Abstract
In many society and natural science fields, some stochastic orders have been established in the literature to compare the variability of two random variables. For a stochastic order, if an individual (or a unit) has some property, sometimes we need to infer that the population (or a system) also has the same property. Then, we say this order has closed property. Reversely, we say this order has reversed closure. This kind of symmetry or anti-symmetry is constructive to uncertainty management. In this paper, we obtain a quantile version of DCPE, termed as the dynamic cumulative past quantile entropy (DCPQE). On the basis of the DCPQE function, we introduce two new nonparametric classes of life distributions and a new stochastic order, the dynamic cumulative past quantile entropy (DCPQE) order. Some characterization results of the new order are investigated, some closure and reversed closure properties of the DCPQE order are obtained. As applications of one of the main results, we also deal with the preservation of the DCPQE order in several stochastic models.
Highlights
In recent years, the concepts of entropies and the various kinds of stochastic order relations have played important roles in the risk management, but have displayed more and more functions in all kinds of social sciences and natural sciences, such as management, physics, astronomy, geography and statistics
The highlights of our research are: (1) We define a new entropy, that is the dynamic cumulative past quantile entropy (DCPQE); (2) We introduce two new nonparametric classes of life distributions, that is decreasing dynamic cumulative past quantile entropy (DDCPQE) and increasing dynamic cumulative past quantile entropy (IDCPQE); (3) We define the dynamic cumulative past quantile entropy (DCPQE) order; (4) We deal with the preservation of the DCPQE order in proportional odds model and record values model; (5) We summarize the research results of this article, and obtained 12 results concerning anti-symmetry
We explore some characterizations of the DCPQE ordering
Summary
The concepts of entropies and the various kinds of stochastic order relations have played important roles in the risk management, but have displayed more and more functions in all kinds of social sciences and natural sciences, such as management, physics, astronomy, geography and statistics. Based on the monotonicity of the PQE function, they defined two nonparametric classes of life distributions, the decreasing (increasing) past quantile entropy (DPQE (IPQE)) classes. Navarro et al [17] thoroughly investigated the dynamic cumulative residual entropy and, using a similar manner to the definition of DCRE, they defined the dynamic cumulative past entropy (DCPE) of X as the CRE of X(t) , and denoted by E X (t). Making use of the DCPQE functions, we define a new stochastic order, termed the DCPQE order This new stochastic order compares the uncertainties of two nonnegative random variables X and Y at the respective age points FX−1 ( p) and GY−1 ( p); here, GY−1 ( p) is the quantile function of Y with distribution function GY (·). Assume that all integrals involved are finite, and ratios are well defined whenever written
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