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Is there a common name to refer to distributions that are bounded on one side, and unbounded on the other side? For example, log-normal distribution, where the minimum value is zero, the maximum is +infinity.

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In at least one article, Wikipedia calls this "one-sided truncation." Although that article focuses on truncated Normal distributions, any continuous distribution whose support does not include the entire real line can be expressed as a truncation of one that is supported on the real line. The Encyclopedia of Mathematics uses the terms "left truncated" and "right truncated" for distributions with semi-infinite support (and cites some authorities). – whuber Aug 26 '14 at 15:18
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I would say no, as there are quite a few distributions in addition to log-normal that have support on $[ 0, \infty)$, such as the $\chi^2$ or gamma. (Wikipedia even has a list devoted to this criterion.)

In practice there are different circumstances in which such distributions are useful to approximate actually observed and measured phenomena. For one example, in some situations you may use the term censored at the boundary to describe how observations can go above and below the boundary, but they can only be measured within a certain support (and when they go outside this support they are recorded as being at the end point and/or beyond). This is typically referred to when you have an instrument that can not measure the numerical value outside of the bounds, it only knows it is at or below the boundary. For log-normal an example is the measurement of $\log(\text{wages})$. I believe for the US census they censor the distribution at $0$, although people who own their own businesses can be in the negative. (Some macro economic variables are well approximated by a log-normal distribution, but the support of the actual micro level units is partly in the negative.)

Another example (as user41315 mentioned) are truncated distributions. Truncated means "chopped off". Sometimes we only observe/record the measurement if it exceeds the boundary. For another economic example of wages, lets say you only had to file taxes if your wages were above $0$. So it is not like the census that just records $0$ even if you have less than $0$, you just simply don't observe the individuals with less than $0$ wages. As whuber stated in the comment, you can take take any distribution and re-express it as a truncated one.

The descriptions of censored or truncated refer to how the data are measured, and not to particular distributions. Not all measurements that are bounded on $[ 0, \infty)$ are necessarily truncated or censored though. For example distances or squares of values we know can not go below $0$.

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I am modeling travel time over a road segment, so this probably does not fall under censored, since travel time cannot be negative by itself? Not like negative wages in the example, where 0 is the lower bound due to post-processing. – inzl Aug 26 '14 at 12:06
Yes exactly @inzl. Even if due to "post-processing" it is not necessarily inappropriate to use log-normal for variables that theoretically have support below 0. The approximation may be good enough, such as macro economic variables are not likely to be anywhere near zero by the time they aggregate up to a polity. – Andy W Aug 26 '14 at 12:09
I would not use the word "censored" as described above, and neither does the discussion in the link provided. Censoring refers to values than can happen but are not observed, not to bounds on the support of a distribution. – rvl Aug 26 '14 at 13:58
To be clear about your comment @RussLenth - you would only describe the measurement process as being censored? Which is not a property of the distributions themselves? – Andy W Aug 26 '14 at 14:06
The use of "censored" in this answer is non-standard. When censoring at a boundary occurs, although the actual values are not recorded, the fact that they fall with a specific interval beyond the boundary is recorded (and therefore they constitute "observations," albeit unquantified ones). This causes the distribution to have positive atoms outside the boundary, which is not the case for the lognormal, $\chi^2$, or Gamma distributions and does not seem to be the subject of the question. – whuber Aug 26 '14 at 15:17

There seems to be no standard term, based on my experience. Some people refer to them as "one-sided", or "of one-sided support". I'd say "supported on [the positive half-line]", even though that is clunky.

Edit: related to Andy W's comment- "truncated" rather than "censored" more appropriate if the values outside a certain range are ignored (without adding mass to the observation window's endpoints). [I don't have enough reputation to comment directly on your answer].

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+1 Thanks for calling attention to the truncated/censored distinction. I added some comments to elaborate on that. – whuber Aug 26 '14 at 15:21

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