Characterizations of bivariate models using dynamic Kullback-Leibler discrimination measures

In this paper, the residual Kullback-Leibler discrimination information measure is extended to conditionally specified models. The extension is used to characterize some bivariate distributions. These distributions are also characterized in terms of proportional hazard rate models and weighted distributions. Moreover, we also obtain some bounds for this dynamic discrimination function by using the likelihood ratio order and some preceding results.     

Non-parametric estimation of the generalized past entropy function with censored dependent data

The generalized past entropy function introduced by Gupta and Nanda (2002) is viewed as a dynamic measure of uncertainty in past life. This measure finds applications in modeling past life time data. In the present work we provide non-parametric kernel-type estimator for the generalized past entropy function based on censored data. Asymptotic properties of the estimator are established under suitable regularity conditions. Simulation studies are carried out using the Monte Carlo method.     

Some characterization results on generalized cumulative residual entropy measure

The cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon entropy, refer to Rao et al. (2004). In this paper we study a generalized cumulative residual information measure based on Verma’s entropy function and a dynamic version of it. The exponential, Pareto and finite range distributions, which are commonly used in reliability modeling, have been characterized using this generalized measure.     

Characterizations based on length-biased weighted measure of inaccuracy for truncated random variables

In survival studies and life testing, the data are generally truncated. Recently, authors have studied a weighted version of Kerridge inaccuracy measure for truncated distributions. In the present paper we consider weighted residual and weighted past inaccuracy measure and study various aspects of their bounds. Characterizations of several important continuous distributions are provided based on weighted residual (past) inaccuracy measure.     

基于双边犹豫模糊非均衡语言优先集成算子的应急响应预案评估方法

在应用多属性决策理论求解应急响应预案评估问题时,问题结构的复杂性往往使决策者评价信息存在高度不确定性且属性相对重要性仅能以优先级关系来表征。为此,本文首先提出了双边犹豫模糊非均衡语言集这种新型信息形式以使决策者能够灵活有效的表征复杂评价信息,并定义了运算法则、熵和距离测度;其次,基于熵测度开发了双边犹豫模糊非均衡语言优先加权集成算子,并构建了能够考虑属性优先关系的多属性决策方法;进一步针对属性相对重要性不能由定性分析获得的情况,设计了客观权重确定方法,并构建了另一种更具实际灵活性的VIKOR决策方法;最后,实例研究表明了方法的有效性与优势。     

模糊熵与距离测度的相互诱导及其应用

模糊信息论就是利用模糊数学这一工具来研究带有模糊不确定性的信息的．模糊熵和距离测度是模糊信息论中两个重要的度量方法．本文主要讨论模糊熵和距离测度之间的相互关系,由此得到几个由模糊熵诱导的距离测度公式和几个由距离测度诱导出的模糊熵公式,说明了模糊熵和距离测度是可以相互诱导的．最后,举例说明距离测度公式在模式识别中的应用．     

信息熵度量风险的探究

本文分析了风险的本质后指出,风险是某一特定行为主体对某一金融投资中损失的不确定性和收益的不确定性的认识。在众多风险度量的方法中,熵函数法有着其独特的度量风险的优势,因此,在本文中重点讨论了熵函数作为风险度量的合理性。同时提出一个新的风险度量模型,剖析其主要的数学特性,阐明该模型可以针对不同行为主体能有效地度量金融风险,并且计算量小,易于操作。     

An extension of weighted generalized cumulative past measure of information

Ricerche di Matematica - In this paper, we consider a shift-dependent measure of generalized cumulative entropy and its dynamic (past) version in the case where the weight is a general non-negative...     

Accessibility in transportation networks probabilistic versus deterministic approach

Existing accessibility measures imply either a perfectly legible structure of the communication network, or a perfectly informed “visitor”. In the study of transportation networks (as well as in the study of large architectural complexes, for example) this implied assumption is extremely unrealistic. Markov chain is interpreted as a model of communication process, and an information-theoretic measure is derived for the measurement of accesibility in conditions of uncertainty of the visitor about the structure of the communication network. A number of elementary network patterns are compared using deterministic and probabilistic measures of accessibility. It is claimed that different patterns differ in the degree of susceptibility to uncertainty in the communication process.     

Combining the data from two normal populations to estimate the mean of one when their means difference is bounded

In this paper we address the problem of estimating θ₁ when , are observed and |θ₁−θ₂|?c for a known constant c. Clearly Y₂ contains information about θ₁. We show how the so-called weighted likelihood function may be used to generate a class of estimators that exploit that information. We discuss how the weights in the weighted likelihood may be selected to successfully trade bias for precision and thus use the information effectively. In particular, we consider adaptively weighted likelihood estimators where the weights are selected using the data. One approach selects such weights in accord with Akaike's entropy maximization criterion. We describe several estimators obtained in this way. However, the maximum likelihood estimator is investigated as a competitor to these estimators along with a Bayes estimator, a class of robust Bayes estimators and (when c is sufficiently small), a minimax estimator. Moreover we will assess their properties both numerically and theoretically. Finally, we will see how all of these estimators may be viewed as adaptively weighted likelihood estimators. In fact, an over-riding theme of the paper is that the adaptively weighted likelihood method provides a powerful extension of its classical counterpart.     

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman?s reverse Brunn–Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman?s deep technology of M-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke–Ruzsa inequalities from additive combinatorics.     

Reversed hazard function of uncertain lifetime

Reversed hazard function is widely applied in reliability analysis. This paper considers the human uncertainty in a system, and employs uncertain variable to model the lifetime of a component. Concepts of mean residual life and residual entropy are proposed to describe a failed system, and their relationships with the reversed hazard function are discussed. In addition, this paper provides some applications of reversed hazard function to the mean past lifetime and past entropy.     

Rényi’s residual entropy: A quantile approach

In the present paper, we introduce a quantile based Rényi’s entropy function and its residual version. We study certain properties and applications of the measure. Unlike the residual Rényi’s entropy function, the quantile version uniquely determines the distribution.     

Entropy and Decision

This paper deals with the irrelevance of entropy to decision. The entropy measure fails to discriminate between states and their significance, as is indicated by a study of a reconnaissance example for which the entropy measure has been suggested. If the reduction of uncertainty is an explicit objective, then it is important to examine the properties of uncertainty measures, and it is queried as to whether some of those uniquely characterizing entropy are relevant. Other measures have some of the properties. The paper then discusses the concept of entropy in hypothesis selection, and the conflict with maximum-likelihood methods, and in Markovian processes, wherein there is some conflict with implied steady-state behaviour.     

Multivariate dynamic information

This paper develops measures of information for multivariate distributions when their supports are truncated progressively. The focus is on the joint, marginal, and conditional entropies, and the mutual information for residual life distributions where the support is truncated at the current ages of the components of a system. The current ages of the components induce a joint dynamic into the residual life information measures. Our study of dynamic information measures includes several important bivariate and multivariate lifetime models. We derive entropy expressions for a few models, including Marshall-Olkin bivariate exponential. However, in general, study of the dynamics of residual information measures requires computational techniques or analytical results. A bivariate gamma example illustrates study of dynamic information via numerical integration. The analytical results facilitate studying other distributions. The results are on monotonicity of the residual entropy of a system and on transformations that preserve the monotonicity and the order of entropies between two systems. The results also include a new entropy characterization of the joint distribution of independent exponential random variables.     

Limit theorems for stochastic measures of the accuracy of density estimators

Stochastic measures of the distance between a density f and its estimate f_n have been used to compare the accuracy of density estimators in Monte Carlo trials. The practice in the past has been to select a measure largely on the basis of its ease of computation, using only heuristic arguments to explain the large sample behaviour of the measure. Steele [11] has shown that these arguments can lead to incorrect conclusions. In the present paper we obtain limit theorems for the stochastic processes derived from stochastic measures, thereby explaining the large sample behaviour of the measures.     

Extension of quantum information by using entropy of deterministic functions

The concept of entropy of random variables first defined by Shannon has been generalized later in various ways by mathematicians who so obtained new measures of uncertainty, again for random variables. Recently, the author suggested another extension which provides a meaningful definition for the entropy of deterministic functions, both in the sense of Shannon and of Renyi. These measures of uncertainty are different from those which are utilized by physicists in the study of chaotic dynamics, like the Kolmogorov entropy for instance.

The aim of this paper is to go a step further, and to derive measures of uncertainty for operators, by using exactly the same rationale. After a short background on the entropies of deterministic functions, one obtains successively the entropy of a constant square matrix operator, the entropy of a varying square matrix operator, the entropy of the kernel of an integral transformation, and the entropy of differential operators defined by square matrices.

Then one carefully exhibits the relation which exists between these results and the quantum mechanical entropy first introduced by Von Neumann, and one so obtains a new generalized quantum mechanical entropy which applies to matrics which are not necessarily density matrices. Finally, some illustrative examples for future applications are outlined.     

Deng entropy

Dempster Shafer evidence theory has been widely used in many applications due to its advantages to handle uncertainty. However, how to measure uncertainty in evidence theory is still an open issue. The main contribution of this paper is that a new entropy, named as Deng entropy, is presented to measure the uncertainty of a basic probability assignment (BPA). Deng entropy is the generalization of Shannon entropy since the value of Deng entropy is identical to that of Shannon entropy when the BPA defines a probability measure. Numerical examples are illustrated to show the efficiency of Deng entropy.     

On extropy of past lifetime distribution

Ricerche di Matematica - Recently Qiu [6] have introduced residual extropy as measure of uncertainty in residual lifetime distributions analogues to residual entropy (see, e.g. [3]). Also, they...