Likelihood-Free Spectral Estimation

• Likelihood-free spectral estimation methods infer frequency domain properties without constructing explicit likelihood functions.
• They employ techniques such as Bayesian SMC, frequency domain empirical likelihood, and random matrix tools to reduce computational costs.
• Robust performance is demonstrated through extensive simulations and real-data applications across diverse high-dimensional settings.

A likelihood-free spectral estimation procedure refers to any statistical method that enables inference on spectral (frequency domain) properties or models without constructing or maximizing an explicit parametric likelihood. Such procedures are essential in high-dimensional settings, models with intractable likelihoods, or where only weak spectral assumptions are defensible. This concept encompasses empirical likelihood in the spectral domain, Bayesian and nonparametric estimation via approximations, and loss-minimization on ensemble spectral functionals or summary statistics. The following sections provide a detailed exposition of the principal methodologies, theoretical underpinnings, and computational strategies of likelihood-free spectral estimation as developed in key contributions.

1. Semiparametric Bayesian Spectral Density Estimation

Traditional maximum likelihood estimation for Gaussian stationary time series is computationally prohibitive due to the $O(n^3)$ cost of Toeplitz matrix operations. The procedure developed by Chopin, Rousseau, and Liseo addresses this with a two-stage likelihood-free framework for the FEXP semi-parametric model. The data vector $x\in\mathbb{R}^n$ is assumed Gaussian with mean $\mu\mathbf{1}$ and Toeplitz covariance $T(f)$. The log-likelihood for $f$ is

$$\ell_n(f) = -\frac n2\log(2\pi) - \frac12\log|T(f)| - \frac12 x^{\top} T(f)^{-1} x,$$

with $T(f)$ determined by $f(\lambda)$, the spectral density. To reduce computational cost, the Toeplitz inverse is approximated via

$$T(f)^{-1} \approx T\big(1/(4\pi^2 f)\big), \quad n\to\infty,$$

yielding an approximate likelihood $\tilde L_n(f)$. The resulting approximate posterior,

$$\widetilde\pi_n(f \mid x) \propto \pi(f) \tilde L_n(f),$$

can be sampled efficiently by SMC using FFT algorithms, avoiding $O(n^3)$ computations.

To recover exact inference, an importance sampling correction is performed using weights

$$w\bigl(f^{(j)}\bigr) = \frac{L_n\bigl(f^{(j)}\bigr)}{\tilde L_n\bigl(f^{(j)}\bigr)},$$

with $\mathrm{Var}_{\widetilde\pi_n}[w(f)] \to 0$ as $n\to\infty$ under mild regularity.

The SMC sampler employed utilizes an annealing sequence with adaptively chosen temperatures,

$$\eta_t(\theta) \propto \pi(\theta)\bigl[\tilde L_n(\theta)\bigr]^{\gamma_t},$$

and combines random-walk and birth-death MCMC moves on model parameters. The procedure is robust to multimodality and adapts automatically to increasingly complex models. Empirical results show that the Bayesian semi-parametric approach outperforms frequentist counterparts in credible quantification for spectral and long-memory parameters (Chopin et al., 2012).

2. Frequency Domain Empirical Likelihood (FDEL)

Empirical likelihood in the frequency domain leverages the approximately exponential distribution of periodogram ordinates under short- and long-range dependence. For observed time series $(X_1, \ldots, X_n)$, the periodogram at frequency $\lambda_j = 2\pi j / n$ is

$$I_n(\lambda_j) = \frac{1}{2\pi n}|d_n(\lambda_j)|^2,$$

with the (mean-corrected) discrete Fourier transform $d_n(\lambda_j)$.

Given a vector of parameter-dependent estimating functions $G(\lambda; \theta)$ that satisfy

$$\int_{-\pi}^\pi G(\lambda; \theta_0) f(\lambda) d\lambda = 0,$$

empirical likelihood assigns weights $w_j \geq 0$ summing to unity such that

$$\sum_{j=1}^N w_j G(\lambda_j; \theta) I_n(\lambda_j) = 0.$$

The maximized empirical likelihood ratio is

$$R_n(\theta) = \prod_{j=1}^N [1 + t^\top G(\lambda_j; \theta) I_n(\lambda_j)]^{-1},$$

with $t$ the vector of Lagrange multipliers. Asymptotically,

$-2\log R_n(\theta_0) \Rightarrow \chi^2_r,$

yielding nonparametric confidence regions for spectral parameters and supporting hypothesis testing and construction of maximum empirical likelihood estimators (MELE). When $G = \partial_\theta \log f(\lambda; \theta)$, the MELE coincides asymptotically with the Whittle estimator (0708.0197).

3. Spectral Estimation for High-Dimensional Linear Processes

In high-dimensional time series $X_t = \sum_{\ell=0}^\infty A_\ell Z_{t-\ell}$, estimation of the joint spectral distribution of $A_\ell$ and $\Sigma_Z$ is computationally intractable through likelihood methods. The likelihood-free spectral estimator proceeds as follows:

• The sample periodogram is formed and weighted integrals are constructed as

$$S_g^{(n)} = \frac{1}{n}\sum_{t=1}^n g(\theta_t)\widetilde X_t\widetilde X_t^*,$$

with $g$ a nonnegative frequency weight and $\widetilde X_t$ the Fourier transform.

• The empirical spectral distribution (ESD) of the weighted periodogram $\widetilde S_g^{(n)}$ is characterized via its Stieltjes transform. Under $p,n\to\infty$ and $p/n\to c$, the ESD converges almost surely to a deterministic limit.
• The estimator minimizes an $L^\kappa$ loss between empirical and theoretical Stieltjes transforms, exploiting the Marčenko–Pastur-type limit for identification: $$D_\kappa(F) = \left\{\sum_{g\in\mathcal G}\sum_{z\in Z} |S_n^g(z) - S^g(z|F)|^\kappa\right\}^{1/\kappa}.$$

Assuming a finite mixture for the spectral law, the minimization is done over mixture weights, and consistency is established under uniform convergence of Stieltjes transforms (Namdari et al., 14 Apr 2025). Simulation studies and empirical applications to S&P 500 returns demonstrate reliable recovery of latent spectral structures.

4. Likelihood-Free Spectral Estimation in Random Matrix Models

Parameter estimation in random matrix models with only a single observed sample exploits free probability tools and Cauchy convolution. For a $d\times d$ self-adjoint matrix $W$, its ESD $\mu_W$ is smoothed by convolution with a Cauchy kernel $P_\gamma(x)$,

$$(P_\gamma * \mu_W)(x) = \frac{1}{\pi} \frac{\gamma}{(x-t)^2+\gamma^2},$$

or equivalently,

$$(P_\gamma * \mu_W)(x) = -\frac{1}{\pi} \Im G_{\mu_W}(x+i\gamma),$$

with $G_{\mu_W}$ the Stieltjes transform.

The free deterministic equivalent (FDE) of the model defines a deterministic limiting spectral density $\tilde\rho_\vartheta(x)$, which is matched to the data by minimizing the Cauchy cross-entropy,

$$L(\vartheta) = \frac{1}{d}\sum_{i=1}^d -\log[\tilde\rho_\vartheta(\lambda_i + T_i)],$$

with $T_i$ independent Cauchy noises and $\lambda_i$ eigenvalues of the observed $W$. Stochastic optimization algorithms (e.g., Adam) are used to minimize $L(\vartheta)$, with gradients computed via implicit differentiation of fixed-point equations that define the FDE. The determination gap between empirical and population losses vanishes as $d \to \infty$.

The methodology extends to dimensionality recovery in signal-plus-noise models, allowing simultaneous estimation of rank and noise parameters (Hayase, 2018).

5. Likelihood-Free Spectral Estimation for Lévy Processes

The fractional order (Blumenthal–Getoor index) of a Lévy process can be estimated in a likelihood-free spectral manner by leveraging the characteristic function of increments. For increments $X_1, \dots, X_N$, estimate

$$\widehat\varphi(u) = \frac{1}{N} \sum_{j=1}^N e^{i u X_j},$$

and analyze the log-modulus decay for large $|u|$ via

$$Y(u) = \log\left(-\log|\varphi(u)|^2\right) \approx \log(2n) + \alpha \log|u| + o(1).$$

A weighted least squares fit over a spectral window gives an estimator for $\alpha$.

Aggregation schemes (Lepski-type) are adopted for data-driven selection of the cut-off parameter $U$, providing robustness to the bias–variance trade-off. This method achieves minimax optimal rates and asymptotic normality, and can be applied to both historical increment data and option-implied settings (Belomestny, 2010).

6. Frequency Domain Empirical Likelihood for Irregular Spatial Data

For spatial processes observed at irregular locations, the lack of orthogonality in the discrete Fourier transform and nontrivial periodogram biases necessitate specialized likelihood-free empirical likelihood methods. The bias-corrected spatial periodogram $\tilde I_n(\omega)$ is used, with empirical likelihood ratios formed over lattices of asymptotically distant frequencies.

Asymptotic Wilks-type results hold under both pure and mixed increasing-domain regimes, after appropriate scaling of the log empirical likelihood ratio $R_n(\theta)$. Confidence regions and tests for spectral parameters can thus be constructed nonparametrically, without explicit variance estimation (Bandyopadhyay et al., 2015).

7. Numerical Performance and Practical Considerations

The effectiveness of likelihood-free spectral estimation is demonstrated through extensive simulation studies and real-data applications across the literature:

• For semi-parametric Bayesian SMC samplers, empirical coverage of credible bands for spectral and long-memory parameters is stable even in challenging regimes (Chopin et al., 2012).
• Frequency domain empirical likelihood yields correct coverage and robust inference for normalized spectral parameters in both short- and long-range dependence regimes (0708.0197).
• High-dimensional spectral estimators recover mixture weights and spectral distributions reliably in synthetic and financial time series (Namdari et al., 14 Apr 2025).
• In random matrix models, Cauchy noise loss-based procedures successfully recover rank and key parameters from single-sample spectra (Hayase, 2018).
• Empirical and calibration settings for Lévy process estimation confirm that adaptive aggregation achieves nearly parametric efficiency for the fractional order (Belomestny, 2010).

This suite of methods demonstrates that likelihood-free estimation in the spectral domain is both theoretically principled and practically effective, despite the absence of tractable or well-specified likelihoods.