Order Statistics: Min, Max, and Range for Actuarial Applications
Study order statistics and their role in actuarial modeling for Exam P.
Definition and Basic Formulas
Given a random sample X1, X2, ..., Xn from a continuous distribution with CDF F and PDF f, the order statistics X_(1) <= X_(2) <= ... <= X_(n) are the sorted values. X_(1) is the sample minimum and X_(n) is the sample maximum.
The CDF of X_(k) is F_{X_(k)}(x) = sum from j=k to n of C(n,j) * [F(x)]^j * [1-F(x)]^(n-j). For the minimum: F_{X_(1)}(x) = 1 - [1 - F(x)]^n. For the maximum: F_{X_(n)}(x) = [F(x)]^n.
PDFs of Extremes
The PDF of the minimum is f_{X_(1)}(x) = n * [1 - F(x)]^(n-1) * f(x). The PDF of the maximum is f_{X_(n)}(x) = n * [F(x)]^(n-1) * f(x). For the general k-th order statistic, f_{X_(k)}(x) = n! / ((k-1)! * (n-k)!) * [F(x)]^(k-1) * [1-F(x)]^(n-k) * f(x).
A useful fact: if X1, ..., Xn are independent Exponential(lambda), then X_(1) ~ Exponential(n*lambda). This follows from the minimum formula and is frequently tested on Exam P.
Actuarial Applications
Order statistics model extreme events in insurance: the largest claim in a portfolio (X_(n)), the smallest loss, or the range X_(n) - X_(1). For uniform order statistics on (0, 1), E[X_(k)] = k/(n+1) and Var(X_(k)) = k(n-k+1)/((n+1)^2(n+2)). These formulas appear in problems about quantile estimation and loss ranking. Practice computing E[X_(1)] and E[X_(n)] for common distributions.