# Distribution of percentile rank of largest value in sample

Let's imagine that I sample 100 values from some probability distribution $$Distribution$$ over the real numbers. Out of these samples, I pick the maximum value $$m$$.

It seems intuitive (and apparently confirmed by some simulations) that $$m$$ will be "around" the 99th percentile of values of $$Distribution$$.

Is there some standard probability distribution that describes this? i.e. some well-characterized distribution for the percentile rank of the maximum value among $$N$$ samples from an arbitrary $$Distribution$$?

• Generalized extreme value distribution. stats.stackexchange.com/questions/158767/… Commented 2 days ago
• The study of extreme values reveals what should be intuitively obvious: there is no "standard probability distribution." When the extreme is suitably standardized and the upper tail of the underlying distribution is eventually continuous, there are three possible limiting distributions. For an example of the complexities, consider a bounded discrete distribution such as the Binomial.
– whuber
Commented 2 days ago

The $$k$$-th smallest value of a sample of size $$n$$ from a distribution is its $$k$$-th order statistic. This term should already be helpful in searching for results. We also have an tag. Specifically, the maximum is the $$n$$-th order statistic.
The distributions of the order statistics of certain sampled distributions are known, so you can compare the expected values of the order statistic distributions to the quantiles of the underlying distribution. As an example, the $$k$$-th order statistic from a $$U[0,1]$$ distribution is $$\text{Beta}(k,n+1-k)$$ distributed, so the $$n$$-th one has a $$\text{Beta(n,1)}$$ distribution. Its expectation is $$\frac{n}{n+1}$$, quite close to the $$(n-1)$$-th quantile at $$\frac{n-1}{n}$$.