# $\mathbb{E}$ and Variance of the maximum of independent $\mathcal{N}(\mu_i, \sigma_i^2)$

I am interested in the expectation and the variance of the maximum of several independent, normal distributed variances. That is, given a set of $I$ different RVs with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, I want to find $$\mathbb{E}[\max~X_i], \\ \text{Var}[\max~X_i].$$

I have found Ross' "Computing Bounds on the Expected Maximum of Correlated Normal Variables", but the method there given requires a numerical integration. I am interested in a closed form and would prefer a closed form approximation over an exact iterative method.

Anyone can point me into the right direction?

• Related: Covariance of INID order statistics. You won't find a closed form in general, even for the expectation. I'm curious why you would find a closed form approximation preferable to an iterative method that could be more accurate: your response might help guide answers to this question.
– whuber
Nov 20 '13 at 13:34
• @Brendon Independent, the article happens to have information on that as well. Nov 20 '13 at 13:35
• @whuber I prefer a closed form so that it can be used as part of an optimization, for which I need a derivative. Thus, what I really mean is that the solution needs to be in a differentiable form. Nov 20 '13 at 13:36
• Are you sure you need a derivative for your optimization? There are several derivative-free methods (ones that don't substitute a finite difference approximation) available, Nelder-Mead being the most widely known for multidimensional problems. Nov 20 '13 at 13:47
• I haven't had time to look at it in detail, but the paper On the Maximum of Bivariate Normal Random Variables, by Alan P. Ker, looks like it might be a useful starting point. Nov 20 '13 at 14:18

You can find a closed form (well, if you accept to use special functions) for the density of $X = \max (X_1, \dots, X_n)$.

Let $F_i, f_i$ be the cdf and the density of $X_i$, for $i=1, ..., n$. The cdf of $X$ is

\begin{aligned} F(x) &= \mathbb P(X \le x) \\ &= \mathbb P(X_1 \le x, \dots, X_n \le x) \\ &= \mathbb P(X_1 \le x) \cdots \mathbb P( X_n \le x) \\ &= F_1(x) \cdots F_n(x). \end{aligned}

Its density is then obtained by derivation: $$f(x) = F(x)\left( {f_1 \over F_1} + \cdots + {f_n \over F_n} \right).$$

Using this, you can find expected value and variance to a good accuracy and reasonable computing time with numerical integration procedures (cf integrate in R).

I bet that in this case, you can permute integral and derivation with respect to the parameter, so you can obtain the derivatives in a similar way.

• Thanks, but I need a closed form; no numerical integration. Nov 21 '13 at 10:07
• The difference is thin. Many 'special functions' are computed through iterative procedures, even if you don’t see it. I understand that you want to avoid doing lots of iteration so that you get something fast enough. In that case, just write the numerical integration as a sum, using, say, 10 trapezoids. This will produce a closed form approximation as requested. Nov 23 '13 at 5:54