On Re-Pairing Observations in a Broken Random Sample

Abstract

It is assumed that a random sample of size $n$ is drawn from a bivariate distribution $f(t, u)$ which possesses a monotone likelihood ratio (MLR). However, before the sample values are observed, the pairs are `broken' into components $t$ and $u$. Therefore, the original sample pairings are unknown, and it is desired to optimally re-pair $t$- and $u$-values in order to reconstruct the original bivariate sample. It is observed that for the maximum likelihood pairing (MLP) to be the `natural' pairing for all $t$- and $u$-values, it is necessary that $f$ has MLR. It is shown that if it is desired to maximize the expected number of correct matches, then the class of procedures $\Phi_{1, n}$, which result in pairing the largest $t$ with the largest $u$ and the smallest $t$ with the smallest $u$, is a complete class. A sufficient condition under which the MLP maximizes the expected number of correct matches is also obtained.