Analytic Approximation to The Standard Error of Swap-Rate

When we select No students from N applicants on the basis of a composite score of subtests, it is important to evaluate the contribution of each subtest. The swap-rate, which is defined as the proportion of the applicants who actually pass the examination but would fail if the j-th subtest were not included in the component to rank the applicants, is one of the measures of the contribution of the j-th subtest.

In this article, first, we derive the characteristics and limiting properties of the population swap-rate. Next, using the properties of the order statistics and the extended hypergeometric distribution, we derive an approximation to the asymptotic variance of the sample swap-rate when the number of applicants is large. Finally, we propose the use of our analytic approximation to the variance of the sample swap-rate in the real data problem and show that it is very efficient.