Consider a normal distribution with unknown μ and unit variance, but with a twist: the sample space has a gap in it between -1 and 3. Accounting for the gap, the resulting probability density becomes:

p\left(x;\mu\right)=\begin{cases} \frac{\phi\left(x-\mu\right)}{1-\Phi\left(3-\mu\right)+\Phi\left(-1-\mu\right)}, & x\in\left(-\infty,-1\right]\cup\left[3,\infty\right),\\ 0, & x\in\left(-1,3\right). \end{cases}

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:

  1. x = -1 (the left boundary)
  2. x = 3 (the right boundary)
For the sake of concreteness suppose our interest is in μ ≤ 0 vs. μ > 0. Should it make a difference to our inference whether we’re in scenario (i) or scenario (ii)?
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