Bivariate extension of generalized cumulative past entropy

The cumulative past entropy (CPE) of order α, a dual measure of cumulative residual entropy (CRE) of order α, has recently been proposed as a suitable extension of CPE. In this article, we extend the definition of (dynamic) CPE of order α (DCPE(α)) to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE(α) for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE(α) are obtained for conditional distributions. Along with some characterization results it is shown that the proposed generalized measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure and prove interrelation with some existing stochastic orders.     

A quantile-based generalized dynamic cumulative measure of entropy

The cumulative residual entropy (CRE), introduced by Rao et al. (2004), is a new measure of uncertainty and viewed as a dynamic measure of uncertainty. Asadi and Zohrevand (2007) proposed a dynamic form of the CRE, namely dynamic CRE. Recently, Kumar and Taneja (2011) introduced a generalized dynamic CRE based on the Varma entropy introduced by Varma (1966) and called it dynamic CRE of order α and type β. In the present article, we introduce a quantile version of the dynamic CRE of order α and type β and study its properties. For this measure, we obtain some characterization results, aging classes properties, and stochastic comparisons.     

Bivariate extension of (dynamic) cumulative past entropy

Recently, the concept of cumulative residual entropy (CRE) has been studied by many researchers in higher dimensions. In this article, we extend the definition of (dynamic) cumulative past entropy (DCPE), a dual measure of (dynamic) CRE, to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure.     

Some new results on the cumulative residual entropy

The residual entropy function is a relevant dynamic measure of uncertainty in reliability and survival studies. Recently, Rao et al. [2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220–1228] and Asadi and Zohrevand [2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931–1941] define the cumulative residual entropy and the dynamic cumulative residual entropy, respectively, as some new measures of uncertainty. They study some properties and applications of these measures showing how the cumulative residual entropy and the dynamic cumulative residual entropy are connected with the mean residual life function. In this paper, we obtain some new results on these functions. We also define and study the dynamic cumulative past entropy function. Some results are given connecting these measures of a lifetime distribution and that of the associated weighted distribution.     

On study of dynamic survival and cumulative past entropies

Abstract

Recently, a new class of measure of uncertainty, called “dynamic survival entropy”, has been defined and studied in the literature. Based on this entropy, DSE(α) ordering, IDSE(α), and DDSE(α) classes of life distributions are defined and some results are studied. In this paper, our main aim is to prove some more results of the ordering and the aging classes of life distributions mentioned above. Some important distributions such as exponential, Pareto, Pareto II, and finite range distributions are also characterized. Here we have defined cumulative past entropy and proved some interesting results.     

An application of information theory to the problem of statistical disclosure

By using prior knowledge it may be possible to deduce pieces of individual information from a frequency distribution of a population. If the prior information is described by a stochastic model, an information-theoretic approach can be applied in order to judge the possibilities for disclosure. By specifying the stochastic model in various ways it is shown how the decrease in entropy caused by the publication of a frequency distribution can be determined and interpreted. The stochastic models are also used to derive formulae for disclosure risks and expected numbers of disclosures.     

Uncertainty Quantification in Case of Imperfect Models: A Non‐Bayesian Approach

The starting point in uncertainty quantification is a stochastic model, which is fitted to a technical system in a suitable way, and prediction of uncertainty is carried out within this stochastic model. In any application, such a model will not be perfect, so any uncertainty quantification from such a model has to take into account the inadequacy of the model. In this paper, we rigorously show how the observed data of the technical system can be used to build a conservative non‐asymptotic confidence interval on quantiles related to experiments with the technical system. The construction of this confidence interval is based on concentration inequalities and order statistics. An asymptotic bound on the length of this confidence interval is presented. Here we assume that engineers use more and more of their knowledge to build models with order of errors bounded by . The results are illustrated by applying the newly proposed approach to real and simulated data.     

Cumulative ratio information based on general cumulative entropy

In this paper, we suggest an extension of the cumulative residual entropy (CRE) and call it generalized cumulative entropy. The proposed entropy not only retains attributes of the existing uncertainty measures but also possesses the absolute homogeneous property with unbounded support, which the CRE does not have. We demonstrate its mathematical properties including the entropy of order statistics and the principle of maximum general cumulative entropy. We also introduce the cumulative ratio information as a measure of discrepancy between two distributions and examine its application to a goodness-of-fit test of the logistic distribution. Simulation study shows that the test statistics based on the cumulative ratio information have comparable statistical power with competing test statistics.     

Gaussian process emulation for second-order Monte Carlo simulations

We consider the use of emulator technology as an alternative method to second-order Monte Carlo (2DMC) in the uncertainty analysis for a percentile from the output of a stochastic model. 2DMC is a technique that uses repeated sampling in order to make inferences on the uncertainty and variability in a model output. The conventional 2DMC approach can often be highly computational, making methods for uncertainty and sensitivity analysis unfeasible. We explore the adequacy and efficiency of the emulation approach, and we find that emulation provides a viable alternative in this situation. We demonstrate these methods using two different examples of different input dimensions, including an application that considers contamination in pre-pasteurised milk.     

On cumulative entropies

In analogy with the cumulative residual entropy recently proposed by Wang et al. [2003a. A new and robust information theoretic measure and its application to image alignment. In: Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 2732, Springer, Heidelberg, pp. 388–400; 2003b. Cumulative residual entropy, a new measure of information and its application to image alignment. In: Proceedings on the Ninth IEEE International Conference on Computer Vision (ICCV’03), vol. 1, IEEE Computer Society Press, Silver Spring, MD, pp. 548–553], we introduce and study the cumulative entropy, which is a new measure of information alternative to the classical differential entropy. We show that the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. Hence, our measure is particularly suitable to describe the information in problems related to ageing properties of reliability theory based on the past and on the inactivity times. Our results include various bounds to the cumulative entropy, its connection to the proportional reversed hazards model, and the study of its dynamic version that is shown to be increasing if the mean inactivity time is increasing. The empirical cumulative entropy is finally proposed to estimate the new information measure.     

Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty

Uncertainty and sensitivity analyses for systems that involve both stochastic (i.e., aleatory) and subjective (i.e., epistemic) uncertainty are discussed. In such analyses, the dependent variable is usually a complementary cumulative distribution function (CCDF) that arises from stochastic uncertainty; uncertainty analysis involves the determination of a distribution of CCDFs that results from subjective uncertainty, and sensitivity analysis involves the determination of the effects of subjective uncertainty in individual variables on this distribution of CCDFs. Uncertainty analysis is presented as an integration problem involving probability spaces for stochastic and subjective uncertainty. Approximation procedures for the underlying integrals are described that provide an assessment of the effects of stochastic uncertainty, an assessment of the effects of subjective uncertainty, and a basis for performing sensitivity studies. Extensive use is made of Latin hypercube sampling, importance sampling and regression-based sensitivity analysis techniques. The underlying ideas, which are initially presented in an abstract form, are central to the design and performance of real analyses. To emphasize the connection between concept and computational practice, these ideas are illustrated with an analysis involving the MACCS reactor accident consequence model a, performance assessment for the Waste Isolation Pilot Plant, and a probabilistic risk assessment for a nuclear power station.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

Stochastic approaches to inversion problems in optics

Image restoration, image smoothing, and probability law estimation are important inverse problems in optics. Within the past 10 or so years, much progress has been made toward their resolution. These problems are often ill-posed mathematically. However, such concepts as maximum entropy, maximum likelihood, binay decision discrimination, median window filtering, MAP estimation, and minimum Fisher information have proven invaluable in regularising, or reducing, the ill-posed nature of the problems. In particular, those concepts which measure uncertainty or disorder have played a central role. Given noise-prone data, concepts that describe maximal disorder (maximum entropy, minimum Fisher information, minimal bina y discrimination) exert a smoothing influence on the solutions that drastically reduces noise propagation into the output. Given insufficient but noise-free data, as in the probability estimation problem, the principle of minimum Fisher information, in particular, creates smooth estimates which, oftentimes, are physically correct as well.     

社会保障基金长期财务随机预测模型的比较与选择

为了测算分析社会保障制度的可持续发展,世界上不少国家建立了社会保障基金长期财务预测模型。与确定性财务预测模型相比,随机预测模型有利于阐明预测结果所面临的不确定。美国在运用随机预测模型对社会保障基金的财务状况做出预测方面走在世界最前端,我国对社会保障基金随机预测模型的研究基本处于空白。本文对美国社会保障署和国会预算办公室采用的社会保障基金长期随机预测模型进行了比较分析,对两种模型的选择给出了建议,最后提出了我国建立社会保障基金长期预测模型的一些建议。     

The Changing Transmission of Uncertainty Shocks in the U.S.

This article investigates if the impact of uncertainty shocks on the U.S. economy has changed over time. To this end, we develop an extended factor augmented vector autoregression (VAR) model that simultaneously allows the estimation of a measure of uncertainty and its time-varying impact on a range of variables. We find that the impact of uncertainty shocks on real activity and financial variables has declined systematically over time. In contrast, the response of inflation and the short-term interest rate to this shock has remained fairly stable. Simulations from a nonlinear dynamic stochastic general equilibrium (DSGE) model suggest that these empirical results are consistent with an increase in the monetary authorities’ antiinflation stance and a “flattening” of the Phillips curve. Supplementary materials for this article are available online.     

Rényi entropy of m-generalized order statistics

Abstract

Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.     

Information content of partially rank-ordered set samples

Partially rank-ordered set (PROS) sampling is a generalization of ranked set sampling in which rankers are not required to fully rank the sampling units in each set, hence having more flexibility to perform the necessary judgemental ranking process. The PROS sampling has a wide range of applications in different fields ranging from environmental and ecological studies to medical research and it has been shown to be superior over ranked set sampling and simple random sampling for estimating the population mean. We study Fisher information content and uncertainty structure of the PROS samples and compare them with those of simple random sample (SRS) and ranked set sample (RSS) counterparts of the same size from the underlying population. We study uncertainty structure in terms of the Shannon entropy, Rényi entropy and Kullback–Leibler (KL) discrimination measures.     

大维不可观测变量的中国宏观经济不确定性测度研究

本文提出利用大型统计数据直接测算中国宏观经济不确定性的方法。通过建立含有潜在不可观测变量的混频动态因子随机波动模型,实现利用月度和季度大型数据测度宏观经济不确定性。利用1994—2017年中国60个月度统计指标和4个季度统计指标,测算了我国宏观经济不确定性,结果表明:（1）中国宏观经济不确定性具有明显的阶段性特征,不确定性的变动受到多种因素的影响,传统的景气监测指标并不是一致最优的同步监测指标。（2）在测算宏观经济不确定性时,有必要将核心指标作为观测到的因子予以保留,不仅提高了结果的解释性,也能得到更符合经济事实的结果。（3）宏观政策变动是引起经济不确定性上升的重要因素,但政策的影响往往具有滞后效应,未预期的政策变动将触发更高的宏观经济不确定。     

On weighted generalized entropy

In the literature of information theory, the concept of generalized entropy has been proposed and the length-based shift dependent information measure has been studied. In this paper, the concept of weighted generalized entropy has been introduced. The properties of weighted generalized residual entropy and weighted generalized past entropy are also discussed.