Application of asymptotic expansions of maximum likelihood estimators errors to gravitational waves from binary mergers: the single interferometer case

TL;DR

This paper introduces a new analytical method to improve error estimates for gravitational wave parameter measurements from binary mergers, extending beyond the traditional Cramer Rao bounds by including second-order terms.

Contribution

It develops an asymptotic expansion approach to calculate the variance and bias of maximum likelihood estimators, providing more accurate error predictions for gravitational wave data analysis.

Findings

Second-order variance and bias improve error estimates.

Predictions align better with numerical simulations.

Identifies SNR thresholds for optimal waveform matching.

Abstract

In this paper we describe a new methodology to calculate analytically the error for a maximum likelihood estimate (MLE) for physical parameters from Gravitational wave signals. All the existing litterature focuses on the usage of the Cramer Rao Lower bounds (CRLB) as a mean to approximate the errors for large signal to noise ratios. We show here how the variance and the bias of a MLE estimate can be expressed instead in inverse powers of the signal to noise ratios where the first order in the variance expansion is the CRLB. As an application we compute the second order of the variance and bias for MLE of physical parameters from the inspiral phase of binary mergers and for noises of gravitational wave interferometers . We also compare the improved error estimate with existing numerical estimates. The value of the second order of the variance expansions allows to get error predictions closer to what is observed in numerical simulations. It also predicts correctly the necessary SNR to approximate the error with the CRLB and provides new insight on the relationship between waveform properties SNR and estimation errors. For example the timing match filtering becomes optimal only if the SNR is larger than the kurtosis of the gravitational wave spectrum.