Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Validating Angular Power Spectral Models for the Stochastic Gravitational-Wave Background Without Distributional Assumptions

Xiangyu Zhang, Erik Floden, Hongru Zhao, Sara Algeri, Galin Jones, Vuk Mandic, Jesse Miller

TL;DR

This work develops a distribution-free inference framework for validating angular power spectral models of the stochastic gravitational-wave background (SGWB), avoiding closed-form likelihoods and Gaussian assumptions for the angular-power estimators. By mapping the problem to a regression with a mean model $\bm{A}_{f_b}(\bm{\theta})$ and a covariance $\bm{\Sigma}_{f_b,s}$, it derives a new consistent estimator for the angular-power covariance and employs sphering and generalized least squares to estimate model parameters. It then constructs distribution-free goodness-of-fit tests using sphered residuals and Khmaladze-2 transforms, with test statistics that converge to a projected Brownian motion under $H_0$, enabling p-values via Monte Carlo without distributional assumptions. Simulation studies and an O3 data application (with simulated signal injections) demonstrate accurate Type I error control and appreciable power to detect model misspecification, offering a practical tool for robust SGWB model validation in future, more sensitive detectors.

Abstract

It is demonstrated that estimators of the angular power spectrum commonly used for the stochastic gravitational-wave background (SGWB) lack a closed-form analytical expression for the likelihood function and, typically, cannot be accurately approximated by a Gaussian likelihood. Nevertheless, a robust statistical analysis can be performed to enable the estimation and testing of angular power spectral models for the SGWB without specifying distributional assumptions. Here, the technical aspects of the method are discussed in detail. Moreover, a new, consistent estimator for the covariance of the angular power spectrum is derived. The proposed approach is applied to data from the third observing run (O3) of Advanced LIGO and Advanced Virgo.

On Validating Angular Power Spectral Models for the Stochastic Gravitational-Wave Background Without Distributional Assumptions

TL;DR

This work develops a distribution-free inference framework for validating angular power spectral models of the stochastic gravitational-wave background (SGWB), avoiding closed-form likelihoods and Gaussian assumptions for the angular-power estimators. By mapping the problem to a regression with a mean model and a covariance , it derives a new consistent estimator for the angular-power covariance and employs sphering and generalized least squares to estimate model parameters. It then constructs distribution-free goodness-of-fit tests using sphered residuals and Khmaladze-2 transforms, with test statistics that converge to a projected Brownian motion under , enabling p-values via Monte Carlo without distributional assumptions. Simulation studies and an O3 data application (with simulated signal injections) demonstrate accurate Type I error control and appreciable power to detect model misspecification, offering a practical tool for robust SGWB model validation in future, more sensitive detectors.

Abstract

It is demonstrated that estimators of the angular power spectrum commonly used for the stochastic gravitational-wave background (SGWB) lack a closed-form analytical expression for the likelihood function and, typically, cannot be accurately approximated by a Gaussian likelihood. Nevertheless, a robust statistical analysis can be performed to enable the estimation and testing of angular power spectral models for the SGWB without specifying distributional assumptions. Here, the technical aspects of the method are discussed in detail. Moreover, a new, consistent estimator for the covariance of the angular power spectrum is derived. The proposed approach is applied to data from the third observing run (O3) of Advanced LIGO and Advanced Virgo.

Paper Structure

This paper contains 16 sections, 4 theorems, 135 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Spherical Harmonics Search Methods for the SGWB
  3. Statistical Methods
  4. Parameter and covariance estimation
  5. Distribution-free testing of the angular power spectum
  6. Model testing via partial sums of the residuals
  7. Construction of distribution-free test statistics
  8. Simulation Studies
  9. Distribution-free Property
  10. Statistical properties
  11. Testing spectral models for the SGWB
  12. Summary and Discussion
  13. The generalized distribution of the angular power spectrum
  14. The covariance of the angular power spectrum
  15. Properties of the plug-in estimator of the covariance of angular power spectrum based on Eq. (ref)
  16. ...and 1 more sections

Key Result

Proposition 1

At a given frequency bin ${f_b}$, and time segment $s$,

Figures (2)

  • Figure 1: Graphs of the simulated null cumulative distribution functions of the Kolmogorov--Smirnov (top) and Cramér--von Mises (bottom) statistics from the four combinations of $\bm{A}^{M1}_{{f_b},s}$ and $\bm{u}^{(1)}_{{f_b},s}$ (blue solid); $\bm{A}^{M1}_{{f_b},s}$ and $\bm{u}^{(2)}_{{f_b},s}$ (red dotted); $\bm{A}^{M2}_{{f_b},s}$ and $\bm{u}^{(1)}_{{f_b},s}$ (yellow dashed); $\bm{A}^{M2}_{{f_b},s}$ and $\bm{u}^{(2)}_{{f_b},s}$ (darkgreen dash-dotted).
  • Figure 2: Graphs of the simulated null cumulative distribution functions of the Kolmogorov-Smirnov (top) and the Cramér-Von Mises (bottom) statistics in Eq. \ref{['eqn:TransTS']} for the four combinations of models and error structures in Eqs. \ref{['eqn:simulated_true_mean']}-\ref{['eqn:simulated_true_err']}, alongside their limiting null distribution.

Theorems & Definitions (8)

  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:cov angular ps']}
  • Theorem 3
  • proof