Optimal level sets for bivariate density representation

Abstract

In bivariate density representation there is an extensive literature on level set estimation when the level is fixed, but this is not so much the case when choosing which level is (or which levels are) of most interest. This is an important practical question which depends on the kind of problem one has to deal with as well as the kind of feature one wishes to highlight in the density, the answer to which requires both the definition of what the optimal level is and the construction of a method for finding it. We consider two scenarios for this problem. The first one corresponds to situations in which one has just a single density function to be represented. However, as a result of the technical progress in data collecting, problems are emerging in which one has to deal with a sample of densities. In these situations, the need arises to develop joint representation for all these densities, and this is the second scenario considered in this paper. For each case, we provide consistency results for the estimated levels and present wide Monte Carlo simulated experiments illustrating the interest and feasibility of the proposed method.

Choosing the most relevant level sets for depicting a sample of densities

“…These graphics show that these six densities are different in location (compare Madrid and Barcelona, for instance), dispersion (Valencia is more concentrated than Madrid, for instance) and shape (Sevilla is quite different from the rest, but also Barcelona, Málaga and Valencia are quite different from Madrid and Zaragoza). These differences where less clear in the upper panel of Figure 6 (here the main findings are the differences in location and the different shape of Sevilla), where α = 0.5 is used, according to the quick rule proposed in [6]. Other procedures proposed in that paper (namely, those based on ranks of distances) lead to choose a probability content of at least 0.9, in agreement with what we obtain here with the additive model (3.1).…”

“…As a matter of illustration, Figure 6 shows the density level sets corresponding to the 6 municipalities with the largest numbers of polling stations. In top plots probability content is α = 0.5 which is the recommended level set by the quick rule given in [6] when using only one level set, while in bottom plots one has α 1 = 0.25 and α 2 = 0.75 as recommended by the quick rule given in [6] when using two level sets.…”

“…In grey, the level sets corresponding to the whole country. Probability contents have been chosen according to the quick rule proposed in [6]. Upper panel: Raw data, the probability content is α = 0.50.…”

“…This simple example shows the importance of the choice of the level(s) when representing a bivariate density function. Data-driven ways for driving this choice for depicting a single density have been studied in the literature and various methods have been proposed (see [6] for general discussion).…”

“…Last but not least point to be highlighted, is the fact that the question of data-driven levels cannot be solved in a simple and naive way just by applying repeatedly existing methods for a single density representation, because in order to detect possible differences and/or similarities between some of the f i it is necessary to find probability contents α j which are common to the whole sample. As far as we know, this question of optimal level sets representation for a family of densities has not been widely studied in the literature (see however [6] for a first approach).…”