Enhanced F-Statistic Framework Overview

• Enhanced F-statistic framework is a collection of extensions to the classical F-statistic, preserving likelihood structure while integrating artifact rejection and robust nuisance parameter handling.
• It employs phase-relaxed statistics, hierarchical follow-up with dense retiling, and deep-learning post-processing to enhance sensitivity and computational efficiency.
• Applications include gravitational-wave continuous and ringdown analyses as well as econometric weak-instrument testing, demonstrating the framework’s versatility across fields.

Searching arXiv for the cited works to ground the article in current records. Taken together, the literature describes an enhanced F-statistic framework as a set of extensions of the classical F-statistic and closely related F-based procedures that preserve explicit likelihood structure or calibrated reference distributions while addressing artifact rejection, partial coherence, Bayesian parameter estimation, heteroskedasticity, computational scaling, and downstream candidate triage. In gravitational-wave data analysis, the F-statistic remains a matched-filter statistic obtained by maximizing over nuisance amplitude parameters; later work extends that core construction to hierarchical follow-up, line-robust detection statistics, transient and ringdown analyses, Bayesian posterior reconstruction, and machine-learning-assisted post-processing. In parallel statistical literatures, F-based methods are generalized to many restrictions under heteroskedasticity, weak-instrument diagnostics, nonparametric empirical Bayes inference, and deep embedding losses (Sieniawska et al., 2019, Keitel et al., 2012, Anatolyev et al., 2020, Windmeijer, 2023).

1. Canonical likelihood structure and profile-statistic formulation

In the time-domain continuous-wave setting, the detector output is modeled as

$s(t)=\sum_{i=1}^4 a_i h_i(t),$

where the basis functions depend on the intrinsic parameters $(\omega,\dot{\omega},\alpha,\delta)$. Maximizing the likelihood over the amplitude, or “extrinsic,” parameters yields the $\mathcal{F}$-statistic as a function only of the intrinsic parameters. One explicit form used in follow-up studies is

$$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$

For candidate post-processing in all-sky searches, the signal-to-noise ratio is related to the statistic by

$\rho=\sqrt{2(\mathcal{F}-2)}.$

This relation is used to threshold candidate events and to construct the structured outputs subsequently analyzed by downstream classifiers (Sieniawska et al., 2019, Morawski et al., 2019).

The same profile-likelihood pattern recurs in later gravitational-wave formulations. For binary black hole signals, the waveform is rewritten as

$$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$

so that the log-likelihood ratio becomes

$$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$

Maximization over $B^\mu$ yields $\hat{B}^\mu=M^{\mu\nu}s_\nu$ and the profile likelihood

$$\ln\Lambda(\bm\theta)=\frac12 s_\mu M^{\mu\nu}s_\nu.$$

An analogous construction is used in ringdown analysis, where the signal is represented as $(\omega,\dot{\omega},\alpha,\delta)$0 and amplitudes and phases of quasinormal modes are analytically eliminated from the sampled space (Wang, 18 Sep 2025, Wang et al., 2024).

A closely related maximum-likelihood reformulation establishes the formal equivalence between the $(\omega,\dot{\omega},\alpha,\delta)$1-statistic and the 5-vector method. In the single-detector case the two methods yield the same detection probability for a given false alarm rate, because their statistics have the same $(\omega,\dot{\omega},\alpha,\delta)$2 and non-central $(\omega,\dot{\omega},\alpha,\delta)$3 distributions. In the multidetector case, the maximum-likelihood derivation prescribes weighting each detector’s 5-vector by its observation time and sensitivity, $(\omega,\dot{\omega},\alpha,\delta)$4, producing statistically efficient estimators and analytical distributions for the detection statistic (D'Onofrio et al., 2024). This suggests that many “enhancements” preserve the original F-statistic’s role as a likelihood-maximized summary while altering how nuisance structure, detector heterogeneity, or model complexity are handled.

2. Robustness to artifacts, phase mismatch, and sparse weak structure

A central line of development targets the F-statistic’s vulnerability to mismatch and instrumental contamination. The “phase-relaxed” statistic $(\omega,\dot{\omega},\alpha,\delta)$5 was introduced for incoherently combining the results of fully coherent searches over short intervals while allowing an independent phase offset $(\omega,\dot{\omega},\alpha,\delta)$6 in each interval. The usual semi-coherent statistic is

$(\omega,\dot{\omega},\alpha,\delta)$7

whereas $(\omega,\dot{\omega},\alpha,\delta)$8 maximizes a coherent quadratic form over the phase offsets. The reported effect is a sensitivity gain of approximately $(\omega,\dot{\omega},\alpha,\delta)$9 over the semi-coherent $\mathcal{F}$0-statistic, together with a rough determination of the time-evolving phase offset between template and signal (Cutler, 2011).

Single-detector monochromatic artifacts motivate a different extension. Under an extended noise model including Gaussian noise plus single-detector lines, the line-veto statistic replaces the standard signal-versus-Gaussian-noise odds ratio with a generalized odds ratio comparing the coherent multi-detector signal hypothesis against both single-detector line hypotheses and Gaussian noise. The key qualitative effect is that candidates with high single-detector $\mathcal{F}$1 but weaker coherent network $\mathcal{F}$2 are downweighted. In the absence of lines, the statistic behaves similarly to the standard $\mathcal{F}$3-statistic; in the presence of lines, it is more robust against line artifacts (Keitel et al., 2012).

A second-pass enhancement based on higher criticism addresses situations in which signal power is weak and sparse across a collection of F- or C-statistic values. For targeted binary searches, higher criticism on C-statistic data is more sensitive by $\mathcal{F}$4 than the C-statistic alone under optimal conditions, and the relative advantage increases as the error in the orbital parameters increases. For phase-wandering sources over multiple time intervals, the method gives a $\mathcal{F}$5 increase in detectability with few assumptions about the frequency evolution. By contrast, for all-sky searches for unknown periodic sources, second-pass higher criticism does not provide any benefits over a first-pass $\mathcal{F}$6 search (Bennett et al., 2013).

These extensions share a common logic: the classical $\mathcal{F}$7-statistic remains the baseline matched-filter object, while robustness is introduced by relaxing cross-segment phase coherence, embedding explicit artifact hypotheses, or exploiting collective departures from the null that are invisible to single-template thresholding.

3. Hierarchical follow-up and coherent candidate refinement

In all-sky searches for isolated neutron stars, the initial scan over frequency, spindown, and sky position produces candidates that must be confirmed or rejected by a dedicated follow-up. The Followup procedure implements the matched-filtering $\mathcal{F}$8-statistic method in a hierarchical coherent framework. Candidates found in short segments are reanalyzed in successively longer coherently joined segments, with the expectation that the SNR of a true signal increases as $\mathcal{F}$9, where $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$0 is the number of joined segments. Each coherent combination refines the parameter estimates and improves discrimination between real signals and noise artifacts (Sieniawska et al., 2019).

The follow-up stage depends on local retiling of parameter space. Reduced Fisher matrix techniques are used to construct optimally dense grids with controlled mismatch, and the grids are recalculated proximate to candidate parameters for zoomed-in searches. Fine maximization is then carried out with derivative-free optimizers. Simplex/Nelder-Mead is effective for initial local maximization but can be trapped by local maxima; MADS explores the space with adaptive mesh size; invMADS, introduced in the follow-up paper, starts from a small mesh and expands, and is reported to have the best SNR recovery though with higher computational cost acceptable in the follow-up context (Sieniawska et al., 2019).

The Polgraw follow-up design describes the same logic in operational terms. Approximate parameters obtained from the coherent search and coincidence stages are first improved on a denser local grid, for example by evaluating all $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$1 steps in each of the four intrinsic parameters, yielding $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$2 points, and then by continuous optimization with Nelder-Mead or MADS. The refined parameters are next used in combined-segment analysis, and the signals from detectors are studied separately together with the network combination, so that the true parameters and signal-to-noise values are established (Sieniawska et al., 2018).

Validation in white Gaussian noise shows agreement with theoretical predictions for the distribution of $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$3, SNR growth, estimator bias, and parameter variance as segment length increases. The follow-up papers therefore position enhancement not as a change to the definition of $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$4 itself, but as a disciplined method for extracting more reliable evidence from a small, computationally manageable subset of candidates (Sieniawska et al., 2019).

4. Post-processing, acceleration, and memory-aware implementations

One prominent downstream enhancement applies deep learning not to raw strain data, but to the structured outputs of the time-domain $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$5-statistic pipeline. In this approach, the object of classification is the distribution of candidate points that survive an $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$6-statistic threshold in a given parameter band. The 1D CNN operates on five sorted vectors: the top 50 SNR points and their associated $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$7, $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$8, $$\mathcal{F}=\frac{|F_a|^2}{\langle a^2(t)\rangle}+\frac{|F_b|^2}{\langle b^2(t)\rangle}, \qquad F_a:=\langle x(t)a(t)e^{-i\phi(t)}\rangle,\quad F_b:=\langle x(t)b(t)e^{-i\phi(t)}\rangle.$$9, and $\rho=\sqrt{2(\mathcal{F}-2)}.$0 values. The 2D CNN operates on four $\rho=\sqrt{2(\mathcal{F}-2)}.$1 histograms in $\rho=\sqrt{2(\mathcal{F}-2)}.$2, $\rho=\sqrt{2(\mathcal{F}-2)}.$3, $\rho=\sqrt{2(\mathcal{F}-2)}.$4, and $\rho=\sqrt{2(\mathcal{F}-2)}.$5 space. The three target classes are Gaussian noise, astrophysical signal, and stationary line artifact. The 1D CNN achieves $\rho=\sqrt{2(\mathcal{F}-2)}.$6 accuracy across broad ranges of $\rho=\sqrt{2(\mathcal{F}-2)}.$7 and injected SNR $\rho=\sqrt{2(\mathcal{F}-2)}.$8, while the 2D CNN reaches $\rho=\sqrt{2(\mathcal{F}-2)}.$9 overall but is less suitable for practical use because of poor line detection. The method is explicitly downstream of matched filtering and is presented as augmenting, not replacing, traditional $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$0-statistic searches (Morawski et al., 2019).

Several papers enhance tractability by altering the implementation rather than the statistical target. In Virgo VSR1 data, a reformulation based on barycentric resampling and FFT-compatible phase factorization allowed the coherent stage of an all-sky $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$1-statistic search to use the Fast Fourier Transform, resulting in a fifty-fold speed-up with respect to the algorithm used in other pipelines. The study reports $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$2 computed $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$3-statistic values and estimates that, in the most sensitive parts of the detector band, more than $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$4 of signals would have been detected with dimensionless gravitational-wave amplitude greater than $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$5 (Aasi et al., 2014).

For long-lived transient signals, the transient $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$6-statistic generalizes the continuous-wave model by introducing window functions $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$7 parameterized by start time $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$8 and duration $$h(f)=B^1 h_1(f)+B^2 h_2(f), \qquad B^1=\frac{\cos2\psi}{d_L},\quad B^2=\frac{\sin2\psi}{d_L},$$9, including rectangular and exponentially decaying windows. A pyCUDA implementation parallelizes the evaluation of the statistic over large $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$0 grids. The reported speedup is at least $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$1, and for realistic exponential-window searches can be much greater, up to $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$2 for a Tesla V100 GPU versus CPU. This makes wider parameter-space searches feasible for glitching pulsars, outlier follow-up, and prospective blind all-sky searches for unknown neutron stars (Keitel et al., 2018).

Memory pressure motivates a different engineering extension: one- and two-bit digitization of detector data prior to $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$3-statistic analysis. Monte Carlo simulations show that, relative to 64-bit data, a one-bit signal needs to be $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$4 stronger and a two-bit signal with optimal thresholds $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$5 stronger to be detected with $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$6 probability in Gaussian noise. The same work reports a factor of 64 reduction in memory for one-bit digitization and 32 for two-bit digitization, with little change in the penalty when the signal frequency decreases secularly or when noise statistics are not Gaussian. Time-series digitization is found to be preferable to direct SFT digitization (Clearwater et al., 2024).

5. Bayesian parameter estimation and evidence calculations

A major recent development is the use of $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$7-statistic-based likelihoods for full Bayesian parameter estimation. For known pulsars, a new method leverages the standard frequency-domain $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$8-statistic code to build posteriors over the amplitude parameters $$\ln\Lambda(\Theta)=B^\mu s_\mu(\theta)-\frac12 B^\mu M_{\mu\nu}(\theta)B^\nu.$$9. The likelihood

$B^\mu$0

can be analytically marginalized over a uniform prior on $B^\mu$1, yielding

$B^\mu$2

The explicit purpose of this marginalization is to remove the $B^\mu$3 degeneracy that otherwise produces bimodal or multimodal posteriors. Simulated signals, hardware injections, and PP tests support posterior calibration, and application to PSR J1526-2744 yields no evidence for a signal and an upper limit $B^\mu$4 of approximately $B^\mu$5 (Ashok et al., 2024).

For binary black hole signals, the same profile-likelihood principle is used to reduce the sampled dimension by analytically maximizing over luminosity distance and polarization angle. In the GW150914 benchmark, the dimension is reduced from 15 to 13, and the reported runtime reduction is approximately $B^\mu$6. The method also introduces a Bayes-factor calculation that properly incorporates priors on the analytically maximized parameters. Across sampler configurations, the $B^\mu$7-statistic implementation shows a log-Bayes factor between runs smaller than $B^\mu$8 and consistently lower Jensen-Shannon divergence values than full frequency-domain inference. Its posteriors can be slightly broader for some parameters, and the authors interpret this as more honest uncertainty quantification in complex high-dimensional spaces (Wang, 18 Sep 2025).

Ringdown analysis extends the same idea to sums of quasinormal modes. The waveform is written as $B^\mu$9, with each mode contributing two basis functions corresponding to cosine and sine phase components. Because amplitudes and phases are analytically integrated out, the number of included QNMs does not increase the marginalized parameter space. The evidence is reduced to a low-dimensional integral over physical parameters such as the remnant mass, spin, and inclination. Applied to GW150914, this method yields results consistent with classical time-domain Bayesian inference and finds no evidence of the first overtone mode (Wang et al., 2024).

These Bayesian extensions retain the F-statistic’s original advantage—analytic treatment of linear nuisance structure—while relocating the method from pure detection or ranking into posterior inference and model comparison.

6. Cross-disciplinary generalizations of F-based inference

Outside gravitational-wave astronomy, the F-statistic is enhanced in econometric hypothesis testing under many restrictions and heteroskedasticity. In this setting, the classical F-distribution is invalid when the number of restrictions is large relative to sample size or when errors are conditionally heteroskedastic. The proposed correction keeps the conventional F statistic but replaces its critical value with a leave-out-adjusted $\hat{B}^\mu=M^{\mu\nu}s_\nu$0-based critical value. Leave-one-out estimation centers the statistic, leave-three-out estimation scales it, and the null is rejected when $\hat{B}^\mu=M^{\mu\nu}s_\nu$1. The method is asymptotically valid whether the number of restrictions is fixed or grows with sample size, can be proportional to the number of observations, and is reported to have non-trivial asymptotic power against the same local alternatives as the exact F test when the latter is valid (Anatolyev et al., 2020).

In instrumental-variables econometrics, the robust F-statistic is generalized through a class of effective F-statistics for linear GMM estimators,

$\hat{B}^\mu=M^{\mu\nu}s_\nu$2

The standard nonhomoskedasticity-robust F-statistic is one member of this class, and the associated GMMf estimator uses a weight matrix constructed from first-stage residuals rather than structural residuals. The central caveat is explicit: the robust F-statistic can be used as a test for weak instruments for the GMMf estimator, not for 2SLS. The recommended reporting practice is therefore to provide both the effective F and the robust F, together with their estimator-specific critical values (Windmeijer, 2023).

A related but distinct strand replaces prior modeling with nonparametric $\hat{B}^\mu=M^{\mu\nu}s_\nu$3-modeling in empirical Bayes inference under unequal and unknown variances. Here the main contribution is a generalized Tweedie-type identity,

$\hat{B}^\mu=M^{\mu\nu}s_\nu$4

which expresses the Bayes estimator in terms of the joint marginal density $\hat{B}^\mu=M^{\mu\nu}s_\nu$5 and its derivatives, and an MGF representation that recovers the full posterior distribution without estimating the prior. The paper frames this as an extension of classical empirical Bayes and F-statistic methodologies to heteroscedastic settings (Zhao et al., 23 Apr 2026).

The F-statistic has also been repurposed as a loss for deep representation learning. In that setting, the statistic measures separation between classes along individual embedding dimensions, and the loss enforces separation only on a subset of dimensions for each class pair. This is intended to support both few-shot learning and disentanglement. Reported outcomes include recall@$\hat{B}^\mu=M^{\mu\nu}s_\nu$6 and few-shot performance matching or exceeding state of the art, together with superior modularity and explicitness on Sprites and small NORB relative to a range of alternatives (Ridgeway et al., 2018).

7. Limitations, caveats, and recurring themes

The enhanced F-statistic framework is not a single universally superior replacement for classical procedures. Several papers emphasize domain-specific limits. The CNN post-processor is trained on candidate distributions rather than raw detector data, and its 1D version generalizes well only when the number of candidates per band exceeds the minimal threshold of 50; frequencies below $\hat{B}^\mu=M^{\mu\nu}s_\nu$7 Hz are problematic because of low candidate count, and the 2D CNN performs worse in identifying stationary lines (Morawski et al., 2019). The hierarchical follow-up studies assume white Gaussian noise and a persistent continuous signal, and invMADS trades better SNR recovery for slower computation (Sieniawska et al., 2019). Higher criticism provides no benefit in all-sky searches dominated by the brightest source (Bennett et al., 2013).

Other caveats are statistical rather than astrophysical. In the many-restrictions heteroskedastic test, leave-three-out estimation can fail or yield negative variance estimates, in which case an upward-biased estimator is used to guarantee conservativeness and an always-defined critical value (Anatolyev et al., 2020). In weak-instrument testing, the robust F-statistic is not a generic diagnostic for 2SLS under heteroskedasticity; it is tied to the bias properties of GMMf (Windmeijer, 2023). In ringdown analysis, the increased efficiency of the $\hat{B}^\mu=M^{\mu\nu}s_\nu$8-statistic formulation does not change the substantive conclusion that GW150914 shows no evidence for the first overtone mode (Wang et al., 2024).

A recurring pattern nonetheless emerges. Across continuous-wave searches, transient analyses, ringdown spectroscopy, Bayesian parameter estimation, econometric testing, and representation learning, the enhanced framework preserves an explicitly analyzable core—analytic maximization over nuisance parameters, calibrated null distributions, or closed-form corrections—while moving complexity into structured hypotheses, posterior reconstruction, detector weighting, or downstream decision rules. This suggests that the enduring value of the F-statistic lies less in any one implementation than in its capacity to support principled extensions without abandoning tractable inference.