RANK INVERSIONS IN SCORING MULTIPART EXAMINATIONS
成果类型：
Article
署名作者：
Berman, Simeon M.
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
1996
页码：
992-1005
关键词：
摘要：
Let (Xi, Y-i), i = 1, . . . , n, be independent random vectors with a standard bivariate normal distribution and let s(X) and s(Y) be the sample standard deviations. For arbitrary p, 0 < p < 1, define T-i = pX(i) + (1-p)Y-i and Z(i) = pX(i)/s(X) + (1-p)Y-i/s(Y,) i = 1, . . . ,n. The couple of pairs (T-i, Z(i)) and (T-j, Z(j)) is said to be discordant if either T-i < T-j and Z(i) > Z(j) or T-i > T-j and Z(i) < Z(j). It is shown that the expected number of discordant couples of pairs is asymptotically equal to n(3/2) times an explicit constant depending on p and the correlation coefficient of X-i and Y-i. By an application of the Durbin-Stuart inequality,this implies an asymptotic lower bound on the expected value of the sum of (rank(Z(i) ) - rank(T-i))(+). The problem arose in a court challenge to a standard procedure for the scoring of multipart written civil service examinations. Here the sum of the positive rank differences represents a measure of the unfairness of the method of scoring.