In this model μ is not quite a location parameter; when it’s far from the gap the density is effectively a normal centered at μ but when it’s close to the gap its shape is distorted. It becomes a half-normal at the gap boundary and then something like an extra-shallow exponential (log-quadratic instead of log-linear like an actual exponential) as μ moves toward the center of the gap. At μ = 1 the probability mass flips from one side of the gap to the other. Here’s a little web app in which you can play around with this statistical model (don’t neglect the play button under the slider on the right hand side).
Now the question; I ask my readers to report their gut reaction in addition to any more considered conclusions in comments.
Suppose μ is unknown and the data is a single observation x. Consider two scenarios:
- x = -1 (the left boundary)
- x = 3 (the right boundary)