Maximum empirical likelihood estimation of time delay in independently and identically distributed noise

The authors apply the maximum empirical likelihood method to the problem of estimating the time delay of a measured digital signal when the signal can be seen as an instance of a stationary random process with additive independently and identically distributed (i.i.d.) noise. It is shown that, under these assumptions, an approximate log-likelihood function can be estimated from the measured data itself, and therefore a maximum likelihood estimate can be obtained without the prior knowledge of the formula for the signal likelihood. The Cramer-Rao lower bounds (CRLB) for two additive noise models (mixed-Gaussian and generalised normal distribution) are derived. The authors also show that the error produced by the maximum log-likelihood estimates (when the likelihood function is estimated from the measured data) better approximates the CRLB than other estimators for noise models other than Gaussian or Laplacian (special case of the generalised normal).