Comparing partial and truncated conglomerates from a concentration theoretic point of view

Abstract

When studying numerical properties of a population (technically: a conglomerate) it often happens that not all data are known. It might be that the total number of objects (persons) in the population is known, but that data on a number of them is missing. It even happens frequently that the total number of objects (N) is unknown. Referring to the population as ‘sources’ and to the property under investigation as ‘items’ or as ‘the production’, the whole dataset of this conglomerate can be represented as an N-vector. In this article N-vectors representing sources and their respective productions are studied from the point of view of concentration theory. Partial vectors (N is known, but data concerning the least productive sources are missing) and truncated vectors (N is unknown) are compared in two ways. First-order comparisons study vectors, while second-order comparisons study differences between vectors. In the case of first-order comparisons, it is shown that truncated vectors may be incomparable, while partial ones are always completely comparable. Similarly for second-order comparisons, partial vectors can be compared and yield a totally ordered double sequence, while truncated ones may be incomparable. Finally, we describe how to make second-order comparisons for vectors with a different number