Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

Universality class of fiber bundles with strong heterogeneities

We study the effect of strong heterogeneities on the fracture of disordered materials using a fiber bundle model. The bundle is composed of two subsets of fibers, i.e. a fraction 0 ≤ α ≤ 1 of fibers is unbreakable, while the remaining 1 - α fraction is characterized by a distribution of breaking thresholds. Assuming global load sharing, we show analytically that there exists a critical fraction of the components αc which separates two qualitatively diferent regimes of the system: below αc the burst size distribution is a power law with the usual exponent Ƭ= 5/2, while above αc the exponent switches to a lower value Ƭ = 9/4 and a cutoff function occurs with a diverging characteristic size. Analyzing the macroscopic response of the system we demonstrate that the transition is conditioned to disorder distributions where the constitutive curve has a single maximum and an inflexion point defining a novel universality class of breakdown phenomena

Are the maximum hardness and minimum polarizability principles always obeyed in nontotally symmetric vibrations?

An overview is given on a study which showed that not only in chemical reactions but also in the favorable case of nontotally symmetric vibrations where the chemical and external potentials keep approximately constant, the generalized maximum hardness principle (GMHP) and generalized minimum polarizability principle (GMPP) may not be obeyed. A method that allows an accurate determination of the nontotally symmetric molecular distortions with more marked GMPP or anti-GMPP character through diagonalization of the polarizability Hessian matrix is introduced

A new chance-constrained maximum capture location problem

The paper presents a new model based on the basic Maximum Capture model,MAXCAP. The New Chance Constrained Maximum Capture modelintroduces astochastic threshold constraint, which recognises the fact that a facilitycan be open only if a minimum level of demand is captured. A metaheuristicbased on MAX MIN ANT system and TABU search procedure is presented tosolve the model. This is the first time that the MAX MIN ANT system isadapted to solve a location problem. Computational experience and anapplication to 55 node network are also presented.

Maximum-likelihood estimation of specific differential phase and attenuation in rain

Precise estimation of propagation parameters inprecipitation media is of interest to improve the performanceof communications systems and in remote sensing applications.In this paper, we present maximum-likelihood estimators ofspecific attenuation and specific differential phase in rain. Themodel used for obtaining the cited estimators assumes coherentpropagation, reflection symmetry of the medium, and Gaussianstatistics of the scattering matrix measurements. No assumptionsabout the microphysical properties of the medium are needed.The performance of the estimators is evaluated through simulateddata. Results show negligible estimators bias and variances closeto Cramer–Rao bounds.

Avaliação de Panicum maximum jacq. para tolerância ao alumínio em solução nutritiva

Em condições controladas de temperatura, umidade e iluminação, 30 genótipos de Panicum maximum foram avaliados para a verificação de tolerância às doses de alumínio de 0, 12 e 24 mg L-1 em solução nutritiva. Os efeitos do alumínio na inibição do alongamento radicular e no índice de tolerância possibilitaram a estratificação dos genótipos em três categorias: tolerantes, intermediários e sensíveis. A maior parte dos genótipos apresentou de média a baixa tolerância ao alumínio, destacando-se, como os mais tolerantes, os genótipos K191, T95, T84, T91 e Centenário e, como os mais sensíveis, os genótipos Centauro, K68, K214 e T46.

The second law of thermodynamics and the global climate system : A review of the maximum entropy production principle

The long-term mean properties of the global climate system and those of turbulent fluid systems are reviewed from a thermodynamic viewpoint. Two general expressions are derived for a rate of entropy production due to thermal and viscous dissipation (turbulent dissipation) in a fluid system. It is shown with these expressions that maximum entropy production in the Earth s climate system suggested by Paltridge, as well as maximum transport properties of heat or momentum in a turbulent system suggested by Malkus and Busse, correspond to a state in which the rate of entropy production due to the turbulent dissipation is at a maximum. Entropy production due to absorption of solar radiation in the climate system is found to be irrelevant to the maximized properties associated with turbulence. The hypothesis of maximum entropy production also seems to be applicable to the planetary atmospheres of Mars and Titan and perhaps to mantle convection. Lorenz s conjecture on maximum generation of available potential energy is shown to be akin to this hypothesis with a few minor approximations. A possible mechanism by which turbulent fluid systems adjust themselves to the states of maximum entropy production is presented as a selffeedback mechanism for the generation of available potential energy. These results tend to support the hypothesis of maximum entropy production that underlies a wide variety of nonlinear fluid systems, including our planet as well as other planets and stars

States of maximum entropy production in a onedimensional vertical model with convective adjustment

We investigate the hypothesis that the atmosphere is constrained to maximize its entropy production by using a one-dimensional (1-D) vertical model. We prescribe the lapse rate in the convective layer as that of the standard troposphere. The assumption that convection sustains a critical lapse rate was absent in previous studies, which focused on the vertical distribution of climatic variables, since such a convective adjustment reduces the degrees of freedom of the system and may prevent the application of the maximum entropy production (MEP) principle. This is not the case in the radiative–convective model (RCM) developed here, since we accept a discontinuity of temperatures at the surface similar to that adopted in many RCMs. For current conditions, the MEP state gives a difference between the ground temperature and the air temperature at the surface ≈10 K. In comparison, conventional RCMs obtain a discontinuity ≈2 K only. However, the surface boundary layer velocity in the MEP state appears reasonable (≈3 m s-¹). Moreover, although the convective flux at the surface in MEP states is almost uniform in optically thick atmospheres, it reaches a maximum value for an optical thickness similar to current conditions. This additional result may support the maximum convection hypothesis suggested by Paltridge (1978)

A fast Bayesian reconstruction algorithm for emission tomography with entropy prior converging to feasible images

The development and tests of an iterative reconstruction algorithm for emission tomography based on Bayesian statistical concepts are described. The algorithm uses the entropy of the generated image as a prior distribution, can be accelerated by the choice of an exponent, and converges uniformly to feasible images by the choice of one adjustable parameter. A feasible image has been defined as one that is consistent with the initial data (i.e. it is an image that, if truly a source of radiation in a patient, could have generated the initial data by the Poisson process that governs radioactive disintegration). The fundamental ideas of Bayesian reconstruction are discussed, along with the use of an entropy prior with an adjustable contrast parameter, the use of likelihood with data increment parameters as conditional probability, and the development of the new fast maximum a posteriori with entropy (FMAPE) Algorithm by the successive substitution method. It is shown that in the maximum likelihood estimator (MLE) and FMAPE algorithms, the only correct choice of initial image for the iterative procedure in the absence of a priori knowledge about the image configuration is a uniform field.

A new maximum likelihood method for luminosity calibrations

A new statistical parallax method using the Maximum Likelihood principle is presented, allowing the simultaneous determination of a luminosity calibration, kinematic characteristics and spatial distribution of a given sample. This method has been developed for the exploitation of the Hipparcos data and presents several improvements with respect to the previous ones: the effects of the selection of the sample, the observational errors, the galactic rotation and the interstellar absorption are taken into account as an intrinsic part of the formulation (as opposed to external corrections). Furthermore, the method is able to identify and characterize physically distinct groups in inhomogeneous samples, thus avoiding biases due to unidentified components. Moreover, the implementation used by the authors is based on the extensive use of numerical methods, so avoiding the need for simplification of the equations and thus the bias they could introduce. Several examples of application using simulated samples are presented, to be followed by applications to real samples in forthcoming articles.

Statistical analysis of maximum likelihood estimator images of human brain FDG PET studies

The work presented evaluates the statistical characteristics of regional bias and expected error in reconstructions of real positron emission tomography (PET) data of human brain fluoro-deoxiglucose (FDG) studies carried out by the maximum likelihood estimator (MLE) method with a robust stopping rule, and compares them with the results of filtered backprojection (FBP) reconstructions and with the method of sieves. The task of evaluating radioisotope uptake in regions-of-interest (ROIs) is investigated. An assessment of bias and variance in uptake measurements is carried out with simulated data. Then, by using three different transition matrices with different degrees of accuracy and a components of variance model for statistical analysis, it is shown that the characteristics obtained from real human FDG brain data are consistent with the results of the simulation studies.