Maximum entropy distribution of order statistics with given marginals

Abstract

Abstract. We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

Geometrical Information Flow Regulated by Time Lengths: An Initial Approach

“…Though, the frequency aspect of consuming water by population was not considered in an exact sense (real time for each trial), but an ideal model reflecting the binary mode system of Bernoulli's method of analysis. Following this way, it is possible to indicate possible sequences of interactions between variables more than the set of intensities in which the variables express themselves [8,17], trying to see if organization of elements in the event through time assumes more relevant outputs regarding expenditure oscillations, than only considering the quantitative aspects of how much water each individual consumed. This approach is not suitable for Granger causality techniques due to particularities of variables and samples range of variance that can reach a broad output and heteroscedasticity [9,11,18] (Figure 2-red data).…”

Fast adaptive estimation of log-additive exponential models in Kullback-Leibler divergence

“…We show that by taking into consideration that f 0 has a product form, we can recover the onedimensional convergence rate for the density estimation, allowing for fast convergence of the estimator even if d is large. The quality of the estimators is measured by the Kullback-Leibler distance, as it has strong connections to the maximum entropy framework of [10].…”

“…f (x) = exp( 0 1 (x 1 ) + 0 2 (x 2 ) − a 0 )1 (x), with 0 i defined as 0 i = log(p i ) − I log(p i ) for i ∈ {1, 2}. According to Corollary 5.7. of [10], f is the density of the maximum entropy distribution of order statistics with marginals f 1 and f 2 given by:…”

“…More generally, in [10], the authors give a necessary and sufficient condition for the existence of a maximum entropy distribution of order statistics with fixed marginal cumulative distribution functions F i , 1 ≤ i ≤ d. See [9] for motivations for this problem. Moreover, its explicit expression is given as a function of the marginal distributions.…”

Geometrical Information Flow Regulated by Time Lengths

“…(Aven, 2016) Thus, it would be more accurate to analyze the variables of the system as static in the sense that approaches on the quantitative aspect of frequency with which the variables interact were not considered in an exact sense (real time for each trial), that is, it is objectively analyzed by variables that reflect the binary mode system as the Bernoulli method of analysis with memory prior to the start of event and constant time length (Lan et al 2012), so that it is possible to indicate possible sequences of interactions between variables more than the set of intensities in which the variables express themselves. (Gibbins, 1987;Uzi Harush & Baruch Barzel, 2017) For all possible sequences of expression between variables in a system there are possible paths in which each variable assumes specific aspects in ordered repeated iterations (Butucea et al, 2015) when it is taken in observation the effect of time lengths within the event, which is observable to the manager in a theoretical and static way, leading him to decide which ways to opt for intervention effects on management risks regarding optimization for resources distribution or containment in a system (Tampieri, 2005). The analysis of this article brings not to trivial ways in calculating by the method of Bernoulli the probabilities of an event in occurrence, but the analysis of static parameters of information inside a system of constant and nonlinear binary sequences, being this last characteristic relative to the number of iterations of the system, since the binary trajectories do not express probabilistic modifications in the time, but the comparison between two binary systems with the same mathematical conditions in itself already reflect oscillations of systems of multivariate and intuitively stochastic mode.…”