Search

AbstractModel checking evaluates the appropriateness of a statistical model based on the observed data, and it is essential to make valid statistical analyses. In this paper, a new procedure for model checking type II censored data is proposed. This procedure combines the Kullback-Leibler divergence, the Dirichlet process, and the relative belief ratio. The method is implemented via a computational algorithm and is explained through several examples. [ABSTRACT FROM AUTHOR]

We investigate the log-concavity on the half-line of the Wright function ϕ (- α , β , - x) , in the probabilistic setting α ∈ (0 , 1) and β ≥ 0. Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for β ≥ α and in the classical case β = 1 - α of the Mittag-Leffler distribution, which exhibits a certain critical parameter α ∗ = 0.771667... defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if β ≥ α or α ≤ 1 / 2 and β = 0. [ABSTRACT FROM AUTHOR]

We give an elementary proof of an inequality of Lin, Kim and Hsieh that implies strong subadditivity of the von Neumann entropy. [ABSTRACT FROM AUTHOR]

We introduce and study the cumulative information generating function, which provides a unifying mathematical tool suitable to deal with classical and fractional entropies based on the cumulative distribution function and on the survival function. Specifically, after establishing its main properties and some bounds, we show that it is a variability measure itself that extends the Gini mean semi-difference. We also provide (i) an extension of such a measure, based on distortion functions, and (ii) a weighted version based on a mixture distribution. Furthermore, we explore some connections with the reliability of k-out-of-n systems and with stress–strength models for multi-component systems. Also, we address the problem of extending the cumulative information generating function to higher dimensions. [ABSTRACT FROM AUTHOR]

Strongly convex functions as a subclass of convex functions, still equipped with stronger properties, are employed through several generalizations and improvements of the Jensen inequality and the Jensen–Mercer inequality. This paper additionally provides applications of obtained main results in the form of new estimates for so-called strong f-divergences: the concept of the Csiszár f-divergence for strongly convex functions f, together with particular cases (Kullback–Leibler divergence, χ 2 -divergence, Hellinger divergence, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence.) Furthermore, new estimates for the Shannon entropy are obtained, and new Chebyshev-type inequalities are derived. [ABSTRACT FROM AUTHOR]

The problem of quantifying the distance between distributions arises in various fields, including cryptography, information theory, communication networks, machine learning, and data mining. In this article, the analogy with the cumulative Jensen–Shannon divergence, defined in Nguyen and Vreeken (2015), we propose a new divergence measure based on the cumulative distribution function and call it the cumulative α-Jensen–Shannon divergence, denoted by CJS (α) . Properties of CJS (α) are studied in detail, and also two upper bounds for CJS (α) are obtained. The simplified results under the proportional reversed hazard rate model are given. Various illustrative examples are analyzed. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of dynamic cumulative residual extropy inaccuracy (DCREI) by expanding on the existing dynamic cumulative residual extropy (DCRE) measure and proposes a weighted version of it. The paper then investigates a characterization problem for the proposed weighted dynamic extropy inaccuracy measure under the proportional hazard model and characterizes some well-known lifetime distributions using the weighted dynamic cumulative residual extropy inaccuracy (WDCREI) measure. Additionally, the study discusses the stochastic ordering of WDCREI and certain results based on it. Non-parametric estimations of the proposed measures based on kernel and empirical estimators are suggested. Results of a simulation study show that the kernel-based estimators perform better than the empirical-based estimator. Finally, applications of the proposed measures on model selection are provided. [ABSTRACT FROM AUTHOR]

We develop a general method to study the Fisher information distance in central limit theorem for nonlinear statistics. We first construct completely new representations for the score function. We then use these representations to derive quantitative estimates for the Fisher information distance. To illustrate the applicability of our approach, explicit rates of Fisher information convergence for quadratic forms and the functions of sample means are provided. For the sums of independent random variables, we obtain the Fisher information bounds without requiring the finiteness of Poincaré constant. Our method can also be used to bound the Fisher information distance in non-central limit theorems. [ABSTRACT FROM AUTHOR]

AbstractGeneralized weighted cumulative residual entropies and generalized weighted cumulative entropies are represented by means of weighted sums of moments of upper and lower order statistics, respectively. A variety of examples is shown by specifying the weight function. [ABSTRACT FROM AUTHOR]

Recently, a new dispersion index, as a measures of information, has been introduced and called varentropy. In this article, we introduce new measures of variability based on two measures of uncertainty, namely, the Kerridge inaccuracy measure and the Kullback-Leibler divergence. Their generating functions are considered and their infinite series representations are given. These new measures and associated properties, bounds and illustrative examples are all presented in detail. Finally, an application of Kullback-Leibler divergence and its dispersion index is illustrated by using the mean-variance rule. [ABSTRACT FROM AUTHOR]

We consider a population of finite size M, where the current number N of entities, N ∈ { 0 , 1 , 2 , ... , M } , determines its states. We geometrically analyze the amounts of information iN and iM−N, carried by the random variables N and (M−N), respectively, and population entropy S = E (i N) = E (i (M − N)). The population is modeled by a family of Birth-Death Processes of size M+1, indexed by the population utilization parameter ρ (i.e., birth/death ratio), which determines its macrostates. Considering the quantities: x = (N − E (N)) & y = (i N (ρ) − S (ρ)) and M x = ((M − N) − E (M − N)) & M y = (i M − N (ρ) − S (ρ)) as vectors, it is shown that the angles: φ = ∠ (x ; y) and ψ = ∠ ( M x ; M y) are supplementary ones, that is φ (ρ) + ψ (ρ) = π , ρ > 0. Expressions for their inner products < x , y > and < M x , M y > , being the covariances of N&iN(ρ) and (M − N) & i M − N (ρ) , respectively, which sum equals zero, with respect to the parameter ρ are also obtained. Further, what is especially important, it is revealed that φ = ψ = π / 2 , that is both inner products equal zero, at the point ρMmax, which is the value of parameter ρ at which the entropy has maximum value. Finally, for an information linear population only, it is shown that N obeys uniform distribution at the point ρM, max, that is that S (ρ M , max ) = ln ⁡ (M + 1). These results are further illustrated on populations described by truncated geometrical, binomial, and truncated Poisson distribution. [ABSTRACT FROM AUTHOR]

Weighted cumulative residual Kullback–Leibler information measure and its dynamic version are proposed and their properties are investigated. Also weighted cumulative Kullback–Leibler information measure along with its dynamic version is introduced. Monotonicity properties of the dynamic information measures are studied. A goodness-of-fit test is developed for exponential distribution using weighted cumulative residual Kullback–Leibler information measure based on complete and censored data. It is observed that proposed tests perform well for monotone decreasing hazard alternatives under censored data. Four data sets are analyzed for illustration. [ABSTRACT FROM AUTHOR]

AbstractThe concepts of entropy and divergence, along with their past, residual, and interval variants are revisited in a reliability theory context and generalized families of them that are based on ϕ-functions are discussed. Special emphasis is given in the parametric family of entropies and divergences of Cressie and Read. For non-negative and absolutely continuous random variables, the dual to Shannon entropy measure of uncertainty, the extropy, is considered and its link to a specific member of the ϕ-entropies family is shown. A number of examples demonstrate the implementation of the generalized entropies and divergences, exhibiting their utility. [ABSTRACT FROM AUTHOR]

We introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society. The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption, we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility. [ABSTRACT FROM AUTHOR]

Non-additive measures are important for many applications. In this paper we propose an extension of the non-additive measure using the concept of Kerridge inaccuracy and cumulative Tsallis entropy. Several properties including non-linear transformation, bounds and characterizations are obtained using this measure. We also study different properties of the quantile version of the proposed measure. Finally, we numerically illustrate the usefulness of the measure and its quantile form using kernel density estimators. [ABSTRACT FROM AUTHOR]

To have accuracy in the extracted information is the goal of the reliability theory investigation. In information theory, varentropy has recently been proposed to describe and measure the degree of information dispersion around entropy. Theoretical investigation on varentropy of past life has been initiated, however details on its stochastic properties are yet to be discovered. In this paper, we propose a novel stochastic order and introduce new classes of life distributions based on past varentropy. Further, we illustrate some of its applications in reliability modeling and in the diversity measure of Boltzmann distribution. [ABSTRACT FROM AUTHOR]

The fuzzy set theory was introduced to handle uncertainty due to imprecision, vagueness and partial information. Then, its extensions such as intuitionistic fuzzy set, intuitionistic interval-valued fuzzy set, Pythagorean fuzzy set were introduced and applied successfully in many fields. Then another extension of orthopair fuzzy set was introduced as Fermatean fuzzy set which is characterized by membership degree and non-membership degree which makes it to provide an excellent tool to present imprecise opinions of humans in decision-making processes. This study is devoted to construct a novel Fermatean fuzzy divergence measure along with its evidence of legitimacy and to deliberate its key properties. The proposed divergence measure for Fermatean fuzzy sets with weighted aggregation operators is applied to fix decision-making problems through numerical illustrations. A comparative study is given between the proposed Fermatean fuzzy divergence measure and the extant methods to test its effectiveness, viability and expediency. Their results were compared in order to check the superiority of the proposed measure. [ABSTRACT FROM AUTHOR]

In this study, Levinson-type inequalities for the class of n-convex (n ≥ 3 ) functions are generalized using new Green functions and the Hermite interpolating polynomial involving two types of data points. Some estimations for novel functionals are derived using f-divergence. Furthermore, different inequalities involving Shannon entropies are presented. [ABSTRACT FROM AUTHOR]

Redundant is used to improve the reliability of a given system and to ensure that it is available to fulfill its duty during a mission time. This paper proposes an algorithmic approach for solving RAPs when component lifetimes are either independent and heterogeneous under a general setting. Two common active and standby schemes for allocations, are considered. Moreover, a new stochastic order is proposed and applied for comparison purposes. Various illustrative examples are also analyzed. This method can be implemented for engineering system networks. [ABSTRACT FROM AUTHOR]

AbstractImpurity functions are crucial in decision trees. These functions help determine the impurity level of a node in a decision tree, guiding the splitting criteria. However, two primary ambiguities have surrounded impurity functions: (1) the question of their non negativity and (2) the debate over their concavity. In this paper, we address these uncertainties by delving into the characteristics of impurity functions. We establish that the non negativity of an impurity function is inconsequential. Through counter examples, we disprove the equivalence between an impurity function and a concave function. We identify an impurity function that is not concave and a concave function that is not an impurity function. Interestingly, we find an impurity function that results in a negative impurity reduction. Furthermore, we validate several significant properties of impurity functions. For example, we demonstrate that when an impurity function is concave, the impurity reduction remains nonnegative for multiway divisions. We also discuss the sufficient conditions for a concave function to be an impurity function. Our numerical results further indicate that a positive linear combination of the two most popular impurity functions, namely Gini Index and Entropy, may surpass the individual performance of each when applied to the well-known German credit dataset. [ABSTRACT FROM AUTHOR]

A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of L 1 -distance. [ABSTRACT FROM AUTHOR]

Recently, a modification of fractional entropy based on the inverse Mittag-Leffler function (MLF) was proposed by Zhang and Shang (2021). In this paper, we present an extension of the fractional cumulative entropy (FCE) and obtain some further results about this measure. We study new equivalent expressions, bounds, stochastic ordering, and properties of dynamic generalized FCE. By using the empirical approach, we give an estimator of this measure and study large sample properties of it. In addition, the validity of this new measure is supported by numerical simulations on logistic map equations. Finally, an application of this measure is proposed in the evaluation of MRI scans for brain cancer. [ABSTRACT FROM AUTHOR]

We study some properties of weighted survival extropy measure. This is a shift dependent measure and put more weights to the large values of the observed random variable. Then we introduce weighted extended survival extropy and its dynamic version. Various properties of the proposed measures are obtained. Some generalized inequalities related to weighted extended survival extropy are studied. Non-parametric estimators of these measures are proposed and their asymptotic properties are investigated. The performance of the estimators is assessed through a simulation study. Finally, it is shown that weighted survival extropy measure can be used in model discrimination and as an alternative risk measure. Analysis of real life data is considered for illustration. [ABSTRACT FROM AUTHOR]

The maximum entropy bootstrap for time series is a technique that creates a large number of replicates, as elements of an ensemble, for inference purposes, which satisfies the ergodic and the central limit theorems. As an alternative to the use of traditional techniques, this work proposes generalized maximum entropy for the estimation of parameters in all the replicated models. An empirical application and a simulated example illustrate the advantages of this two-stage maximum entropy approach for time series regression modeling, where maximum entropy is used both in data replication and in parameter estimation. [ABSTRACT FROM AUTHOR]

We solve for the first time a longstanding puzzle of quantitative finance that has often been described as the holy grail of volatility modelling: build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures and VIX options. We use a nonparametric discrete-time approach: given a VIX future maturity T 1 , we consider the set P of all probability measures on the SPX at T 1 , the VIX at T 1 and the SPX at T 2 = T 1 + 30 days which are perfectly calibrated to the full SPX smiles at T 1 and T 2 and the full VIX smile at T 1 , and which also satisfy the martingality constraint on the SPX as well as the requirement that the VIX is the implied volatility of the 30-day log-contract on the SPX. By casting the superreplication problem as a dispersion-constrained martingale optimal transport problem, we first establish a strong duality theorem and prove that the absence of joint SPX/VIX arbitrage is equivalent to P ≠ ∅ . Should they arise, joint arbitrages are identified using classical linear programming. In their absence, we then provide a solution to the joint calibration puzzle by solving a dispersion-constrained martingale Schrödinger problem: we choose a reference measure and build the unique jointly calibrating model that minimises the relative entropy. We establish several duality results. The minimum-entropy jointly calibrating model is explicit in terms of the dual Schrödinger portfolio, i.e., the maximiser of the dual problem, should the latter exist, and is numerically computed using an extension of the Sinkhorn algorithm. Numerical experiments show that the algorithm performs very well in both low and high volatility regimes. [ABSTRACT FROM AUTHOR]

The study of Shannon differential entropy function and its associated measures using quantile function are of recent interest among researchers. However, a bivariate extension of the same have not been considered so far. Recently, a new method of deriving bivariate quantile function and their properties in the context of reliability modeling is available in literature. Motivated by this, the present paper introduces and obtain some properties of quantile-based entropy function in the bivariate case. We also examine various properties of bivariate quantile entropy function for residual random variables. Finally, we illustrate the bivariate quantile-based residual entropy function using a suitable lifetime data. [ABSTRACT FROM AUTHOR]

We investigate random variables for which the variance and the information potential satisfy a preservation law. [ABSTRACT FROM AUTHOR]

In this article, we present a novel approach to measuring inaccuracy, introducing a two-parametric weighted generalized inaccuracy measure of order α and type β, along with its residual version. Our proposed measure depends on the proportional hazard rate model (PHRM) to uniquely determine the survival function, and we have derived a characterization result for this measure. Through our analysis under the PHRM framework, we have studied various properties of the proposed measure and their interrelationships. [ABSTRACT FROM AUTHOR]

We generalize cyclic Jensen's inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence. [ABSTRACT FROM AUTHOR]

Calculating the entropy for complex systems is a significant problem in science and engineering problems. However, this calculation is usually computationally expensive when the entropy is computed directly. This paper introduces three classes of information sources that for all members of each class, the entropy value is the same. These classes are characterized according to special dynamics created by three kinds of self-mappings on Ω, and A, where Ω is a probability space and A is a finite set. An approximation of rank variables of the product of information sources is made, and it is proved that the topological entropy of the product of two information sources is equal to the summation of their topological entropies. [ABSTRACT FROM AUTHOR]

We consider a quasi-classical version of the Alicki–Fannes–Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called “quasi-classical”. Several applications of the proposed method are described. Among others, we obtain the universal continuity bound for the von Neumann entropy under the energy-type constraint which in the case of one-mode quantum oscillator is close to the specialized optimal continuity bound presented recently by Becker, Datta and Jabbour. We obtain semi-continuity bounds for the quantum conditional entropy of quantum-classical states and for the entanglement of formation in bipartite quantum systems with the rank/energy constraint imposed only on one state. Semi-continuity bounds for entropic characteristics of classical random variables and classical states of a multi-mode quantum oscillator are also obtained. [ABSTRACT FROM AUTHOR]