Question

2. Show that W can be written as where U is the number of pairs (Xi, Yj) with X, < Y,. In other words n m U=ΣΣ1," where ,j -(0 otherwise. i=1 j=1 Hint: Let Yi),Ya),... , Ym) be the order statistics for the y-sample. Then U is the number of pairs (Xi,Yu)) with Xi 〈 YG). For fixed j , the number of Xi with Xi 〈 Yu) is just the rank of Y (j) minus the number of Y k with Y k 〈Y(j), that is Rj-3-Now sum over j.]

Answer 1

$Let~X_1,X_2,...,X_n~be~a~random~sample~drawn~from~Population~with~cdf~F_X\\ Y_1,Y_2,...,Y_m~be~another~random~sample~from~the~population~with~continuous~cdf~F_Y.\\ We~arrange~n~X's~and~m~Y's~in~increasing~order.~Let~N=n&plus;m. \\Define~Z_i=1~if~the~ith~random~variable~in~the~combined~ordered~sample~is~an~Y\\ =0~if~it~is~a~X,~i=1(1)N\\ Then~W=\sum_{i=1}^NiZ_i\\ Let~Y_{(1)},~Y_{(2)},...,Y_{(m)}~be~the~order~statistics~for~the~Y-sample.~For~fixed~j,~the~number~of\\ X_i~with~X_i<Y_{(j)}\\ =rank~of~Y_{(j)}~in~combined~sample-the~number~of~Y_k\leq Y_{(j)}\\ =jZ_j-j\\ Then,~U=\sum_{j=1}^mno.~of~(X_1,...,X_n<Y_{(j)})=\sum_{j=1}^m(jZ_j-j)=W-\frac{m(m&plus;1)}{2}\\ Hence~W=U&plus;\frac{m(m&plus;1)}{2}.$