Testing models for angular power spectra: A distribution-free approach

A novel goodness-of-fit strategy is introduced for testing models of angular power spectra with unknown parameters. Using this strategy, it is possible to assess the validity of such models without specifying the distribution of the angular power spectrum estimators. This holds under general conditions, ensuring the method’s applicability in diverse applications. Moreover, the proposed solution overcomes the need for case-by-case simulations when testing different models, leading to notable computational advantages.

💡 Research Summary

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The paper introduces a novel, distribution‑free goodness‑of‑fit (GoF) methodology for testing theoretical models of angular power spectra (APS) when the underlying distribution of the APS estimators is unknown. Angular power spectra, defined as the variance of spherical harmonic coefficients aℓm across multipole ℓ, are central to many fields—cosmic microwave background (CMB) analysis, galaxy clustering, geodesy, and the emerging study of stochastic gravitational‑wave (GW) backgrounds. In practice, the estimators Ĉℓ are averages of squared aℓm and therefore follow a generalized χ² distribution rather than a Gaussian. Existing approaches either approximate this distribution with a mixture of Gaussian and log‑normal components or resort to case‑by‑case Monte‑Carlo simulations, both of which are computationally intensive and can be inaccurate.

The authors propose a two‑stage framework that eliminates the need to know the distribution of Ĉℓ. First, they estimate the model parameters θ by solving a generalized nonlinear least‑squares (GNLS) problem:

θ̂ = arg minθ (Ĉ – C_M(θ))ᵀ Σ⁻¹ (Ĉ – C_M(θ)),

where Σ is the (block‑diagonal) covariance matrix of the APS estimates. This estimator is consistent regardless of the true distribution of Ĉ, provided the model C_M(θ) is correctly specified. The residual vector ε = Σ⁻¹ᐟ² (Ĉ – C_M(θ̂)) is then decorrelated and has mean zero and identity covariance.

A naïve GoF test could be built from the partial‑sum process w_N(t) = N⁻¹ᐟ² Σ_{k≤t} ε_k, using functionals such as the Kolmogorov‑Smirnov (KS) or Cramér‑von‑Mises statistics. However, the asymptotic covariance of w_N(t) depends on the model through vectors μ_j that involve the Jacobian ∂C_M/∂θ and Σ. Consequently, the null distribution of any test statistic derived from w_N(t) would have to be recomputed for each candidate model via Monte‑Carlo or parametric bootstrap, negating the computational advantage.

To overcome this, the authors adapt the Khmaladze transformation, specifically a unitary operator U_{a,b} that maps one unit‑norm vector a to another unit‑norm vector b while preserving orthogonal components. They construct a sequence of orthonormal vectors {r_j} that are completely independent of the model, and iteratively apply U operators to map the model‑dependent μ_j vectors onto the chosen r_j basis. The resulting transformed residuals e = U_T ε – Σ_j r_j r_jᵀ U_T ε (with U_T the composition of all U operators) have a partial‑sum process v_N(t) whose asymptotic distribution is that of a standard Gaussian process u_N(t) with covariance I_N – Σ_j r_j r_jᵀ, which depends only on the arbitrarily selected orthonormal set {r_j}. Thus, any functional h