I have data which can be well described by the sum of two distributions, $f(x,\theta) + g(x,\theta)$, where $\theta$ are the parameters to be estimated. However, the data range is limited to $x_0 < x < x_1$. Let's assume $f$ and $g$ are normalised to 1 over all space.
In order to perform the MLE, the total distribution must be normalized to 1 over $[x_0,x_1]$. Unfortunately, the normalization is non-trivial to compute as $f$ is marginalized over 3 other parameters. I'm finding that my fit results for $\theta$ are extremely sensitive to that normalization factor. Is there any way to mitigate this effect? Or is there any way to perform MLE on data in a restricted range without having to compute the new normalization factors?