initial layer for the homogenization of a conservation law with vanishing viscosity we study the limit as e 0 of the solutions of the equation partial t u varepsilon rm div x left a left frac x varepsilon u varepsilon right right varepsilon delta x u varepsilon 0 this problem has already been addressed in a previous article in the case of well prepared initial data i e when the microscopic profile of the solution is adapted to the medium at time t 0 here we prove that when the initial data is not well prepared there is an initial layer during which the solution adapts itself to match the profile dictated by the environment the typical size of the initial layer is of order e the proof relies strongly on the parabolic form of the equation in particular no condition of nonlinearity on a is required

homogenization of non linear scalar conservation laws we study the limit as e 0 of the entropy solutions of the equation partial t u varepsilon rm div x left a left frac x varepsilon u varepsilon right right 0 we prove that the sequence ue two scale converges toward a function u t x y and u is the unique solution of a limit evolution problem the remarkable point is that the limit problem is not a scalar conservation law but rather a kinetic equation in which the macroscopic and microscopic variables are mixed we also prove a strong convergence result in l rm loc 1

homogenization of linear transport equations with oscillatory vector fields the l 2 weak limit as varepsilon downarrow 0 of the solutions of the linear transport equation is studied begin gathered u t bf a left frac x varepsilon right cdot nabla x u 0 u t 0 u 0 left x frac x varepsilon right end gathered where varepsilon 0 x in r 2 t in 0 t for any t 0 bf a is a smooth divergence free vector field on the two dimensional unit torus and has no equilibrium points the initial condition u 0 is assumed to be smooth compactly supported in x and periodic in x varepsilon using a weak l 2 formulation and the idea of rotation number for flows on the two dimensional torus the effective equations for the weak limit are found they depend sensitively on the underlying flow on the two dimensional torus and its ergodicity

homogenization of scalar conservation laws with oscillatory forcing terms the homogenization of a scalar conservation law with a highly oscillatory forcing term is studied effective equations are derived for the local averages of the oscillatory solutions together with approximations that correctly represent the phase and the amplitude of the oscillations a random choice method is also designed which as demonstrated by our numerical results gives the correct local averages without necessarily resolving the small scales finally the homogenization problem with an additional small viscosity term is studied

goodness of fit and related inference processes for quantile regression abstract we introduce a goodness of fit process for quantile regression analogous to the conventional r2 statistic of least squares regression several related inference processes designed to test composite hypotheses about the combined effect of several covariates over an entire range of conditional quantile functions are also formulated the asymptotic behavior of the inference processes is shown to be closely related to earlier p sample goodness of fit theory involving bessel processes the approach is illustrated with some hypothetical examples an application to recent empirical models of international economic growth and some monte carlo evidence

statistical methods for eliciting probability distributions elicitation is a key task for subjectivist bayesians although skeptics hold that elicitation cannot or perhaps should not be done in practice it brings statisticians closer to their clients and subject matter expert colleagues this article reviews the state of the art reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information in a discussion of the elicitation process the first issue to address is what it means for an elicitation to be successful that is what criteria should be used our answer is that a successful elicitation faithfully represents the opinion of the person being elicited it is not necessarily true in some objectivistic sense and cannot be judged in that way we see that elicitation as simply part of the process of statistical modeling indeed in a hierarchical model at which point the likelihood ends and the pr

modeling tangential vector fields on a sphere abstractphysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences these naturally occurring vector fields are often subject to physical constraints such as being curl free or divergence free we start with constructing parametric models for curl free and divergence free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere using the helmholtz hodge decomposition we then construct a class of simple but flexible parametric models for general tangential vector fields which are represented as a sum of a curl free and a divergence free components we propose a likelihood based parameter estimation procedure and show that fast computation is possible even for large data sets when the observations are on a regular latitude longitude grid characteristics and practical utility of the proposed methodology are illustrated th

evaluating probabilities asymmetric scoring rules proper scoring rules are over evaluation measures that reward accurate probabilities specific rules encountered in the literature and used in practice are invariably symmetric in the sense that the expected score for a perfectly calibrated probability assessor or model generating probabilities is minimized at a probability of one half a family of asymmetric scoring rules that provide better measures of the degree of skill inherent in the probabilities and render scores that are more comparable in different situations is developed here one member of this family a quadratic asymmetric rule is applied to evaluate an extensive set of precipitation probability forecasts from the u s national weather service connections to previous characterizations of proper scoring rules are investigated and some relevant issues pertaining to the design of specific asymmetric rules for particular inferential and decision making problems are discussed briefly

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

matrix free subcell residual distribution for bernstein finite element discretizations of linear advection equations abstract in this work we introduce a new residual distribution rd framework for the design of bound preserving high resolution finite element schemes the continuous and discontinuous galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing led approximations to that end we perform mass lumping and redistribute the element residuals in a manner which guarantees the led property the hierarchical correction procedure for high order bernstein finite element discretizations involves localization to subcells and definition of bound preserving weights for subcell contributions using strong stability preserving ssp runge kutta methods for time integration we prove the validity of discrete maximum principles under cfl like time step restrictions the low order version of our method has roughly the same accuracy as the one derived from a piecewise multi linear approximation on a submesh with the same nodal points in high order extensions we use an element based flux corrected transport fct algorithm which can be interpreted as a nonlinear rd scheme the proposed led corrections are tailor made for matrix free implementations which avoid the rapidly growing cost of matrix assembly for high order bernstein elements the results for 1d 2d and 3d test problems compare favorably to those obtained with the best matrix based approaches

differentiable monotonicity preserving schemes for discontinuous galerkin methods on arbitrary meshes this work is devoted to the design of interior penalty discontinuous galerkin dg schemes that preserve maximum principles at the discrete level for the steady transport and convection diffusion problems and the respective transient problems with implicit time integration monotonic schemes that combine explicit time stepping with dg space discretization are very common but the design of such schemes for implicit time stepping is rare and it had only been attained so far for 1d problems the proposed scheme is based on a piecewise linear dg discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas in order to define the new shock detector we have introduced the concept of discrete local extrema the diffusion operator is a graph laplacian instead of the more common finite element discretization of the laplacian operator which is essential to keep monotonicity on general meshes and in multi dimension the resulting nonlinear stabilization is non smooth and nonlinear solvers can fail to converge as a result we propose a smoothed twice differentiable version of the nonlinear stabilization which allows us to use newton with line search nonlinear solvers and dramatically improve nonlinear convergence a theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties further the resulting operator is lipschitz continuous and there exists at least one solution of the discrete problem even in the non smooth version we provide a set of numerical results to support our findings

powering up with space time wind forecasting the technology to harvest electricity from wind energy is now advanced enough to make entire cities powered by it a reality high quality short term forecasts of wind speed are vital to making this a more reliable energy source gneiting et al 2006 have introduced a model for the average wind speed two hours ahead based on both spatial and temporal information the forecasts produced by this model are accurate and subject to accuracy the predictive distribution is sharp that is highly concentrated around its center however this model is split into nonunique regimes based on the wind direction at an offsite location this paper both generalizes and improves upon this model by treating wind direction as a circular variable and including it in the model it is robust in many experiments such as predicting wind at other locations we compare this with the more common approach of modeling wind speeds and directions in the cartesian space and use a skew t distribution for the errors the quality of t

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

a new probabilistic approach in rank regression with optimal bayesian partitioning in this paper we consider the supervised learning task which consists in predicting the normalized rank of a numerical variable we introduce a novel probabilistic approach to estimate the posterior distribution of the target rank conditionally to the predictors we turn this learning task into a model selection problem for that we define a 2d partitioning family obtained by discretizing numerical variables and grouping categorical ones and we derive an analytical criterion to select the partition with the highest posterior probability we show how these partitions can be used to build univariate predictors and multivariate ones under a naive bayes assumption we also propose a new evaluation criterion for probabilistic rank estimators based on the logarithmic score we show that such criterion presents the advantage to be minored which is not the case of the logarithmic score computed for probabilistic value estimator a first set of experimentations on synthetic data shows the good properties of the proposed criterion and of our partitioning approach a second set of experimentations on real data shows competitive performance of the univariate and selective naive bayes rank estimators projected on the value range compared to methods submitted to a recent challenge on probabilistic metric regression tasks our approach is applicable for all regression problems with categorical or numerical predictors it is particularly interesting for those with a high number of predictors as it automatically detects the variables which contain predictive information it builds pertinent predictors of the normalized rank of the numerical target from one or several predictors as the criteria selection is regularized by the presence of a prior and a posterior term it does not suffer from overfitting

bayesian calibration of inexact computer models bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias the parameter in the predominant methodology is left undefined this results in an issue where the posterior of the parameter is suboptimally broad there has been no generally accepted alternatives to date this article proposes using bayesian calibration where the prior distribution on the bias is orthogonal to the gradient of the computer model problems associated with bayesian calibration are shown to be mitigated through analytic results in addition to examples supplementary materials for this article are available online

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

non gaussian spatiotemporal modelling through scale mixing we construct non gaussian processes that vary continuously in space and time with nonseparable covariance functions starting from a general and flexible way of constructing valid nonseparable covariance functions through mixing over separable covariance functions the resulting models are generalized by allowing for outliers as well as regions with larger variances we induce this through scale mixing with separate positive valued processes smooth mixing processes are applied to the underlying correlated processes in space and in time thus leading to regions in space and time of increased spread an uncorrelated mixing process on the nugget effect accommodates outliers posterior and predictive bayesian inference with these models is implemented through a markov chain monte carlo sampler an application to temperature data in the basque country illustrates the potential of this model in the identification of outliers and regions with inflated variance and shows that this improves the predictive performance

multivariate type g matern stochastic partial differential equation random fields for many applications with multivariate data random field models capturing departures from gaussianity within realizations are appropriate for this reason we formulate a new class of multivariate non gaussian models based on systems of stochastic partial differential equations with additive type g noise whose marginal covariance functions are of matern type we consider four increasingly flexible constructions of the noise where the first two are similar to existing copula based models in contrast with these the last two constructions can model non gaussian spatial data without replicates computationally efficient methods for likelihood based parameter estimation and probabilistic prediction are proposed and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications less

a view of margin losses as regularizers of probability estimates regularization is commonly used in classifier design to assure good generalization classical regularization enforces a cost on classifier complexity by constraining parameters this is usually combined with a margin loss which favors large margin decision rules a novel and unified view of this architecture is proposed by showing that margin losses act as regularizers of posterior class probabilities in a way that amplifies classical parameter regularization the problem of controlling the regularization strength of a margin loss is considered using a decomposition of the loss in terms of a link and a binding function the link function is shown to be responsible for the regularization strength of the loss while the binding function determines its outlier robustness a large class of losses is then categorized into equivalence classes of identical regularization strength or outlier robustness it is shown that losses in the same regularization class can be parameterized so as to have tunable regularization strength this parameterization is finally used to derive boosting algorithms with loss regularization boostlr three classes of tunable regularization losses are considered in detail canonical losses can implement all regularization behaviors but have no flexibility in terms of outlier modeling shrinkage losses support equally parameterized link and binding functions leading to boosting algorithms that implement the popular shrinkage procedure this offers a new explanation for shrinkage as a special case of loss based regularization finally tunable losses enable the independent parameterization of link and binding functions leading to boosting algorithms of great exibility this is illustrated by the derivation of an algorithm that generalizes both adaboost and logitboost behaving as either one when that best suits the data to classify various experiments provide evidence of the benefits of probability regularization for both classification and estimation of posterior class probabilities

a general method for comparing probability assessors a probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur in this paper a general method for comparing forecasters after a finite number of trials is introduced the general method is proven to include calculating all proper scoring rules as special cases it also includes comparison of forecasters in all simple two decision problems as special cases the relationship between the general method and calibration is also explored the general method is also translated into a method for deciding who will give better forecasts in the future

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

time varying combinations of predictive densities using nonlinear filtering abstract we propose a bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights several specifications of multivariate time varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms in the proposed approach the model set can be incomplete meaning that all models can be individually misspecified a sequential monte carlo method is proposed to approximate the filtering and predictive densities the combination approach is assessed using statistical and utility based performance measures for evaluating density forecasts of simulated data us macroeconomic time series and surveys of stock market prices simulation results indicate that for a set of linear autoregressive models the combination strategy is successful in selecting with probability close to one the true model when the model set is complete and it is able to detect parameter instability when the model set includes the true model that has generated subsamples of data also substantial uncertainty appears in the weights when predictors are similar residual uncertainty reduces when the model set is complete and learning reduces this uncertainty for the macro series we find that incompleteness of the models is relatively large in the 1970 s the beginning of the 1980 s and during the recent financial crisis and lower during the great moderation the predicted probabilities of recession accurately compare with the nber business cycle dating model weights have substantial uncertainty attached with respect to returns of the s p 500 series we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 1990 s and switches to giving more weight to the professional forecasts over time information on the complete predictive distribution and not just on some moments turns out to be very important above all during turbulent times such as the recent financial crisis more generally the proposed distributional state space representation offers great flexibility in combining densities

uniqueness for scalar conservation laws with discontinuous flux via adapted entropies we prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes to do so we introduce a partial adaptation of kruzkov s entropies which naturally takes into account the space dependency of the flux the advantage of this approach is that the proof turns out to be a simple variant of kruzkov s original method especially we do not need traces interface condition bounded variation assumptions neither on the solution nor on the flux or convex fluxes however we use a special local uniform invertibility structure of the flux which applies to cases where different interface condiftions are known to yield different solutions

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

on the optimality of conditional expectation as a bregman predictor we consider the problem of predicting a random variable x from observations denoted by a random variable z it is well known that the conditional expectation e x z is the optimal l sup 2 predictor also known as the least mean square error predictor of x among all borel measurable functions of z in this orrespondence we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor we show that e x z is the optimal predictor for all bregman loss functions blfs of which the l sup 2 loss function is a special case moreover under mild conditions we show that the blfs are exhaustive i e if for every random variable x the infimum of e f x y over all constants y is attained by the expectation e x then f is a blf

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

multivariate postprocessing techniques for probabilistic hydrological forecasting hydrologic ensemble forecasts driven by atmospheric ensemble prediction systems need statistical postprocessing in order to account for systematic errors in terms of both location and spread runoff is an inherently multivariate process with typical events lasting from hours in case of floods to weeks or even months in case of droughts this calls for multivariate postprocessing techniques that yield well calibrated forecasts in univariate terms and ensure a realistic temporal dependence structure at the same time to this end the univariate ensemble model output statistics emos postprocessing method is combined with two different copula approaches that ensure multivariate calibration throughout the entire forecast horizon the domain of this study covers three subcatchments of the river rhine that represent different sizes and hydrological regimes the upper rhine up to the gauge maxau the river moselle up to the gauge trier and the river lahn up to the gauge kalkofen in this study the two approaches to model the temporal dependence structure are ensemble copula coupling ecc which preserves the dependence structure of the raw ensemble and a gaussian copula approach gca which estimates the temporal correlations from training observations the results indicate that both methods are suitable for modeling the temporal dependencies of probabilistic hydrologic forecasts

weather forecasting for weather derivatives we take a simple time series approach to modeling and forecasting daily average temperature in u s cities and we inquire systematically as to whether it may prove useful from the vantage point of participants in the weather derivatives market the answer is perhaps surprisingly yes time series modeling reveals conditional mean dynamics and crucially strong conditional variance dynamics in daily average temperature and it reveals sharp differences between the distribution of temperature and the distribution of temperature surprises as we argue it also holds promise for producing the long horizon predictive densities crucial for pricing weather derivatives so that additional inquiry into time series weather forecasting methods will likely prove useful in weather derivatives contexts

a valid matern class of cross covariance functions for multivariate random fields with any number of components we introduce a valid parametric family of cross covariance functions for multivariate spatial random fields where each component has a covariance function from a well celebrated matern class unlike previous attempts our model indeed allows for various smoothnesses and rates of correlation decay for any number of vector components we present the conditions on the parameter space that result in valid models with varying degrees of complexity we discuss practical implementations including reparameterizations to reflect the conditions on the parameter space and an iterative algorithm to increase the computational efficiency we perform various monte carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging the application of the proposed multivariate matern model is illustrated on two meteorological datasets temperature pressure over the pacific northwest bivariate and wind temperature pressure in oklahoma trivariate in the latter case our flexi

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

composite binary losses we study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function we characterise when margin losses can be proper composite losses explicitly show how to determine a symmetric loss in full from half of one of its partial losses introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and classification calibrated losses we also consider the question of the best surrogate binary loss we introduce a precise notion of best and show there exist situations where two convex surrogate losses are incommensurable we provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss this characterisation suggests new ways of surrogate tuning as well as providing an explicit characterisation of when bregman divergences on the unit interval are convex in their second argument finally in an appendix we present some new algorithm independent results on the relationship between properness convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non robust to misclassification noise

staggered grid residual distribution scheme for lagrangian hydrodynamics this paper is focused on the residual distribution rd interpretation of the dobrev kolev and rieben scheme siam j sci comput 34 2012 pp b606 b641 for the numerical solution of the euler equations in lagrangian form the first ingredient of the original scheme is the staggered grid formulation which uses continuous node based finite element approximations for the kinematic variables and cell centered discontinuous finite elements for the thermodynamic parameters the second ingredient of the dobrev et al scheme is an artificial viscosity technique applied in order to make possible the computation of strong discontinuities the aim of this paper is to provide an efficient mass matrix diagonalization method in order to avoid the inversion of the global sparse mass matrix while keeping all the accuracy properties and to construct a parameter free stabilization of the scheme to get rid of the artificial viscosity in addition we study the conservation and entropy properties of the constructed

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

large time behavior and homogenization of solutions of two dimensional conservation laws we study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data under a very weak nonlinearity condition we prove that the solutions converge to constants as time goes to infinity even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data we then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data

bregman divergence bounds and universality properties of the logarithmic loss a loss function measures the discrepancy between the true values and their estimated fits for a given instance of data in classification problems a loss function is said to be proper if a minimizer of the expected loss is the true underlying probability we show that for binary classification the divergence associated with smooth proper and convex loss functions is upper bounded by the kullback leibler kl divergence to within a normalization constant this implies that by minimizing the logarithmic loss associated with the kl divergence we minimize an upper bound to any choice of loss from this set as such the logarithmic loss is universal in the sense of providing performance guarantees with respect to a broad class of accuracy measures importantly this notion of universality is not problem specific enabling its use in diverse applications including predictive modeling data clustering and sample complexity analysis generalizations to arbitary finite alphabets are also developed the derived inequalities extend several well known f divergence results

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

mixtures of g priors for bayesian model averaging with economic application this paper examines the issue of variable selection in linear regression modeling where there is a potentially large amount of possible covariates and economic theory offers insufficient guidance on how to select the appropriate subset in this context bayesian model averaging presents a formal bayesian solution to dealing with model uncertainty the main interest here is the effect of the prior on the results such as posterior inclusion probabilities of regressors and predictive performance the authors combine a binomial beta prior on model size with a g prior on the coefficients of each model in addition they assign a hyperprior to g as the choice of g has been found to have a large impact on the results for the prior on g they examine the zellner siow prior and a class of beta shrinkage priors which covers most choices in the recent literature the authors propose a benchmark beta prior inspired by earlier findings with fixed g and show it leads to consistent model selection inference is conducted through a markov chain monte carlo sampler over model space and g the authors examine the performance of the various priors in the context of simulated and real data for the latter they consider two important applications in economics namely cross country growth regression and returns to schooling recommendations for applied users are provided

strict definiteness of integrals via complete monotonicity of derivatives let k be a nonnegative integer and let 0 r be a c function with k k completely monotone and not constant if 6 0 is a signed measure on any euclidean space rd with vanishing moments up to order k 1 then the integral rd rd x y 2 d x d y is strictly positive whenever it exists for general d no larger class of continuous functions seems to admit the same conclusion examples and applications are indicated a section on bilinear integrability might be of independent interest

of quantiles and expectiles consistent scoring functions choquet representations and forecast rankings summary in the practice of point prediction it is desirable that forecasters receive a directive in the form of a statistical functional for example forecasters might be asked to report the mean or a quantile of their predictive distributions when evaluating and comparing competing forecasts it is then critical that the scoring function used for these purposes be consistent for the functional at hand in the sense that the expected score is minimized when following the directive we show that any scoring function that is consistent for a quantile or an expectile functional can be represented as a mixture of elementary or extremal scoring functions that form a linearly parameterized family scoring functions for the mean value and probability forecasts of binary events constitute important examples the extremal scoring functions admit appealing economic interpretations of quantiles and expectiles in the context of betting and investment problems the choquet type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function in empirical settings it suffices to compare the average scores for only a finite number of extremal elements plots of the average scores with respect to the extremal scoring functions which we call murphy diagrams permit detailed comparisons of the relative merits of competing forecasts

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

high order multi material ale hydrodynamics we present a new approach for multi material arbitrary lagrangian eulerian ale hydrodynamics simulations based on high order finite elements posed on high order curvilinear meshes the method builds on and extends our previous work in the lagrangian v a dobrev t v kolev and r n rieben siam j sci comput 34 2012 pp b606 b641 and remap r w anderson et al internat j numer methods fluids 77 2015 pp 249 273 phases of ale and depends critically on a functional perspective that enables subzonal physics and material modeling v a dobrev et al internat j numer methods fluids 82 2016 pp 689 706 curvilinear mesh relaxation is based on node movement which is determined through the solution of an elliptic equation the remap phase is posed in terms of advecting state variables between two meshes over a fictitious time interval the resulting advection equation is solved by a discontinuous galerkin dg formulation combined with a customized flux corrected transpor

effective scoring rules for probabilistic forecasts this paper studies the use of a scoring rule for the elicitation of forecasts in the form of probability distributions and for the subsequent evaluation of such forecasts given a metric distance function on a space of probability distributions a scoring rule is said to be effective if the forecaster s expected score is a strictly decreasing function of the distance between the elicited and true distributions two simple well known rules the spherical and the quadratic are shown to be effective with respect to suitable metrics examples and a practical application in foreign exchange rate forecasting are also provided

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

on the consistency of ordinal regression methods many of the ordinal regression models that have been proposed in the literature can be seen as methods that minimize a convex surrogate of the zero one absolute or squared loss functions a key property that allows to study the statistical implications of such approximations is that of fisher consistency fisher consistency is a desirable property for surrogate loss functions and implies that in the population setting i e if the probability distribution that generates the data were available then optimization of the surrogate would yield the best possible model in this paper we will characterize the fisher consistency of a rich family of surrogate loss functions used in the context of ordinal regression including support vector ordinal regression orboosting and least absolute deviation we will see that for a family of surrogate loss functions that subsumes support vector ordinal regression and orboosting consistency can be fully characterized by the derivative of a real valued function at zero as happens for convex margin based surrogates in binary classification we also derive excess risk bounds for a surrogate of the absolute error that generalize existing risk bounds for binary classification finally our analysis suggests a novel surrogate of the squared error loss we compare this novel surrogate with competing approaches on 9 different datasets our method shows to be highly competitive in practice outperforming the least squares loss on 7 out of 9 datasets

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

should scoring rules be effective a scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions a rule is strictly proper if it encourages the forecaster to state his true subjective probabilities and effective if it is associated with a metric on the set of probability distributions recently the property of effectiveness which is stronger than strict properness has been proposed as a desideratum for scoring rules for continuous forecasts for reasons of monotonicity in keeping the forecaster close to his true probabilities since in practice the forecast must be chosen from a low dimensional set of admissible distributions it is shown in this paper that what effectiveness implies beyond strict properness is not a monotonicity property but a transitivity property which is difficult to justify behaviorally the logarithmic scoring rule is shown to violate the transitivity property and hence is not effective the l1 and l metrics are shown to allow no effective scoring rules some potential difficulties in interpreting admissible forecasts are also discussed

analysis of algebraic flux correction schemes a family of algebraic flux correction afc schemes for linear boundary value problems in any space dimension is studied these methods main feature is that they limit the fluxes along each one of the edges of the triangulation and we suppose that the limiters used are symmetric for an abstract problem the existence of a solution existence and uniqueness of the solution of a linearized problem and an a priori error estimate are proved under rather general assumptions on the limiters for a particular but standard in practice choice of the limiters it is shown that a local discrete maximum principle holds the theory developed for the abstract problem is applied to convection diffusion reaction equations where in particular an error estimate is derived numerical studies show its sharpness

discrimination slope and integrated discrimination improvement properties relationships and impact of calibration discrimination slope defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status has recently gained popularity as a measure of model performance it is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis calibrated models whereas others have raised concerns about the ability of improving discrimination slope without adding new information in this paper we show that under certain assumptions the discrimination slope is asymptotically related to two other r squared measures one of which is a rescaled version of the brier score known to be proper furthermore we illustrate how a simple recalibration makes the slope equal to the rescaled brier r squared metric we also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the gini brier regret function using theoretical and practical examples we illustrate how all of these metrics are affected by different levels of model mis calibration in particular we demonstrate that simple recalibration ascertaining calibration in the large and calibration slope equal to 1 are not sufficient to correct for some forms of mis calibration we conclude that r squared metrics including the discrimination slope offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration copyright 2016 john wiley sons ltd

conditional transformation models the ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables this goal is however seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not aected by the regressors the underlying reason for such a restriction is the assumption of additivity of signal and noise we propose to relax this common assumption in the framework of transformation models the novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables these transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts e g the continuous ranked probability score the corresponding estimated conditional distribution functions are consistent based on applications from dierent domains we show that these conditional transformation models are useful for describing possible heteroscedasticity comparing spatially varying distributions identifying extreme events deriving prediction intervals and selecting variables beyond mean regression eects an empirical investigation based on a heteroscedastic varying coecient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more benecial than kernel based non parametric approaches or parametric generalised additive models for location scale and shape

flux corrected transport algorithms for continuous galerkin methods based on high order bernstein finite elements abstract this work extends the flux corrected transport fct methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes using bernstein polynomials as local basis functions we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the bezier net the design of accuracy preserving fct schemes for high order bernstein bezier finite elements requires the development of new algorithms and or generalization of limiting techniques tailored for linear and multilinear lagrange elements in this paper we propose i a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils ii high order variational stabilization based on the difference between two gradient approximations and iii new localized limiting techniques for antidiffusive element contributions the optional use of a smoothness indicator based on a second derivative test makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions the accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1d and 2d

estimating value at risk and expected shortfall using the intraday low and range data abstract value at risk var is a popular measure of market risk to convey information regarding potential exceedances beyond the var expected shortfall es has become the risk measure for trading book bank regulation however the estimation of var and es is challenging as it requires the estimation of the tail behaviour of daily returns in this paper we take advantage of recent research that develops joint scoring functions for var and es using these functions we present a novel approach to estimating the two risk measures based on intraday data we focus on the intraday range which is the difference between the highest and lowest intraday log prices in contrast to intraday observations the intraday low and high are widely available for many financial assets to alleviate the challenge of modelling extreme risk measures we propose the use of the intraday low series we draw on a theoretical result for brownian motion to show that a quantile of the daily returns can be estimated as the product of a constant term and a less extreme quantile of the intraday low returns which we define as the difference between the lowest log price of the day and the log closing price of the previous day in view of this we use estimates of the var and es of the intraday low returns to estimate the var and es of the daily returns we provide empirical support for the new proposals using data for five stock indices and five individual stocks

monotonicity preserving finite element schemes based on differentiable nonlinear stabilization in this work we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping the method relies on an artificial diffusion method based on a graph laplacian operator it is nonlinear since it depends on a shock detector further the resulting method is linearity preserving the same shock detector is used to gradually lump the mass matrix the resulting method is led positivity preserving and also satisfies a global dmp lipschitz continuity has also been proved however the resulting scheme is highly nonlinear leading to very poor nonlinear convergence rates we propose a smooth version of the scheme which leads to twice differentiable nonlinear stabilization schemes it allows one to straightforwardly use newton s method and obtain quadratic convergence in the numerical experiments steady and transient linear transport and transient burgers equation have been considered in 2d using the newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of anderson acceleration with the original non smooth scheme in any case these properties are only true for the converged solution but not for iterates in this sense we have also proposed the concept of projected nonlinear solvers where a projection step is performed at the end of every nonlinear iterations onto a fe space of admissible solutions the space of admissible solutions is the one that satisfies the desired monotonic properties maximum principle or positivity

a partition of unity approach to adaptivity and limiting in continuous finite element methods abstract the partition of unity finite element method pufem proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner the lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures in contrast to traditional h p adaptivity for finite elements the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells in the context of space discretizations a continuous blending function is used to combine finite element bases corresponding to high order polynomials and piecewise linear approximations based on the same set of nodes in a similar vein spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable the design of blending functions may be based on a priori knowledge e g in applications to problems with singularities or boundary layers local error estimates smoothness indicators and or discrete maximum principles in adaptive methods changes of the finite element approximation exhibit continuous dependence on the data the presented numerical examples illustrate the typical behavior of local h 1 and l 2 errors

embedded discontinuous galerkin transport schemes with localised limiters motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction we introduce locally bounded transport schemes for partially continuous finite element spaces the underlying high order transport scheme is constructed by injecting the partially continuous field into an embedding discontinuous finite element space applying a stable upwind discontinuous galerkin dg scheme and projecting back into the partially continuous space we call this an embedded dg transport scheme we prove that this scheme is stable in l 2 provided that the underlying upwind dg scheme is we then provide a framework for applying limiters for embedded dg transport schemes standard dg limiters are applied during the underlying dg scheme we introduce a new localised form of element based flux correction which we apply to limiting the projection back into the partially continuous space so that the whole transport scheme is bounded we provide details in the specific case of tensor product finite element spaces on wedge elements that are discontinuous p1 q1 in the horizontal and continuous p2 in the vertical the framework is illustrated with numerical tests

high order local maximum principle preserving mpp discontinuous galerkin finite element method for the transport equation abstract in this work we present a fct like maximum principle preserving mpp method to solve the transport equation we use high order polynomial spaces in particular we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension the method combines the concepts of positive basis functions for discontinuous galerkin finite element spatial discretization locally defined solution bounds element based flux correction and non linear local mass redistribution we consider a simple 1d problem with non smooth initial data to explain and understand the behavior of different parts of the method convergence tests in space indicate that high order accuracy is achieved numerical results from several benchmarks in two and three dimensions are also reported

probabilistic coherence and proper scoring rules this paper provides self contained proof of a theorem relating probabilistic coherence of forecasts to their non domination by rival forecasts with respect to any proper scoring rule the theorem recapitulates insights achieved by other investigators and clarifies the connection of coherence and proper scoring rules to bregman divergence

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

subcell flux limiting for high order bernstein finite element discretizations of scalar hyperbolic conservation laws abstract this work extends the concepts of algebraic flux correction and convex limiting to continuous high order bernstein finite element discretizations of scalar hyperbolic problems using an array of adjustable diffusive fluxes the standard galerkin approximation is transformed into a nonlinear high resolution scheme which has the compact sparsity pattern of the piecewise linear or multilinear subcell discretization the representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under cfl like conditions in contrast to predictor corrector approaches based on the flux corrected transport methodology the proposed flux limiting strategy is monolithic i e limited antidiffusive terms are incorporated into the well defined residual of a nonlinear semi discrete problem a stabilized high order galerkin discretization is recovered if no limiting is performed in the limited version the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors a formal proof of sparsity is provided for simplicial and box elements the involved element contributions can be calculated efficiently making use of matrix free algorithms and precomputed element matrices of the reference element numerical studies for q 2 discretizations of linear and nonlinear two dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions

bayesian generalized additive models for location scale and shape for zero inflated and overdispersed count data frequent problems in applied research preventing the application of the classical poisson log linear model for analyzing count data include overdispersion an excess of zeros compared to the poisson distribution correlated responses as well as complex predictor structures comprising nonlinear effects of continuous covariates interactions or spatial effects we propose a general class of bayesian generalized additive models for zero inflated and overdispersed count data within the framework of generalized additive models for location scale and shape where semiparametric predictors can be specified for several parameters of a count data distribution as standard options for applied work we consider the zero inflated poisson the negative binomial and the zero inflated negative binomial distribution the additive predictor specifications rely on basis function approximations for the different types of effects in combination with gaussian smoothness priors we develop bayesian inference based on markov c

bayesian leave one out cross validation approximations for gaussian latent variable models the future predictive performance of a bayesian model can be estimated using bayesian cross validation in this article we consider gaussian latent variable models where the integration over the latent values is approximated using the laplace method or expectation propagation ep we study the properties of several bayesian leave one out loo cross validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data our main objective is to assess the accuracy of the approximative loo cross validation estimators that is for each method laplace and ep we compare the approximate fast computation with the exact brute force loo computation secondarily we evaluate the accuracy of the laplace and ep approximations themselves against a ground truth established through extensive markov chain monte carlo simulation our empirical results show that the approach based upon a gaussian approximation to the loo marginal distribution the so called cavity distribution gives the most accurate and reliable results among the fast methods

spectral density estimation for random fields via periodic embeddings we introduce methods for estimating the spectral density of a random field on a formula see text dimensional lattice from incomplete gridded data data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function the imputations are convenient computationally in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in formula see text time and formula see text memory however these so called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates in addition we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias the paper contains theoretical results on properties of the imputed data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios we present an application to a gridded satellite surface temperature dataset with missing values

asymptotically optimal prediction for time varying data generating processes we develop a methodology referred to as kinetic prediction for predicting time series undergoing unknown changes in their data generating distributions based on kolmogorov tikhomirov s varepsilon entropy we propose a concept called varepsilon predictability that quantifies the size of a model class which can be parametric or nonparametric and the maximal number of abrupt structural changes that guarantee the achievability of asymptotically optimal prediction moreover for parametric distribution families we extend the aforementioned kinetic prediction with discretized function spaces to its counterpart with continuous function spaces and propose a sequential monte carlo based implementation we also extend our methodology for predicting smoothly varying data generating distributions under reasonable assumptions we prove that the average predictive performance converges almost surely to the oracle bound which corresponds to the case that the data generating distributions are known in advance the results also shed some light on the so called prediction inference dilemma various examples and numerical results are provided to demonstrate the wide applicability of our methodology

probabilistic forecasts calibration and sharpness summary probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions we propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only a simple theoretical framework allows us to distinguish between probabilistic calibration exceedance calibration and marginal calibration we propose and study tools for checking calibration and sharpness among them the probability integral transform histogram marginal calibration plots the sharpness diagram and proper scoring rules the diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the stateline wind energy centre in the us pacific northwest in combination with cross validation or in the time series context our proposal provides very general nonparametric alternatives to the use of information criteria for model diagnostics and model selection

a kinetic equation with kinetic entropy functions for scalar conservation laws we construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws in particular we prove that it is well posed uniformly in e the microscopic scale we also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to boltzmann s microscopich function such that they recover krushkov type entropy inequality on the macroscopic scale finally we prove by both bv compactness arguments in the multidimensional case and by compensated compactness arguments in the one dimensional case that the local density of kinetic particles admits a continuum limit as it converges strongly with e 0 to the unique entropy solution of the corresponding conservation law

economic interpretation of probabilities estimated by maximum likelihood or score the conventional method of estimating a probability prediction model by maximum likelihood mle is a form of maximum score estimation with economic meaning of all the probabilities that a given model might have produced those obtained by mle yield maximum in sample betting return to a log utility investor recognition of this affinity between mle and log utility begs the wider methodological question of whether different decision makers benefit in different degrees from different probabilities probabilities produced by mle can be either too conservative or too bold relative to those found by maximizing utility under more risk tolerant or risk averse score functions a very not very risk averse user who bets characteristically small large fractions of wealth based on a conservative forecast is bound to make a rapidly slowly increasing bet as the forecast probability becomes progressively bolder or more distant from the market probability the effect of this interaction between risk aversion and forecast is that a highly risk averse user may need a much bolder forecast to obtain the same certainty equivalent as a more risk tolerant investor it follows more broadly that professional forecasters should anticipate how a client with given risk aversion expects to gain from any given forecast or forecast revision before committing resources toward making a better informed but still honest forecast this paper was accepted by peter wakker decision analysis