4. (a) Non-parametric statistics tries to make minimal assumption on the underlying dis- tributions. Let X...

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4. (a) Non-parametric statistics tries to make minimal assumption on the underlying dis- tributions. Let X ~ F. Do we need to make any assumptions (e.g. normality) on F to study the distribution of F(X)? If yes, state the assumption(s) and result. If no, state the reason. 1 (b) For a sample of n observations {X, , Xn}, let {R,, Rn} be its rank statistics and F(x) be its empirical distribution function. Show that Ri n(X) = n for i = 1,..., n. (You may assume that there is no tie.) (c) In view of (a) and (b), comment on the advantage(s) of using the ranks as a statistical tool. 4. (a) Non-parametric statistics tries to make minimal assumption on the underlying dis- tributions. Let X ~ F. Do we need to make any assumptions (e.g. normality) on F to study the distribution of F(X)? If yes, state the assumption(s) and result. If no, state the reason. 1 (b) For a sample of n observations {X, , Xn}, let {R,, Rn} be its rank statistics and F(x) be its empirical distribution function. Show that Ri n(X) = n for i = 1,..., n. (You may assume that there is no tie.) (c) In view of (a) and (b), comment on the advantage(s) of using the ranks as a statistical tool.