Mathematical Statistics II · Rank and distribution-free methods
Let X₁,…,Xₙ be iid from an arbitrary continuous distribution
Problem
Let \(X_1,\ldots,X_n\) be iid from an arbitrary continuous distribution. Let \(R_i\) be the ascending rank of \(X_i\), with rank 1 assigned to the smallest observation. Show that \((R_1,\ldots,R_n)\) is uniform over the \(n!\) permutations of \((1,\ldots,n)\). Deduce the marginal distribution, mean, and variance of \(R_i\). Does that make the ranks independent? Show that a common strictly increasing transformation of the observations preserves all ranks. Finally, explain why exchangeability together with almost-sure absence of ties is enough for the uniform-permutation conclusion, even without independence.
Hint
List the possible strict orderings of the observation labels.
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