It has been known for a long time that for bootstrapping the distribution of the extremes under the traditional linear normalization of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. In this paper, we show that the same is true if we use the bootstrap for estimating a central, or an intermediate quantile under power normalization. A simulation study illustrates and corroborates theoretical results.
The full consistency of Saaty's matrix of preference intensities used in AHP is practically unachievable for a large number of objects being compared. There are many procedures and methods published in the literature that describe how to assess whether Saaty's matrix is "consistent enough". Consistency is in these cases measured for an already defined matrix (i.e. ex-post). In this paper we present a procedure that guarantees that an acceptable level of consistency of expert information concerning preferences will be achieved. The proposed method is based on dividing the process of inputting Saaty's matrix into two steps. First, the ordering of the compared objects with respect to their significance is determined using the pairwise comparison method. Second, the intensities of preferences are defined for the objects numbered in accordance with their ordering (resulting from the first step). In this paper the weak consistency of Saaty's matrix is defined, which is easy to check during the process of inputting the preference intensities. Several propositions concerning the properties of weakly consistent Saaty's matrices are proven in the paper. We show on an example that the weak consistency, which represents a very natural requirement on Saaty's matrix of preference intensities, is not achieved for some matrices, which are considered "consistent enough" according to the criteria published in the literature. The proposed method of setting Saaty's matrix of preference intensities was used in the model for determining scores for particular categories of artistic production, which is an integral part of the Registry of Artistic Results (RUV) currently being developed in the Czech Republic. The Registry contains data on works of art originating from creative activities of Czech art colleges and faculties. Based on the total scores achieved by these institutions, a part of the state budget subsidy is being allocated among them.