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Let's consider a general signal processing estimation problem where the measurements are modeled as $${\bf x}[n]={\bf s(\theta)}+{\bf w}[n],$$ where ${\bf w}$ is a non-Gaussian r.v. (noise term) and ${\bf s(\theta)}$ is a deterministic nonlinear signal model. Since the noise is non-Gaussian I am wondering how one could design a Maximum Likelihood estimator for the parameter vector ${\bf \theta}$ if an estimate of the pdf of ${\bf w}$ was obtained by density estimation or another nonparametric technique. I have looked into empirical likelihood but I am not sure if it applies to this problem. Is this a standard problem in the context on non-parametric statistics? References are welcome.

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