Minimum of i.i.d ~gamma random variables

[Pere Callahan]

Hi,

Another question...

I know that the minimum of n i.i.d $$\lambda$$-exponentially distributed random variables is again exponentially distributed (with parameter $$n\lambda$$). Is something similar true for $$\Gamma(k,\theta)$$ ...? that is, is the minimum of n i.i.d Gamma distributed random variables again Gamma distributed... or is it some other well known distribution?

I also know about the extreme-value theorem which might be of use if I were only interested in large n (which is actually the case) but an explicit distribution seems always better to me.

Thanks for any answers

-Pere

 

[mmwave]

Hello Pere,

Thank you for your question. The minimum of n i.i.d Gamma distributed random variables is not necessarily Gamma distributed. It can follow a different distribution depending on the values of the shape and scale parameters (k and \theta). In general, the minimum of n i.i.d Gamma distributed random variables follows a distribution known as the generalized extreme value distribution, which is a type of extreme-value distribution.

However, if the scale parameter \theta is small enough, the minimum of n i.i.d Gamma distributed random variables can be approximated by a Gamma distribution with a different shape parameter k' and a smaller scale parameter \theta'. This approximation is known as the minimum order statistic and is commonly used in statistical analysis.

I hope this helps answer your question. Let me know if you have any further questions.