Moment Constraints Research Articles

On cumulative residual Kullback–Leibler information

Some extensions of entropy and KL information to the survival function have been recently proposed. We first compare some extensions of KL information and provide a criterion in choosing one among those extensions. Then we study moment constraints for maximum cumulative residual entropy distribution (Rao et al., 2004) in view of the relation between the cumulative residual entropy difference and cumulative residual Kullback–Leibler (KL) information. We further discuss the estimation methods and suggest a weighted cumulative residual entropy. Finally, we discuss the application of the estimated cumulative residual KL information as a goodness of fit test statistic with numerical examples.

On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions

In the present paper, we would like to draw attention to a possible generalized Fisher information that fits well in the formalism of nonextensive thermostatistics. This generalized Fisher information is defined for densities on \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{n}.$\end{document}Rn. Just as the maximum Rényi or Tsallis entropy subject to an elliptic moment constraint is a generalized q-Gaussian, we show that the minimization of the generalized Fisher information also leads a generalized q-Gaussian. This yields a generalized Cramér-Rao inequality. In addition, we show that the generalized Fisher information naturally pops up in a simple inequality that links the generalized entropies, the generalized Fisher information, and an elliptic moment. Finally, we give an extended Stam inequality. In this series of results, the extremal functions are the generalized q-Gaussians. Thus, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure in nonextensive thermostatistics.

Bounds on the Bayes Error Given Moments

We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes use of Curto and Fialkow's solutions for the truncated moment problem. The lower bound shows that the popular Gaussian assumption is not robust in this regard. We also construct an upper bound for the supremum Bayes error by constraining the decision boundary to be linear.

A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and [formula omitted]-Gaussians

We give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution. When the parameter of the distribution varies, it defines a parametric curve that we call an escort-path. The Rényi divergence appears as a natural by-product of the setting. We study the dynamics of the Fisher information on this path, and show in particular that the thermodynamic divergence is proportional to Jeffreys’ divergence. Next, we consider the problem of inferring a distribution on the escort-path, subject to generalized moment constraints. We show that our setting naturally induces a rationale for the minimization of the Rényi information divergence. Then, we derive the optimum distribution as a generalized q-Gaussian distribution.

Variations of the specific barrier model—part I: effect of subevent size distributions

The specific barrier model (SBM), introduced and developed by Papageorgiou and Aki (1983a; 1983b; 1985) and recently re-calibrated by Halldorsson and Papageorgiou (2005) for earthquakes of different tectonic regions, is a particular case of a composite seismic source model according to which the seismic moment is distributed in a deterministic manner on the fault plane on the basis of moment and area constraints. Namely, it assumes that a rectangular fault surface is filled with an aggregate of non-overlapping subevents modeled as circular cracks of equal diameter, the ‘barrier interval’, on which a ‘local stress drop’ takes place. In the present work, we relax the basic assumption regarding subevent size of the SBM. Subevents, still modeled as circular cracks, are allowed to vary in size according to various prescribed probability density functions controlling the frequency of occurrence of subevent sizes. Closed form expressions of the ‘far-field’ spectra of the composite source are derived using an approach proposed by Joyner and Boore (1986). The seismic energy radiated by individual subevents arrives at a site in a time window the duration of which is related to the duration of rupture and the source-site geometry. The effect of the distribution of ‘arrival times’ on the spectral amplitudes of the composite source is investigated in a companion paper, referred to as Part II. The type of subevent size distribution and its allowed size range directly affects the number of subevents required to satisfy the moment constraint. A larger number of subevents leads to increased ‘complexity’ of the earthquake which generally results in relatively higher source acceleration spectral levels at high-frequencies. The level of the plateau of the high-frequency acceleration spectral amplitudes, corresponding to different size-distributions, does not differ significantly from that of the far-field spectrum of the SBM, for a constant local stress drop. Furthermore, the differences observed in the spectral amplitudes of the SBM and its variants are likely to be less than the expected uncertainty associated with local stress drop values determined from strong-motion data. Thus, despite its simplifying assumptions, the SBM appears to provide the most simple, yet effective, description that captures the essential characteristics of a composite seismic source. This is especially advantageous for consistent strong-motion modeling in the ‘near-fault’, as well as in the ‘far-field’ region for earthquake engineering applications.

Variations of the specific barrier model—part II: effect of isochron distributions

The specific barrier model (SBM) is a particular case of a composite earthquake source model where the seismic moment is distributed in a deterministic manner on a rectangular fault plane on the basis of moment and area constraints. It is assumed that the fault surface is composed of an aggregate of subevents of equal diameter, the ‘barrier interval’. Furthermore, the subevents are assumed to rupture randomly and statistically independent of one another as the rupture front sweeps the fault plane. In the formulation of the far-field source spectrum of the SBM the ‘arrival time’ of the seismic radiation emitted by each subevent is specified via a probability density function (PDF). In the SBM the subevents are assumed to be of equal sizes (an assumption relaxed in a companion paper, referred to as Part I) and the PDF of ‘arrival times’ is assumed to be uniform. In this study we investigate the effects of different PDFs of ‘arrival times’ on the far-field source spectrum of the SBM. Different PDFs of ‘arrival times’ affect the source spectra primarily at the intermediate frequency range (between the first and second corner frequencies). Such effects become more pronounced as the earthquake magnitude increases. The far-field spectrum of seismic energy observed/recorded at a site depends on the location of the site relative to the causative fault plane, the location of rupture initiation (hypocenter) and the onset times of the rupturing subevents. All the above factors are effectively taken into account by the ‘isochrons’, which vary with source-site geometry. We investigate the selection of the appropriate PDF of seismic energy arrival times at a given site by computing isochrons for a grid of stations surrounding the earthquake fault, represented by the SBM. We show that only for stations located in a direction normal to the fault plane is the assumption of uniform PDF of ‘arrival times’ valid. At other sites non-uniform PDFs of ‘arrival times’ are observed. We identify and categorize the prevalent types of PDFs by directivity (forward vs. backward vs. neutral) and source-site distance (near-fault vs. far-field), show examples in which we group the stations accordingly. We investigate the effects of the different PDF-groups on the SBM source spectrum. Selection of the appropriate PDF for a given source-site configuration when simulating strong ground motions using the SBM in the context of the stochastic method is expected to yield more self-consistent, and physically realistic simulations.

Empirical Likelihood for Threshold Autoregressive Models

This article develops empirical likelihood for threshold autoregressive models. We propose general estimating equations based on moment constraint. Under some suitable conditions, we show the empirical likelihood estimators for parameter are asymptotically normally distributed, and the proposed log empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution.

On the Choice of Nonparametric Entropy Estimator in Entropy-Based Goodness-of-Fit Test Statistics

Entropy-based goodness-of-fit test statistics can be established by estimating the entropy difference or Kullback–Leibler information, and several entropy-based test statistics based on various entropy estimators have been proposed. In this article, we first give comments on some problems resulting from not satisfying the moment constraints. We then study the choice of the entropy estimator by noting the reason why a test based on a better entropy estimator does not necessarily provide better powers.

Efficient Estimation of Conditional Risk Measures in a Semiparametric GARCH Model

This paper proposes efficient estimators of risk measures in a semiparametric GARCH model defined through moment constraints. Moment constraints are often used to identify and estimate the mean and variance parameters and are however discarded when estimating error quantiles. In order to prevent this efficiency loss in quantile estimation we propose a quantile estimator based on inverting an empirical likelihood weighted distribution estimator. It is found that the new quantile estimator is uniformly more efficient than the simple empirical quantile and a quantile estimator based on normalized residuals. At the same time, the efficiency gain in error quantile estimation hinges on the efficiency of estimators of the variance parameters. We show that the same conclusion applies to the estimation of conditional Expected Shortfall. Our comparison also leads to interesting implications of residual bootstrap for dynamic models. We find that these proposed estimators for conditional Value-at-Risk and expected shortfall are asymptotically mixed normal. This asymptotic theory can be used to construct confidence bands for these estimators by taking account of parameter uncertainty. Simulation evidence as well as empirical results are provided.

Multivariate Variance Targeting in the BEKK-GARCH Model

This paper considers asymptotic inference in the multivariate BEKK model based on (co-)variance targeting (VT). By de finition the VT estimator is a two-step estimator and the theory presented is based on expansions of the modifi ed likelihood function, or estimating function, corresponding to these two steps. Strong consistency is established under weak moment conditions, while sixth order moment restrictions are imposed to establish asymptotic normality. Included simulations indicate that the multivariately induced higher order moment constraints are indeed necessary.

Derivation of a BGK model for mixtures

The aim of this article is to construct a BGK operator for gas mixtures starting from the true Navier–Stokes equations. That is the ones with transport coefficients given by the hydrodynamic limit of the Boltzmann equation(s). Here the same hydrodynamic limit is obtained by introducing relaxation coefficients on certain moments of the distribution functions. Next the whole model is set by using entropy minimization under moment constraints as in Brull and Schneider (2008, 2009) [23,24]. In our case the BGK operator allows to recover the exact Fick and Newton laws and satisfy the classical properties of the Boltzmann equations for inert gas mixtures.

Maximum relative entropy-based probabilistic inference in fatigue crack damage prognostics

A general probabilistic inference procedure is proposed in this paper based on the Maximum relative Entropy (MrE) approach which generalizes both Bayesian and Maximum Entropy (MaxEnt) inference methodologies. The construction of the conditional probability (likelihood function) for general model-based inference problems is discussed in detail to systematically manage uncertainties from mechanism modeling, model parameters, and measurements. Analytical and numerical examples are used to investigate the sequence effect in the probabilistic inference using point observations and moment constraints. The developed methodology is applied to the engineering fatigue crack growth problem with experimental data for demonstration and validation. Following this, a detailed comparison between the classical Bayesian inference and the MrE inference is given.

Signals ofCPviolation beyond the MSSM in Higgs and flavor physics

We study an extension of the Higgs sector of the minimal supersymmetric standard model (MSSM), considering the effects of new degrees of freedom at the TeV scale and allowing for sources of $CP$ violation beyond the MSSM. We analyze the impact of the beyond-the-MSSM sources of $CP$ violation on the Higgs collider phenomenology and on low energy flavor and $CP$-violating observables. We identify distinct Higgs collider signatures that cannot be realized, either in the case without $CP$-violating phases or in the $CP$-violating MSSM, and investigate the prospects to probe them at the Tevatron and the LHC. The most striking benchmark scenario has three neutral Higgs bosons that all decay dominantly into $W$ boson pairs and that are well within the reach of the 7 TeV LHC run. On the other hand, we also present scenarios with three Higgs bosons that have masses ${M}_{{H}_{i}}\ensuremath{\gtrsim}150\text{ }\text{ }\mathrm{GeV}$ and decay dominantly into $b\overline{b}$. Such scenarios are much more challenging to probe and can even lie completely outside the reach of the 7 TeV LHC run. We explore complementary scenarios with standard MSSM Higgs signals that allow us to accommodate a nonstandard ${B}_{s}$ mixing phase as indicated by D0, as well as the excess in ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ candidates recently reported by CDF. We find that, in contrast to the MSSM, a minimal flavor-violating soft sector is sufficient to generate significant corrections to $CP$-violating observables in meson mixing, compatible with Electric Dipole Moment constraints. In particular, a ${B}_{s}$ mixing phase ${S}_{\ensuremath{\psi}\ensuremath{\phi}}\ensuremath{\lesssim}0.15$ can be achieved for specific regions of parameter space, compatible with all the presently available experimental constraints on flavor observables. Such a nonstandard ${B}_{s}$ mixing phase would unambiguously imply a sizable suppression of ${S}_{\ensuremath{\psi}{K}_{S}}$ with respect to the standard model prediction and a $\mathrm{BR}({B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})$ close to its 95% C.L. upper bound of $1.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$.

Minimax and risk averse multistage stochastic programming

In this paper we study relations between the minimax, risk averse and nested formulations of multistage stochastic programming problems. In particular, we discuss conditions for time consistency of such formulations of stochastic problems. We also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation. Finally, we discuss a minimax approach with moment constraints to the classical inventory model.

2족 보행 로봇을 위한 허용 ZMP (Zero Moment Point) 영역의 활용을 통한 에너지 효율적인 기준 보행 궤적 생성

An energy-efficient reference walking trajectory generation algorithm is suggested utilizing allowable ZMP (Zero-Moment-Point) region, which maxmizes the energy efficiency for cyclic gaits, based on three-dimensional LIPM (Linear Inverted Pendulum Model) for biped robots. As observed in natural human walking, variable ZMP manipulation is suggested, in which ZMP moves within the allowable region to reduce the joint stress (i.e., rapid acceleration and deceleration of body), and hence to reduce the consumed energy. In addition, opimization of footstep planning is conducted to decide the optimal step-length and body height for a given forward mean velocity to minimize a suitable energy performance - amount of energy required to carry a unit weight a unit distance. In this planning, in order to ensure physically realizable walking trajectory, we also considered geometrical constraints, ZMP stability condition, friction constraint, and yawing moment constraint. Simulations are performed with a 12-DOF 3D biped robot model to verify the effectiveness of the proposed method.

Entropic Least-Squares Valuation of American Options Subject to Moment Constraints

We propose a new nonparametric method for valuing American options. We extract the risk neutral moments using a set of option data and incorporate them into the entropy framework as constraints to recover the risk-neutral pricing measure. With the recovered risk-neutral measure, we generate paths of the underlying asset returns and determine the optimal exercising strategies via the least-squares algorithm of Longstaff and Schwartz (2001).The results, based on simulations and the IBM option contracts, demonstrate that our method consistently outperforms several alternative methods. Implementation of our method is simple and straightforward.

Flavourful production at hadron colliders

We ask what new states may lie at or below the TeV scale, with sizable flavour-dependent couplings to light quarks, putting them within reach of hadron colliders via resonant production, or in association with Standard Model states. In particular, we focus on the compatibility of such states with stringent flavour-changing neutral current and electric-dipole moment constraints. We argue that the broadest and most theoretically plausible flavour structure of the new couplings is that they are hierarchical, as are Standard Model Yukawa couplings, although the hierarchical pattern may well be different. We point out that, without the need for any more elaborate or restrictive structure, new scalars with "diquark" couplings to standard quarks are particularly immune to existing constraints, and that such scalars may arise within a variety of theoretical paradigms. In particular, there can be substantial couplings to a pair of light quarks or to one light and one heavy quark. For example, the latter possibility may provide a flavour-safe interpretation of the asymmetry in top quark production observed at the Tevatron. We thereby motivate searches for diquark scalars at the Tevatron and LHC, and argue that their discovery represents one of our best chances for new insight into the Flavour Puzzle of the Standard Model.

Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-)equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over-dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.

Low-energy observables and general gauge mediation in the MSSM and NMSSM

We study constraints on the general gauge mediation (GGM) parameter space arising from low-energy observables in the MSSM and NMSSM. Specifically, we look at the dependence of the spectra and observables on the correlation function ratios in the hidden sector where supersymmetry is presumably broken. Since these ratios are not a priori constrained by theory, current results from the muon anomalous magnetic moment and flavor physics can potentially provide valuable intuition about allowed possibilities. It is found that the muon anomalous magnetic moment and flavor-physics observables place significant constraints on the GGM parameter space with distinct dependences on the hidden sector correlation function ratios. The particle spectra arising in GGM, with the possibility of different correlation function ratios, is contrasted with common intuition from regular gauge mediation (RGM) schemes (where the ratios are always fixed). Comments are made on precision gauge coupling unification, topography of the NLSP space, correlations of the muon anomalous magnetic moment with other observables, and approximate scaling relations in sparticle masses with respect to the high-scale correlation function ratios.

CP violation in supersymmetry with Effective Minimal Flavour Violation

We analyze CP violation in supersymmetry with Effective Minimal Flavour Violation, as recently proposed in arXiv:1011.0730. Unlike the case of standard Minimal Flavour Violation, we show that all the phases allowed by the flavour symmetry can be sizable without violating existing Electric Dipole Moment constraints, thus solving the SUSY CP problem. The EDMs at one and two loops are precisely analyzed as well as their correlations with the expected CP asymmetries in B physics.