Order Statistics and Sample Quantiles

Every risk report that quotes a 95% Value-at-Risk, a median daily return, or an interquartile range is quoting a function of the sorted data — and everything derived so far in this unit ($\bar X$ and $S^2$ from Populations, Samples, and Sampling Distributions, the $t$-statistic) assumed normality to get an exact sampling distribution. Order statistics give up that crutch: the theory in this note applies to any continuous population, at the cost of messier-looking (but fully explicit) densities. This note derives the density of a single order statistic, the joint density of the extremes, and uses both to characterise the sampling behaviour of the sample median and the interquartile range (IQR) — the two most commonly quoted "robust" statistics in a risk report, precisely because, unlike $\bar X$ and $S^2$, they barely react to a handful of extreme outliers.

Learning objectives. By the end of this note you should be able to: