Second, the favorable-case conservativeness shown here is the price of the protection demonstrated in the tail-mismatch experiment; the baseline table makes the trade-off explicit rather than hiding it. This supports the intended interpretation of the method as a protection mechanism with a quantified cost: it retreats toward ML under harmful regularization, and in favorable settings the trace-corrected or SURE-weighted variants recover most of the remaining benefit. The next two public benchmarks are deliberately interpreted differently from the synthetic experiments. They use the same linear FIR estimator on nonlinear input-output systems, so they are out-of-model robustness checks rather than tests of state-of-the-art nonlinear identification accuracy. Two protocol details apply to both benchmarks and differ from the synthetic experiments. First, the noise variance is not known: it is estimated from the ML residuals as σ b2 = ∥y − ΦθbML ∥22 /(N − n) on the training segment, and this estimate enters the regularized estimator, the plug-in components, and the trace correction. Because the linear FIR model is misspecified for these nonlinear systems, σ b2 absorbs unmodeled dynamics in addition to measurement noise. Second, the FIR regression matrix is built from the zero-initialized input sequence, so the first n training rows contain the initial transient; the test-side predictions discard a 50-sample initialization window, while the training side does not. Both choices affect all estimators identically, and we disclose them for reproducibility. In the tables, the mixing weight is a useful diagnostic: weights near zero indicate that the plug-in XMSE criterion detects harmful shrinkage and returns ML, whereas weights near one in rows where EB is worse than ML indicate finite-sample calibration error in the estimated XMSE components. The trace-corrected weight ω bc and the corresponding RMSE/FIT columns test whether that calibration error is explained by the trace bias of Remark 5. G. Public Silverbox benchmark As an archival real-data check, we use the Silverbox benchmark [36], an electronic Duffing-oscillator system with an official train/test split. Because the benchmark is nonlinear, the purpose is not to obtain a state-of-the-art Silverbox model with a linear FIR estimator. Instead, we ask whether the XMSEaware rule retreats from harmful shrinkage, and where its plugin calibration limits appear, on a public input-output dataset outside the synthetic data-generating assumptions. We fit a length-50 FIR model using the first 500 samples of the official training record and evaluate one-step FIR predictions on the three official test records (multisine, arrow_full, and

arrow_no_extrap) after a 50-sample initialization window. Inputs and outputs are standardized using the training record. Table XIII gives both a conservative-success case and a visible calibration failure mode. The weight is selected once on the training data and reused across the three test records. When the training-side plug-in criterion flags the finite-sample GCV regularized fit as harmful, the XMSE-aware weight is zero and the mixed estimator returns ML exactly on every test record. In the EB rows, the rule uses a small amount of shrinkage and is indistinguishable from ML at the displayed precision on the arrow tests (differences in the fourth decimal of RMSE on a single test record, for which no uncertainty quantification is available), while avoiding the larger degradation of the base EB estimator. We therefore do not claim improvement over ML on this benchmark; the supported claim is that the safeguard avoids most of the base estimator’s degradation. The SURE-type rows are the failure mode: the plug-in criterion assigns a large weight (e.g., 0.945 for TC) and inherits most of the base estimator’s error. The trace-corrected variant does not repair this; instead, ω bc = 1 in every row, because the misspecification-inflated σ b2 makes the subtracted traces as large as the plug-in quadratic forms, so the corrected squaredbias component collapses to its zero floor and the corrected rule returns full EB. The benchmark calibration failure is thus not explained by the trace bias alone: under model misspecification, both the uncorrected and the corrected plugin components are operating outside the theory, and the mixed rule degrades gracefully to one of its endpoints rather than failing unboundedly. This is the precise sense in which the benchmark is a robustness check rather than evidence of improvement outside the linear FIR setting. H. Public Cascaded Tanks benchmark As a second public input-output benchmark, we use the Cascaded Tanks dataset [37], accessed through the official nonlinear-benchmarks dataloader. The data consist of one training record and one validation record of length 1024. We fit the same length-50 FIR model using the first 700 training samples and evaluate one-step predictions on the validation record after a 50-sample initialization window. Inputs and outputs are standardized using the training segment. Table XIV shows the same split between protective behavior and calibration limits. The regularized EB estimates are slightly worse than ML in this linear FIR evaluation, and the mixed rule either retreats to ML, as in the GCV rows, or uses only a partial shrinkage weight. The SURE-type rows remain the limitation: the plug-in criterion assigns full EB weight for

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test multisine multisine multisine multisine multisine multisine arrow full arrow full arrow full arrow full arrow full arrow full arrow no extrap arrow no extrap arrow no extrap arrow no extrap arrow no extrap arrow no extrap

kernel TC TC TC SS SS SS TC TC TC SS SS SS TC TC TC SS SS SS

hpe EB SURE GCV EB SURE GCV EB SURE GCV EB SURE GCV EB SURE GCV EB SURE GCV

ω b 0.219 0.945 0.000 0.044 0.352 0.000 0.219 0.945 0.000 0.044 0.352 0.000 0.219 0.945 0.000 0.044 0.352 0.000

ω bc 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

RMSE ML 0.2198 0.2198 0.2198 0.2198 0.2198 0.2198 0.2955 0.2955 0.2955 0.2955 0.2955 0.2955 0.1815 0.1815 0.1815 0.1815 0.1815 0.1815

RMSE EB 0.2230 0.2216 0.2234 0.2229 0.2208 0.2257 0.2968 0.2963 0.2952 0.2971 0.2961 0.2972 0.1828 0.1822 0.1826 0.1830 0.1821 0.1842

RMSE mix 0.2203 0.2215 0.2198 0.2199 0.2202 0.2198 0.2954 0.2962 0.2955 0.2955 0.2956 0.2955 0.1814 0.1822 0.1815 0.1815 0.1817 0.1815

RMSE corr 0.2230 0.2216 0.2234 0.2229 0.2208 0.2257 0.2968 0.2963 0.2952 0.2971 0.2961 0.2972 0.1828 0.1822 0.1826 0.1830 0.1821 0.1842

FIT ML 77.5 77.5 77.5 77.5 77.5 77.5 69.3 69.3 69.3 69.3 69.3 69.3 76.5 76.5 76.5 76.5 76.5 76.5

FIT EB 77.1 77.3 77.1 77.2 77.4 76.9 69.1 69.2 69.3 69.1 69.2 69.1 76.4 76.4 76.4 76.3 76.4 76.2

FIT mix 77.4 77.3 77.5 77.5 77.4 77.5 69.3 69.2 69.3 69.3 69.2 69.3 76.5 76.4 76.5 76.5 76.5 76.5

FIT corr 77.1 77.3 77.1 77.2 77.4 76.9 69.1 69.2 69.3 69.1 69.2 69.1 76.4 76.4 76.4 76.3 76.4 76.2

TABLE XIII P UBLIC S ILVERBOX BENCHMARK WITH A LENGTH -50 FIR MODEL TRAINED ON THE FIRST 500 SAMPLES OF THE OFFICIAL TRAINING RECORD . RMSE AND FITy ARE COMPUTED ON STANDARDIZED OFFICIAL TEST RECORDS AFTER A 50- SAMPLE INITIALIZATION WINDOW. T HE NOISE VARIANCE IS ESTIMATED FROM ML TRAINING RESIDUALS . ω b IS THE PLUG - IN WEIGHT AND ω bc , RMSE CORR , FIT CORR THE TRACE - CORRECTED VARIANT. test cascaded tanks cascaded tanks cascaded tanks cascaded tanks cascaded tanks cascaded tanks

kernel TC TC TC SS SS SS

hpe EB SURE GCV EB SURE GCV

ω b 0.490 1.000 0.000 0.111 0.558 0.000

ω bc 1.000 1.000 1.000 1.000 1.000 1.000

RMSE ML 0.4719 0.4719 0.4719 0.4719 0.4719 0.4719

RMSE EB 0.4766 0.4751 0.4823 0.4743 0.4729 0.4831

RMSE mix 0.4742 0.4751 0.4719 0.4722 0.4725 0.4719

RMSE corr 0.4766 0.4751 0.4823 0.4743 0.4729 0.4831

FIT ML 54.7 54.7 54.7 54.7 54.7 54.7

FIT EB 54.3 54.4 53.8 54.5 54.6 53.7

FIT mix 54.5 54.4 54.7 54.7 54.7 54.7

FIT corr 54.3 54.4 53.8 54.5 54.6 53.7

TABLE XIV P UBLIC C ASCADED TANKS BENCHMARK WITH A LENGTH -50 FIR MODEL TRAINED ON THE FIRST 700 SAMPLES OF THE TRAINING RECORD . RMSE AND FITy ARE COMPUTED ON STANDARDIZED VALIDATION DATA AFTER A 50- SAMPLE INITIALIZATION WINDOW. N OISE VARIANCE ESTIMATED FROM ML TRAINING RESIDUALS ; CORRECTED COLUMNS AS IN TABLE XIII.

the TC kernel and therefore inherits the base estimator’s small degradation (0.4751 versus 0.4719 for ML). As on Silverbox, the trace-corrected variant sets ω bc = 1 throughout and cannot repair this, for the same σ b2 -inflation reason, which localizes the failure in the misspecification-driven calibration of the plug-in components rather than in the oracle mixing argument or in the trace bias alone. The worst case across all rows is bounded by the base estimator’s own error, which is the graceful-degradation property of the convex mixture. Thus the benchmark strengthens the robustness evidence without claiming that the XMSE plug-in rule is uniformly optimal on nonlinear real-data problems. IX. L IMITATIONS AND F UTURE W ORK The present theory separates the fixed-weight XMSE expansion from the plug-in selection of the mixing weight. We prove consistency of the plug-in weight and derive a second-order oracle regret rate for the corresponding design criterion. The transfer result (Theorem 9) carries this regret rate to the deterministic fixed-weight risk curve evaluated at the selected weight, and its uniform-convergence ingredient is proved from the inherited expansion rather than assumed. The remaining gap is explicit: the XMSE of the data-dependent estimator θbmix (b ωXMSE ) itself, with the random weight inside the expectation, differs from the evaluated risk curve by cross-moments between the weight and the EB correction, which are generated by their common dependence on the data (Remark 6). Bounding these cross-moments—plausibly by a delta-method expansion of the smooth scaled-kernel weight

map away from the projection boundary, with extra care near the projection and threshold boundaries where the weight map is nonsmooth—is the main open theoretical step, and none of our experimental claims rely on it. We also provide a thresholded decision result at the zero-bias boundary and a finite-set oracle inequality for kernel selection, with an extension to growing candidate dictionaries under uniform plug-in approximation and a high-probability oracle bound, including a sub-Gaussian rate specialization, and a compact multi-parameter plug-in consistency result based on uniform convergence and an argmin argument. Two design choices deserve a limitation note of their own. First, the projection of the weight onto [0, 1] deliberately forgoes the strictly negative criterion values that an extrapolated weight could attain when CEB > 0 or CEB < −2BEB (Remark 3); the experiments quantify how often the unconstrained ratio leaves [0, 1], but we do not study extrapolated rules. Second, the plug-in components inherit an O(σ 2 /N ) trace bias (Remark 5). The trace-corrected variant removes the identified bias terms and repairs the over-conservative weights in the well-specified semi-real setting, but it does not remove the variance of the plug-in quadratic forms, and under model misspecification, where σ b2 also absorbs model error, it over-corrects and returns full EB weight on the public benchmarks. Finite-sample and misspecification-robust calibration of the plug-in components therefore remains the main practical limitation. All asymptotic statements are for fixed parameter dimension. The growing dictionary results allow the candidate menu

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to expand, but they do not by themselves prove XMSE validity when the FIR order grows with N . A growing-order extension would need uniform control of the design eigenvalues, kernel inverses, scale-estimation objectives, and plug-in component errors as both the sample size and parameter dimension vary. We view this as a separate high-dimensional extension rather than an assumption hidden in the present fixed-order theory. The simulations focus on scaled kernels P (η) = ηK and on controlled tail-mismatch settings that make kernel misalignment explicit, supplemented by a semi-real fixed impulse-response benchmark and limited-training Silverbox and Cascaded Tanks benchmarks. This provides a clean stress test, a fixed-response check, and two public real-data benchmarks, with SURE-tuned, hard-selection, trace-corrected, and marginal-likelihood baselines. One natural competitor is not yet included: the closed-form biased estimators of [35], which match the XMSE of an EB estimator without hyperparameter estimation and would make an informative third anchor besides ML and EB. A broader experimental study should include this comparison, additional input designs, unknown noise variance in the synthetic settings, non-Gaussian disturbances, more archival system-identification benchmarks, and model classes beyond linear FIR when the data are strongly nonlinear. Although we provide finite-set, growing-dictionary, and compact multi-parameter consistency guarantees, plus a 100-system numerical validation, a larger automatic kernelselection study across additional kernels and operating conditions is left for future work. Finally, the finite-sample plug-in approximation inherits the limitations of second-order asymptotic XMSE. For very small samples, higher-order terms can be visible, as seen in the N = 30 sensitivity experiment. Bootstrap or higher-order corrections may improve the plug-in rule in this regime. X. C ONCLUSION XMSE is not only a diagnostic tool for explaining empirical Bayes failures. It can also serve as a constructive design criterion for adaptive estimators. The mixed estimator studied here has a scalar oracle rule with second-order dominance over both ML and the base EB estimator at the XMSE scale. Its plug-in version is supported by consistency, a boundary-robust oracle inequality, an oracle-regret rate, and a transfer of that rate to the fixed-weight risk curve evaluated at the selected weight, and the primitive plug-in route extends to compact multi-parameter kernel families. The same criterion also extends naturally to finite and growing candidate dictionaries with high-probability and sub-Gaussian oracle guarantees. The result is a simple safeguard: it preserves useful regularization when the kernel is appropriate and retreats toward ML when the regularizer is harmful. R EFERENCES [1] B. Efron and C. Morris. Limiting the risk of Bayes and empirical Bayes estimators—Part II: The empirical Bayes case. Journal of the American Statistical Association, 67(337):130–139, 1972. [2] J. S. Maritz and T. Lwin. Empirical Bayes Methods with Applications. Chapman and Hall/CRC, 2018.

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