I am having a bunch of trouble figuring out the likelihood function for this problem.
Let $\eta \sim \mathcal{N}(0,\sigma^2)$ and $\epsilon \sim U(-A,A)$ independently, and $K,P_1,P_2$ are constants :
Define:
$ f(\theta)= \left\{ \begin{array} a \theta P_1 + \eta & \text{ if } \theta P_1 + \eta\leq K-A \\ \theta P_2 + \eta & \text{ if } \theta P_1 + \eta> K-A \\ \end{array} \right\} + \epsilon$
Through convolution of the uniform and normal distributions, I've been able to figure out that the unconditional probability of observing $f(\theta)$ is (where $i$ indexes $P$):
$Convo(i) = \frac{1}{2 A}\big[\Phi(\frac{f(\theta) - \theta P_i + A}{\sigma}) - \Phi(\frac{f(\theta) - \theta P_i - A}{\sigma})\big]$
The next step that I'm having trouble with is figuring out how to deal with the different cases. It seems to make sense to me to divide $f(\theta)$ into regions that are associated with only first part of the piecewise function (below $K-2A$), both parts of the piecewise function (between $K-2A$ and $K$), and then the last part of the piecewise function (above $K$). In each of these segments, I figured I'd multiply the unconditional probability of each $f(\theta)$ with the probability that $\eta$ results in observations in this range. Here's what I've been working with so far but it doesn't seem right based on the simulations; any tips would be very much appreciated!
$L(\theta) = \Pi_{j=1}^{n} \Big[ \Phi( \frac{K-2A-\theta P_1}{\sigma}) Convo(1) \Big]^{ \mathbb{1}\{f(\theta)\leq K-2A\} } \Big[ \Big(\Phi( \frac{K-A-\theta P_1}{\sigma}) - \Phi( \frac{K-2A-\theta P_1}{\sigma}) \Big) Convo(1) + \Big(\Phi( \frac{K+A-\theta P_2}{\sigma}) - \Phi( \frac{K-A-\theta P_1}{\sigma}) \Big) Convo(2) \Big]^{ \mathbb{1}\{f(\theta)>K-2A \hspace{.2cm} \& \hspace{.2cm} f(\theta)\leq K \} } \Big[ (1- \Phi( \frac{K-A-\theta P_1}{\sigma})) Convo(2) \Big]^{ \mathbb{1}\{f(\theta)>K\} } $