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Asymptotic expansions of Maximum Likelihood estimators errors, with an application to gravitational waves generated in the inspiral phase of binary mergers |
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Le 5 février 2010, à 11h
Salvatore Vitale (LPTMC)
In this talk we describe a new methodology to calculate analytically the error for a maximum likelihood estimate (MLE) of physical parameters from Gravitational Wave (GW) signals. All the existing literature 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 SNRs 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. The examples are limited to a single, optimally oriented, interferometer. 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. The power expansions can also be used for multiple measurements of the same observable. In that case the SNR should be replaced with SNR times the number of measurements.
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