ON THE AVERAGE DIFFERENCE BETWEEN CONCOMITANTS AND ORDER-STATISTICS

For a sequence of bivariate pairs (X(i), Y(i)), the concomitant Y[i] of the ith largest x-value X(i) equals that value of Y paired with X(i). In assessing the quality of a file-merging or file-matching procedure, the penalty for incorrect matching may often be expressed as the average value of a function of the difference Y[i] - Y(i). We establish strong laws and central limit theorems for such quantities. Our proof is based on the observation that if G(x)(.) denotes the distribution function of Y given X = x, then G(X)(Y) is stochastically independent of X, even though G(x)(.) depends numerically on x.