In some sensitive domains where data imperfections are present, standard classification techniques reach their limits. To avoid misclassification that has serious consequences, recent works propose cautious classification algorithms to handle the problem. Despite of the presence of uncertainty, a point prediction classifier is forced to decide which single class to associate to a sample. In such a case, a cautious classifier proposes the appropriate subset of candidate classes that can be assigned to the sample in the presence of imperfect information. On the other hand, cautiousness should not override relevance and a trade-off has to be made between these two criteria. Among the existing cautious classifiers, two classifiers propose to manage this trade-off in the decision step of the classifier algorithm by the mean of a parametrized objective function. The first one is the non-deterministic classifier (ndc) proposed within the framework of probability theory and the second one is eclair (evidential classifier based on imprecise relabelling) proposed within the framework of belief functions. The theoretical aim of the mentioned parameter is to control the size of predictions for both classifiers. This paper proposes to study this parameter in order to select the “best" value in a classification task. First the gain for each prediction candidate is studied related to the values of the hyper-parameter. In the illustration section, we propose a method to choose this hyper-parameter base on the training data and we show the classification results on randomly generated data and we present some comparisons with two other imprecise classifiers on 11 UCI datasets based on five measures of imprecise classification performances used in the state of the art.