Cumulative Entropies Research Articles

Cumulative entropies and sums of moments of order statistics

Generalized 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.

Phase-space quantum distributions and information theory

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.

Entropy Interpretation of Hadamard Type Fractional Operators: Fractional Cumulative Entropy.

Interpretations of Hadamard-type fractional integral and differential operators are proposed. The Hadamard-type fractional integrals of function with respect to another function are interpreted as an generalization of standard entropy, fractional entropies and cumulative entropies. A family of fractional cumulative entropies is proposed by using the Hadamard-type fractional operators.

Asymptotic results for linear combinations of spacings generated by i.i.d. exponential random variables

We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous functions defined on [0, 1] which satisfy some suitable conditions. In this way we generalize some recent results by Giuliano et al. (J Statist Plann Inference 157–158:77–89, 2015) which concern the empirical cumulative entropies defined in Di Crescenzo et al. (J Statist Plann Inference 139:4072–4087, 2009a).

Entropy analysis of human death uncertainty

Uncertainty about the time of death is part of one’s life, and plays an important role in demographic and actuarial sciences. Entropy is a measure useful for characterizing complex systems. This paper analyses death uncertainty through the concept of entropy. For that purpose, the Shannon and the cumulative residual entropies are adopted. The first may be interpreted as an average information. The second was proposed more recently and is related to reliability measures such as the mean residual lifetime. Data collected from the Human Mortality Database and describing the evolution of 40 countries during several decades are studied using entropy measures. The emerging country and inter-country entropy patterns are used to characterize the dynamics of mortality. The locus of the two entropies gives a deeper insight into the dynamical evolution of the human mortality data series.

On Cumulative Entropies in Terms of Moments of Order Statistics

In this paper, relations between some kinds of cumulative entropies and moments of order statistics are established. By using some characterizations and the symmetry of a non-negative and absolutely continuous random variable X, lower and upper bounds for entropies are obtained and illustrative examples are given. By the relations with the moments of order statistics, a method is shown to compute an estimate of cumulative entropies and an application to testing whether data are exponentially distributed is outlined.

(Generalized) Maximum Cumulative Direct, Residual, and Paired Φ Entropy Approach.

A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents so-called contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily S&P500 returns and time durations between mine explosions.

A family of weighted distributions based on the mean inactivity time and cumulative past entropies

In this paper, a family of mean past weighted ($$\hbox {MPW}_{\alpha }$$) distributions of order $$\alpha $$ is introduced. For the construction of this family, the concepts of the mean inactivity time and cumulative $$\alpha $$-class past entropy are used. Distributional properties and stochastic comparisons with other known weighted distributions are given. Furthermore, an upper bound for the k-order moment of the random variables associated with the new family and a characterization result are obtained. Generalized discrete mixtures that involve $$\hbox {MPW}_{\alpha }$$ distributions and other weighted distributions are also explored.

Notes on Cumulative Entropy as a Risk Measure

Abstract Di Crescenzo and Longobardi [Di Crescenzo and Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 2009, 12, 4072–4087] proposed the cumulative entropy (CE) as an alternative to the differential entropy. They presented an estimator of CE using empirical approach. In this paper, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference for some distributions. We also make empirical comparisons of these measures using samples from stock market in members of the Organization for Economic Co-operation and Development (OECD) countries.

Stochastic comparisons of interfailure times under a relevation replacement policy

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

Quantile-based cumulative entropies

ABSTRACTIt is well known that Shannon’s entropy plays an important role in the measurement of uncertainty of probability distributions. However, in certain situations Shannon entropy is not appropriate to measure the uncertainty and therefore an alternative measure has been introduced called cumulative residual entropy, based on the survival function (sf) instead of the probability density function (pdf) f(x) used in Shannon’s entropy. In the present paper, we introduce and study quantile versions of the cumulative entropy functions in the residual and past lifetimes. Unlike the cumulative entropies based on sf, the quantile-based cumulative entropy measures uniquely determine the underlying probability distribution.

On Shift-Dependent Cumulative Entropy Measures

Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in the studies of reliability and life testing. In this paper, cumulative distribution and survival function are used to develop weighted forms of CRE and CE. These new measures are denominated as weighted cumulative residual entropy (WCRE) and weighted cumulative entropy (WCE) and the connections of these new measures with hazard and reversed hazard rates are assessed. These information-theoretic uncertainty measures are shift-dependent and various properties of these measures are studied, including their connections with CRE, CE, mean residual lifetime, and mean inactivity time. The notions of weighted mean residual lifetime (WMRL) and weighted mean inactivity time (WMIT) are defined. The connections of weighted cumulative uncertainties with WMRL and WMIT are used to calculate the cumulative entropies of some well-known distributions. The joint versions of WCE and WCRE are defined which have the additive properties similar to those of Shannon entropy for two independent random lifetimes. The upper boundaries of newly introduced measures and the effect of linear transformations on them are considered. Finally, empirical WCRE and WCE are proposed by virtue of sample mean, sample variance, and order statistics to estimate the new measures of uncertainty. The consistency of these estimators is studied under specific choices of distributions.

On cumulative residual (past) inaccuracy for truncated random variables

To overcome the drawbacks of Shannon's entropy, the concept of cumulative residual and past entropy has been proposed in the information theoretic literature. Furthermore, the Shannon entropy has been generalized in a number of different ways by many researchers. One important extension is Kerridge inaccuracy measure. In the present communication we study the cumulative residual and past inaccuracy measures, which are extensions of the corresponding cumulative entropies. Several properties, including monotonicity and bounds, are obtained for left, right and doubly truncated random variables.

Asymptotic results for runs and empirical cumulative entropies

We prove large and moderate deviations for two sequences of estimators based on the order statistics and, more precisely, on spacings. In the first case we deal with runs, and we have sums of independent Bernoulli distributed random variables. In the second case we deal with empirical cumulative entropies, and we have linear combinations of independent exponentially distributed random variables.

Cumulative Measure of Uncertainty for Conditionally Specified Models

Abtsrcat Recently, Rao (2005) introduced an alternate measure of uncertainty called cumulative entropy that parallels Shannon entropy (see Asadi (2007)). Motivated by this, in the present paper, we introduce cumulative entropies for two types of conditionally specied models. Finally, an applica- tion of the cumulative entropy using the maximum entropy principle is also illustrated.

Thermodynamic properties of discotic mesogens: heat capacities and phase transitions of benzene-hexa- n-alkanoates

Molar heat capacities of disk-like compounds benzene-hexa- n-nonanoate (BH9) and benzene-hexa- n-decanoate (BH10) have been measured with an adiabatic calorimeter between 13 K and 390 K. Three and two different phases appeared in the solid states of BH9 and BH10, respectively. Molar entropies and transition entropies were determined and compared with those of other benzene-hexa- n-alkanoates BH n ( n: 5 to 10). The odd–even effect with respect to the number of carbon atoms in a side chain of a disc-like molecule was observed in the cumulative transition entropies. The molar entropies exhibited a peculiar ‘pairing phenomenon’ between BH(2 m−1) and BH(2 m), where m is integer. This phenomenon is well accounted for in terms of characteristic molecular structure and packing in the crystal lattice.

Thermal properties of a disc-like compound benzene-hexa-n-pentanoate: a precursor of a discotic mesogen

The heat capacity of a disk-like compound, benzene-hexa-n-pentanoate (BH5), a precursor of a discotic mesogen, has been measured by adiabatic calorimetry between 13 and 390 K. BH5 showed four different crystalline phases, but no liquid crystalline phase. Molar entropy and transition entropies were determined and compared with those of other homologues (BH6, BH7, and BH8). An odd-even effect with respect to the number of carbon atoms in the alkyl chain was observed for the cumulative entropies of the phase transitions occurring in the solid state. This effect is discussed by comparing with data for the n-alkanes. The molar entropies of the crystalline state, especially below 250 K, exhibited a peculiar 'pairing effect' between BH(2m- 1) and BH(2m), where m is an integer.

Solvation of halide ions with water and acetonitrile in the gas phase

The gas-phase solvation reactions X/sup -/(S)/sub n-1/ + S = X/sup -/(S)/sub n/, for S = H/sub 2/O and CH/sub 3/CN and X/sup -/ = F/sup -/, Cl/sup -/, Br/sup -/, and I/sup -/, were studied by using a pulsed electron beam high-pressure mass spectrometer. For hydration reactions, the values of -..delta..G/sup 0//sub n-1,n/ are in the order F/sup -/ > Cl/sup -/ > Br/sup -/ > I/sup -/ for all n. However, the -..delta..H/sup 0//sub n-1,n/ values for Cl/sup -/ decrease more rapidly than those for Br/sup -/, and a crossover is observed between n = 3 and 4. For the solvation reactions with CH/sub 2/CN, the -..delta..H/sup 0//sub 0,1/ value for F/sup -/ was found to be much larger than those for other halide ions, indicating the participation of covalent bonding in the cluster F/sup -/ ...H/sub 3/CCN. The covalent bond formation leads to the rapid decrease in the -..delta..H/sup 0//sub n-1,n/ values and -..delta..H/sup 0//sub n-1,n/ for F/sup -/ becomes smaller than that for Cl/sup -/ at n = 7. Much more gradual decreases in -..delta..H/sup 0//sub n-1,n/ values for Cl/sup -/, Br/sup -/, and I/sup -/ are observed up to the highest values of n measured;more » i.e., the completion of the first solvation shell does not seem to occur up to n = 8. Due to crowding in the inner shell, the -..delta..S/sup 0//sub n-1,n/ values increase with n. The difference between the cumulative enthalpies and entropies (i.e., -..delta..H/sup 0//sub 0,n/ and -..delta..S/sup 0//sub 0,n/) does not approach the differential bulk solvation enthalpies and entropies, respectively.« less

Thermodynamische Funktionen der komplexbildung und ionenradien — II: Berechnung der kumulativen freien bildungsenthalpien und Bildungsentropien der glykolatokomplexe der Seltenen Erden aus ionenradien und polarisierbarkeiten

By extending the electrostatic arguments for the calculation of the free formation enthalpies regarding polarizable donors the cumulative free enthalpies and entropies for the 1:1, 1:2, 1:3 and 1:4 compounds of the rare earths and glycollic acid have been calculated. Comparing these values with experimental dates it is to be seen, that the free enthalpies are in good agreement. The calculated entropies are 3–4 cal/mol degree higher than the experimental values. Eventually it is necessary to respect the freezing in of the two inner rotations during coordination of the ligand.