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I have the two following questions:

  1. Imagine I have two sets of observations, and both sets have a lognormal distribution. Now, given I look at the union of the two sets - is the distribution still a lognormal?

  2. Given now I have again a set of observations with lognormal distribution. Given I add or subtract a scalar to $LogN(\mu, \sigma^2)$, what happens to each observation. Let's say I have a right shift of +2. Does that imply, that the value of each observation is bigger by 2?

Thanks

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  1. This is true only if the parameters of both populations are the same. Otherwise you may end up with a bimodal sample. In order to check this you may want to use a test of homogeneity. See

http://www.ltcconline.net/greenl/Courses/201/Regression/HomogeneityCollaborative/homogeneity.html

This can be done either by using parametric or nonparametric methods. See also page 92 of "Statistical Inference in Science" by D. Sprott.

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  • $\begingroup$ thank you..is this true for the union of arbitrary distributions, that the union's distribution is the same as the original distribution given the parameters of the original distributions are (approximately) the same? But the parameters of the distributions union is not the same as the orignal parameters I assume? And after all: Why exactly is this the case? Thank you^1000 $\endgroup$ – Pugl Mar 12 '13 at 17:50
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    $\begingroup$ When two sets of observations appear to have the same distribution, they can be viewed as two independent sets of iid draws from that common distribution, whence the unions of these observations will appear to have the same distribution. (A union of distributions makes no sense; the concept you may be referring to is a mixture.) $\endgroup$ – whuber Mar 12 '13 at 18:20
  • $\begingroup$ I didn't mean "union of distributions" but "union of observations". I thought that would make some sense. Anyway: Thank you! what about the question regarding the shift? $\endgroup$ – Pugl Mar 12 '13 at 18:26

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