Calculating in R The Expected Maximum of a Gaussian Sample using Order Statistics

In generating a sample of n datapoints drawn from a normal/Gaussian distribution, how big on average the biggest datapoint is will depend on how large n is. I implement in the R programming language & compare some of the approaches to estimate how big on average.
topics: statistics, computer science, R, bibliography
created: 22 Jan 2016; modified: 19 May 2019; status: finished; confidence: highly likely; importance: 5

In generating a sample of n datapoints drawn from a normal/Gaussian distribution with a particular mean/SD, how big on average the biggest datapoint is will depend on how large n is. Knowing this average is useful in a number of areas like sports or breeding or manufacturing, as it defines how bad/good the worst/best datapoint will be (eg the score of the winner in a multi-player game).

The order statistic of the mean/average/expectation of the maximum of a draw of n samples from a normal distribution has no exact formula, unfortunately, and is generally not built into any programming language’s libraries.

I implement & compare some of the approaches to estimating this order statistic in the R programming language, for both the maximum and the general order statistic. The overall best approach is to calculate the exact order statistics for the n range of interest using numerical integration via lmomco and cache them in a lookup table, rescaling the mean/SD as necessary for arbitrary normal distributions; next best is a polynomial regression approximation; finally, the Elfving correction to the Blom 1958 approximation is fast, easily implemented, and accurate for reasonably large n such as n>100.

Visualizing maxima/minima in order statistics with increasing n in each sample (1-100).
Visualizing maxima/minima in order statistics with increasing n in each sample (1-100).


Monte Carlo

Most simply and directly, we can estimate it using a Monte Carlo simulation with hundreds of thousands of iterations:

But in R this can take seconds for small n and gets worse as n increases into the hundreds as we need to calculate over increasingly large samples of random normals (so one could consider this ); this makes use of the simulation difficult when nested in higher-level procedures such as anything involving resampling or simulation. In R, calling functions many times is slower than being able to call a function once in a ‘vectorized’ way where all the values can be processed in a single batch. We can try to vectorize this simulation by generating random normals, group it into a large matrix with n columns (each row being one n-sized batch of samples), then computing the maximum of the i rows, and the mean of the maximums. This is about twice as fast for small n; implementing using rowMaxs from the R package matrixStats, it is anywhere from 25% to 500% faster (at the expense of likely much higher memory usage, as the R interpreter is unlikely to apply any optimizations such as Haskell’s stream fusion):

Each simulate is too small to be worth parallelizing, but there are so many iterations that they can be split up usefully and run with a fraction in a different process; something like

We can treat the simulation estimates as exact and use memoization such as provided by the R package memoise to cache results & never recompute them, but it will still be slow on the first calculation. So it would be good to have either an exact algorithm or an accurate approximation: for one of analyses, I want accuracy to ±0.0006 SDs, which requires large Monte Carlo samples.

Upper bounds

To summarize the Cross Validated discussion: the simplest upper bound is , which makes the diminishing returns clear. Implementation:

Most of the approximations are sufficiently fast as they are effectively with small constant factors (if we ignore that functions like /qnorm themselves may technically be or for large n). However, accuracy becomes the problem: this upper bound is hopelessly inaccurate in small samples when we compare to the Monte Carlo simulation. Others (also inaccurate) include and :


Blom 1958, Statistical estimates and transformed beta-variables provides a general approximation formula , which specializing to the max () is and is better than the upper bounds:

Elfving 1947, apparently, by way of Mathematical Statistics, Wilks 1962, demonstrates that Blom 1958’s approximation is imperfect because actually , so:

(Blom 1958 appears to be more accurate for n<48 and then Elfving’s correction dominates.)

Harter 1961 elaborated this by giving different values for , and Royston 1982 provides computer algorithms; I have not attempted to provide an R implementation of these.

probabilityislogic offers a 2015 derivation via the beta-F compound distribution1: and an approximate (but highly accurate) numerical integration as well:

The integration can be done more directly as

Another approximation comes from Chen & Tyler 1999: . Unfortunately, while accurate enough for most purposes, it is still off by as much as 1 IQ point and has an average mean error of -0.32 IQ points compared to the simulation:

Error in using the Chen & Tyler 1999 approximation to estimate the expected value (gain) from taking the maximum of n normal samples
Error in using the Chen & Tyler 1999 approximation to estimate the expected value (gain) from taking the maximum of n normal samples

Polynomial regression

From a less mathematical perspective, any regression or machine learning model could be used to try to develop a cheap but highly accurate approximation by simply predicting the extreme from the relevant range of n=2-300 - the goal being less to make good predictions out of sample than to overfit as much as possible in-sample.

Plotting the extremes, they form a smooth almost logarithmic curve:

This has the merit of utter simplicity (function(n) {0.658802439 + 0.395762956*log(n)}), but while the R2 is quite high by most standards, the residuals are too large to make a good approximation - the log curve overshoots initially, then undershoots, then overshoots. We can try to find a better log curve by using polynomial or spline regression, which broaden the family of possible curves. A 4th-order polynomial turns out to fit as beautifully as we could wish, R2=0.9999998:

This has the virtue of speed & simplicity (a few arithmetic operations) and high accuracy, but is not intended to perform well past the largest datapoint of n=300 (although if one needed to, one could simply generate the additional datapoints, and refit, adding more polynomials if necessary), but turns out to be a good approximation up to n=800 (after which it consistently overestimates by ~0.01):

So this method, while lacking any kind of mathematical pedigree or derivation, provides the best approximation so far.


The R package lmomco (Asquith 2011) calculates a wide variety of order statistics using numerical integration & other methods. It is fast, unbiased, and generally correct (for small values of n2) - it is close to the Monte Carlo estimates even for the smallest n where the approximations tend to do badly, so it does what it claims to and provides what we want (a fast exact estimate of the mean gain from selecting the maximum from n samples from a normal distribution). The results can be memoized for a further moderate speedup (eg calculated over n=1-1000, 0.45s vs 3.9s for a speedup of ~8.7x):

Error in using Asquith 2011’s L-moment Statistics numerical integration package to estimate the expected value (gain) from taking the maximum of n normal samples
Error in using Asquith 2011’s L-moment Statistics numerical integration package to estimate the expected value (gain) from taking the maximum of n normal samples


With lmomco providing exact values, we can visually compare the presented methods for accuracy; there are considerable differences but the best methods are in close agreement:

Comparison of estimates of the maximum for n=2-300 for 12 methods, showing Chen 1999/polynomial approximation/Monte Carlo/lmomco are the most accurate and Blom 1958/upper bounds highly-inaccurate.
Comparison of estimates of the maximum for n=2-300 for 12 methods, showing Chen 1999/polynomial approximation/Monte Carlo/lmomco are the most accurate and Blom 1958/upper bounds highly-inaccurate.

And micro-benchmarking them quickly (excluding Monte Carlo) to get an idea of time consumption shows the expected results (aside from Pil 2015’s numerical integration taking longer than expected, suggesting either memoising or changing the fineness):

Rescaling for generality

The memoised function has three arguments, so memoising on the fly would seem to be the best one could do, since one cannot precompute all possible combinations of the n/mean/SD. But actually, we only need to compute the results for various n!

We can default to assuming the standard normal distribution () without loss of generality because it’s easy to rescale any normal to another normal: to scale to a different mean , one simply adds to the expected extreme, so one can assume ; and to scale to a different standard deviation, we simply multiply appropriately. So if we wanted the extreme for n=5 for , we can calculate it simply by taking the estimate for n=5 for and multiplying by and then adding :

So in other words, it is unnecessary to memoize all possible combinations of n, mean, and SD - in reality, we only need to compute each n and then rescale it as necessary for each caller. And in practice, we only care about n=2-200, which is few enough that we can define a lookup table using the lmomco results and use that instead (with a fallback to lmomco for , and a fallback to Chen et al 1999 for to work around lmomco’s bug with large n):

exactMax <- function (n, mean=0, sd=1) {
if (n>2000) {
    chen1999 <- function(n,mean=0,sd=1){ mean + qnorm(0.5264^(1/n), sd=sd) }
    chen1999(n,mean=mean,sd=sd) } else {
    if(n>200) { library(lmomco)
        exactMax_unmemoized <- function(n, mean=0, sd=1) {
            expect.max.ostat(n, para=vec2par(c(mean, sd), type="nor"), cdf=cdfnor, pdf=pdfnor) }
        exactMax_unmemoized(n,mean=mean,sd=sd) } else {

 lookup <- c(0,0,0.5641895835,0.8462843753,1.0293753730,1.1629644736,1.2672063606,1.3521783756,1.4236003060,

 return(mean + sd*lookup[n+1]) }}}

This gives us exact computation at (with an amortized when ) with an extremely small constant factor (a conditional, vector index, multiplication, and addition, which is overall ~10x faster than a memoised lookup), giving us all our desiderata simultaneously & resolving the problem.

General order statistics for the normal distribution

One might also be interested in computing the general order statistic.

Some available implementations in R:

See also

  1. Exploiting the beta transformation, where the order statistics of a simple 0-1 interval turns out to follow a beta distribution, which can then be easily transformed into the equivalent order statistics of more useful distributions like the normal distribution. The beta transformation is not just computationally useful, but critical to order statistics in general.↩︎

  2. lmomco is accurate for all values I checked with Monte Carlo n<1000, but appears to have some bugs n>2000: there are occasional deviations from the quasi-logarithmic curve, such as n=2225-2236 (which are off by 1.02SD compared to the Monte Carlo estimates and the surrounding lmomco estimates), another cluster of errors n~=40,000, and then after n>60,000, the estimates are totally incorrect. The maintainer has been notified & verified the bug.↩︎

  3. A previous version of EnvStats described the approximation thus:

    The function evNormOrdStatsScalar computes the value of for user-specified values of r and n. The function evNormOrdStats computes the values of for all values of r for a user-specified value of n. For large values of n, the function evNormOrdStats with approximate=FALSE may take a long time to execute. When approximate=TRUE, evNormOrdStats and evNormOrdStatsScalar use the following approximation to , which was proposed by Blom (1958, pp. 68-75, [“6.9 An approximate mean value formula” & formula 6.10.3-6.10.5]):

    This approximation is quite accurate. For example, for , the approximation is accurate to the first decimal place, and for it is accurate to the second decimal place.