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I am curious about a normal distribution with no probability mass in certain regions, sort of like the complement of the truncated normal. In particular, it will have zero mass in a circular region.

Here is its pdf:

$$P(X | \mu, \Sigma) \propto I_{\{X \in C\}} exp\{-\frac{1}{2}(X - \mu)'\Sigma^{-1}(X - \mu)\}$$

Where $C$ is the set of all points that lie within the circle of interest, $I_{X \in C}$ is a function that takes value 1 when $X \in C$ and 0 otherwise (sometimes called the indicator function), $\mu$ is a mean vector, and $\Sigma$ is a positive definite covariance matrix.

Let me show some graphics illustrating what I mean.

First in 1 dimension. This is the standard normal density:

standard normal density

And now the swiss cheese density, with a circle centered at 0.5 with radius 0.5:

swiss cheese density

I would also like to contrast this with the truncated normal distribution over the same region:

truncated normal density

(Notice that the y axis is incorrect on the last two charts as I have failed to renormalize this density; this is simply an illustrative tool)

In 1D, the Swiss Cheese distribution may be thought of as a mixture of truncated normal distributions. However, this is no longer the case in higher dimensions.

Let's consider a 2D case so that it may be visualized.

I will sample from a 2D gaussian and bin the results (since my ggplot skills are inadequate at best, I am unsure how to plot a 3D density):

2d Gaussian

And likewise for a 2D swiss cheese distribution:

2d swiss cheese

I am interested in the higher D case, as well as the case where there are several circles cut out of the distribution.

Now my questions:

  • Is there a name for such a distribution?
  • Is there a source for results about such distributions?
John Madden
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    When you recognize that "truncated" really just means "restricted to a subset," then it becomes obvious that all these are truncated distributions, too. Most results you can find for truncated distributions, such as computing their moments and forms of Maximum Likelihood estimators, apply in this general sense. – whuber May 23 '17 at 20:39
  • @whuber thanks for your reply. Perhaps this is true in a general sense, but much of the material I find on truncated distributions assumes that the R.V.'s are restricted to live inside of some interval. For instance, the wiki article on truncated normals makes this assumption (https://en.wikipedia.org/wiki/Truncated_normal_distribution), the wiki article on truncated distributions avoids this assumption in the introduction, but assumes it elsewhere (https://en.wikipedia.org/wiki/Truncated_distribution)... – John Madden May 23 '17 at 20:51
  • ..., this article from FSU mentions only the upper/lower bound case (https://people.sc.fsu.edu/~jburkardt/presentations/truncated_normal.pdf), and indeed this CV question (https://stats.stackexchange.com/questions/48897/maximum-likelihood-estimators-for-a-truncated-distribution) also makes that assumption. I suppose I was hoping that truncation to some other type of region had a distinct name, or perhaps there was a more general word than truncation. – John Madden May 23 '17 at 20:51
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    My point is that it would be impossible to lay out all the possible configurations even in one dimension, so any treatment of the subject has to be understood as *illustrating the techniques*. You can adapt just about all of the material in those articles to the general case without making any substantial changes to the methods. As that Wikipedia article says, "other possibilities can be handled straightforwardly." – whuber May 23 '17 at 20:53
  • @whuber I suppose. I thought this situation might be standard enough that the possibility might already have been handled straightforwardly by someone else. But the lack of an answer informs me otherwise. – John Madden May 23 '17 at 21:00
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    I like your vivid name for the case of internal regions of zero probability; I am not aware of a specific name for that case (I might have called it -- rather more prosaically -- *internal truncation*). – Glen_b May 24 '17 at 00:50
  • @Glen_b I see, thanks for your reply. I think us statisticians could learn to be a little more imaginative in our naming schemes (cf *deep learning*). – John Madden May 24 '17 at 00:57
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    Tukey was a master of the colourful name. A lot of them caught on ("bit" for example, was one of his more famous contributions) but quite a few didn't (*quefrency* doesn't seem to come up so much). – Glen_b May 24 '17 at 01:04

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