Calculate order statistics of a sample.

A claims auditor pulls 50 settled losses and needs the 90th-percentile severity to set a large-loss threshold. That number is an order statistic, and its sampling distribution is what tells you whether the threshold is stable or noisy.

Definition. Take an iid sample $X_1,\dots ,X_n$ from a continuous distribution with cdf $F$ and pdf $f$. Sort the observations to get the order statistics $X_{(1)}\le X_{(2)}\le \cdots \le X_{(n)}$. The index in parentheses is rank, not the original subscript. Ties have probability zero in the continuous case, so the ordering is strict almost surely.

Extremes are easy. The maximum is below $x$ only if every observation is below $x$. Independence gives a clean product.

$$F_{(n)}(x)=[F(x){]}^n,F_{(1)}(x)=1-[1-F(x){]}^n$$

Differentiate to get densities: $f_{(n)}(x)=n[F(x){]}^{n-1}f(x)$ and $f_{(1)}(x)=n[1-F(x){]}^{n-1}f(x)$.

KEY: For exponential samples, $X_{(1)}\sim \text{Exponential}(n\lambda )$. The minimum of $n$ iid exponentials has rate $n\lambda$, mean $1/(n\lambda )$. This is the most-tested extreme-value identity.

Common mistakes

• Using $k/n$ as the plotting position. The correct convention is $k/(n+1)$; the alternative pushes the max to probability 1 and breaks Q-Q alignment.
• Forgetting to include $f(x)$ in the density formula. The factor $F^{k-1}(1-F{)}^{n-k}$ without $f$ is a probability weight, not a density. The full density requires the additional $f(x)$.
• Confusing the minimum's rate. For iid exponentials at rate $\lambda$, the minimum has rate $n\lambda$ (mean $1/(n\lambda )$), not $\lambda /n$ and not $\lambda$ itself.

Bottom line

• Order statistics $X_{(1)}\le X_{(2)}\le \cdots \le X_{(n)}$ are the sorted sample; $X_{(1)}$ is the minimum, $X_{(n)}$ the maximum.
• CDF method: $F_{(k)}(x)={\sum }_{j=k}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)F(x{)}^j[1-F(x){]}^{n-j}$; the max has CDF $[F(x){]}^n$ and the min has CDF $1-[1-F(x){]}^n$.
• Density: $f_{(k)}(x)=\frac{n!}{(k-1)!(n-k)!}F(x{)}^{k-1}[1-F(x){]}^{n-k}f(x)$; the trailing $f(x)$ is mandatory.
• Uniform(0,1): $U_{(k)}\sim \text{Beta}(k,n-k+1)$ with mean $k/(n+1)$; the probability-integral transform extends this to any continuous $F$.

Exam shortcut

If the problem asks about the minimum or maximum directly, skip the binomial sum and write down $[F(x){]}^n$ or $1-[1-F(x){]}^n$; differentiate only if you need the density. For interior order statistics of a uniform sample, recognize the beta immediately and pull moments from $k/(n+1)$ and $k(n-k+1)/[(n+1{)}^2(n+2)]$ rather than integrating.

The full lesson (about 1,642 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B7

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