Unforced errors in the matching problem

The solution of the Montmort's matching problem can be seen as the probability mass function of the number of matches in an experiment for assessing the agreement between nominal variables and gold standard classifications. [Vidal I, de Castro M. A Bayesian analysis of the matching problem. J Stat Plan Inference. 2021;212:194–200] presented a generalization of the Montmort's matching problem by considering the chronological order in what assignments are made and counting the number of unforced errors additionally to the number of matches. These authors carried out a Bayesian analysis of the problem, but in this paper we found a solution of this new approach from a frequentist point of view. We found the bivariate probability mass function of the number of matches and the number of unforced errors. The marginal distribution of the number of unforced errors is computed and expressed in terms of the Stirling numbers of the second kind. Also, some elementary properties were proven, including that the distribution of the unforced errors is equal to that of forced errors. As a practical consequence, we propose a new experiment and a new way of collecting and analysing the data in order to assess the agreement between nominal variables and gold standard classifications. Two real data sets were analysed using the proposed methodology.