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Let (X1, Y,),...,(X,, Y,,) be independent and identically distributed pairs of continuous random variables. (a) Show that if (X,, Y,) have a linear relation Y, = aX, +b with constant a > 0 and be (-00, 00), i= 1,...,n, then the Kendall correlation coefficient r =1. Is the converse also true (i.e., r =1 implies Y, = aX, + b for some constant a > 0 and be (-00,00), i= 1,...,n )? Explain why. [4] (b) Let D, = R; - S; , i=1,...,n, where (R1,..., R,, ) denote the ranks of (X,,...,X, ) and (S,,...,S,, ) the ranks of (Y1,..., Y,, ). Assume no ties in the ranks. Prove that 12 ER,S, - 3- n directly (not using equations (7.47) and (7.48) in Lecture Notes). [4] (c) The following data are observed from (X], Y,),...,(X,,, Y,, ) with n =12 and have been rearranged in increasing order of (X] ,..., X,,) : 5 7 11 15 16 19 25 30 34 42 58 12 5 9 33 26 12 64 18 42 78 47 56 Test the null hypothesis Ho of independent (X,, Y, ) against alternative H, : 7 > 0 at appropriate level of significance based on the Kendall statistic K using the normal approximation. [6] (d) Find an approximate 95% confidence interval of the Kendall correlation coefficient r based on the data in part (b) (formula (7.44) in Lecture Nots can still be used). [6] (e) Let r represent the correlation coefficient of (X,, Y,) . Test the null hypothesis Ho of independent (X;, Y,) against H, : r > 0 at appropriate level of significance based on the Spearman rank correlation coefficient r, and the data in part (b) above using the normal approximation. [5]