# If $X$ has a log-normal distribution, does $X-c$ also have a log-normal distribution?

Let's say $X$ has a log-normal distribution and there is one real positive number $c$. then is it right to say that $(X -c)$ also has some log-normal distribution? My feeling is that, it can't be, because $(X - c)$ may take negative value whereas a log-normal distribution is only defined on the positive domain. Can somebody disprove that?

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I think you are correct. I had to add 1 to my data to be able to use the Zipf distribution. –  Damien Jul 12 '12 at 18:18
Please consider accepting answers to some of your previous questions. This can be done by clicking on the checkmark next to the answer that best responds to your question. See the FAQ for more info. –  cardinal Jul 13 '12 at 13:21

The answer to your question is (essentially) no and your argument has the right idea. Below, we formalize it a bit. (For an explanation of the caveat above, see @whuber's comment below.)

If $X$ has a lognormal distribution this means that $\log(X)$ has a normal distribution. Another way of saying this is that $X = e^{Z}$ where $Z$ has a $N(\mu, \sigma^2)$ distribution for some $\mu \in \mathbb{R}, \sigma^2 >0$. Note that by construction, this implies that $X \geq 0$ with probability one.

Now, $X-c = e^Z - c$ cannot have a lognormal distribution because

$$P(e^Z - c < 0 ) = P(e^Z < c) = P(Z < \log(c)) = \Phi \left( \frac{ \log(c) - \mu }{\sigma} \right)$$

which is strictly positive for any $c > 0$. Therefore, $e^Z - c$ has a positive probability of taking on negative values, which precludes $e^Z - c$ from being lognormally distributed.

In summary, the lognormal distribution is not closed under subtraction of a positive constant. It is, however, closed under multiplication by a (positive) constant, but that's an entirely different question.

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+1 It may be worth noting that in some circles the term "lognormal distribution" may comprise the three-parameter version in which an additive location parameter is included. In this case, the answer--by explicit construction--is yes. –  whuber Jul 12 '12 at 18:49
I've asked about robust parameter estimation for the shifted LogNormal distribution. Maybe you can help me? –  Erich Schubert Feb 7 at 15:14