Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or similar concepts? Do we know that $\max(X,Y)$ is normally distributed, maybe a formula for mean and standard deviation in terms of those for $X$ and $Y$? I checked the usual places (wikipedia, google) but didn't find anything.
Solved – the distribution for the maximum (minimum) of two independent normal random variables
extreme valuenormal distribution
Related Solutions
The correct answer here is hierarchical modeling (also called multilevel modeling). What you want to do is have the variance parameters drawn from a common prior distribution who's parameters are also estimated. Something like
CountryVariance_i ~ D(Location, Scale)
Location ~ D_2(LocationPriorParams)
Scale ~ D_3(ScalePriorParams)
Where D, D_2, D_3 are whatever distributions would like.
This formalizes the notion that the variance parameters should be similar, and will do the correct kind of shrinkage. I believe that Gelman's books (Bayesian Data Analysis and Data Analysis Using Regression and Multilevel/Hierarchical Models) talk quite a bit about this kind of thing (though perhaps not a lot for variance parameters).
I don't have a great introduction on hand, so I recommend looking at the wikipedia page and searching for "Hierarchical modeling" or something and clicking on various introductions (there are many). Look for graphs as these are helpful for understanding hierarchical models.
According to Nadarajah and Kotz, 2008, Exact Distribution of the Max/Min of Two Gaussian Random Variables, the PDF of $X = \max(X_1, X_2)$ appears to be
$$f(x) = 2 \cdot \phi(x) \cdot \Phi\left( \frac{1 - r}{\sqrt{1 - r^2}} x\right),$$
where $\phi$ is the PDF and $\Phi$ is the CDF of the standard normal distribution.
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Best Answer
The max of two non-identical Normals can be expressed as an Azzalini skew-Normal distribution. See, for instance, a 2007 working paper/presentation by Balakrishnan
A recent paper by (Nadarajah and Kotz - viewable here) gives some properties of max$(X,Y)$:
For earlier work, see:
One can also use a computer algebra system to automate the calculation. For example, given $X \sim N(\mu_1, \sigma_1^2)$ with pdf $f(x)$, and $Y \sim N(\mu_2, \sigma_2^2)$ with pdf $g(y)$:
... the pdf of $Z = max(X,Y)$ is:
where I am using the
Maximumfunction from the mathStatica package of Mathematica, andErfdenotes the error function.