I want to estimate unknown vector $\boldsymbol\theta$ from observations $$x_n = a_n \cdot f_n(\boldsymbol\theta) + w_n$$ with $n = 1 \ldots N$. The $f_n$ are simple, known nonlinear functions and $w_n$ is i.i.d. gaussian noise. The $a_n$ are unknown nuisance parameters with value range $[-1,1]$ and can be modeled as dependent random variables.

Now we can just write nonlinear LS $$(\hat{\boldsymbol\theta},\hat{\boldsymbol a}) = \text{argmin}_{\tilde{\boldsymbol\theta},\tilde{\boldsymbol a}} \sum_{n=1}^N \left(x_n - \tilde{a}_n \cdot f_n(\tilde{\boldsymbol\theta}) \right)^2$$ but clearly there is a dimensionality problem (trying to estimate $N+\text{dim}\,\boldsymbol\theta$ dimensions from $N$ measurements). However, for my specific problem, the covariance matrix $\text{E}(\boldsymbol a \boldsymbol a^\text{T})$ is known (even given by a simple closed-form expression). Now I wonder how this (and the bounded and linear nature of $a_n$) could be incorporated into the estimation, in order to successfully estimate $\boldsymbol\theta$ from the $x_n$ ?

If helpful, I could kick out the gaussian noise from the model.


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