identification and estimation of statistical functionals using incomplete data incomplete data due to missing observations or interval measurement of variables usually cause parameters of interest in applications to be unidentified except under untestable and often controversial assumptions however it is often possible to identify sharp bounds on parameters without making untestable assumptions about the process through which data become incomplete the bounds contain all logically possible values of the parameters and can be estimated consistently by replacing the population distribution of the data with the empirical distribution this is straightforward in some circumstances but computationally burdensome in others this paper describes the general problem and presents an empirical illustration

estimation of tail risk based on extreme expectiles we use tail expectiles to estimate value at risk var expected shortfall es and marginal expected shortfall mes three instruments of risk protection of utmost importance in actuarial science and statistical finance the concept of expectiles is a least squares analogue of quantiles both expectiles and quantiles were embedded in the more general class of m quantiles as the minimizers of an asymmetric convex loss function it has been proved very recently that the only m quantiles that are coherent risk measures are the expectiles moreover expectiles define the only coherent risk measure that is also elicitable the elicitability corresponds to the existence of a natural backtesting methodology the estimation of expectiles did not however receive yet any attention from the perspective of extreme values the first estimation method that we propose enables the usage of advanced high quantile and tail index estimators the second method joins together the least asymmetrically weighted squares estimation with the tail restrictions of extreme value theory we establish the limit distributions of the proposed estimators when they are located in the range of the data or near and even beyond the maximum observed loss a main tool is to first estimate the intermediate large expectile based var es and mes and then extrapolate these estimates to the very far tails we show through a detailed simulation study the good performance of the procedures and also present concrete applications to medical insurance data and three large us investment banks

on the early exercise boundary of the american put option we study the short time behavior of the early exercise boundary for american style put options in the black scholes theory we develop an asymptotic expansion which shows that the simple lower bound of barles et al is a more accurate approximation to the actual boundary than the more complex upper bound our expansion is obtained through iteration using a boundary integral equation this integral equation is derived from the time derivative of the option value function which closely resembles the classical stefan free boundary value problem for melting ice our analytical results are supported by numerical computations designed for very short times

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

penalized bregman divergence for large dimensional regression and classification regularization methods are characterized by loss functions measuring data fits and penalty terms constraining model parameters the commonly used quadratic loss is not suitable for classification with binary responses whereas the loglikelihood function is not readily applicable to models where the exact distribution of observations is unknown or not fully specified we introduce the penalized bregman divergence by replacing the negative loglikelihood in the conventional penalized likelihood with bregman divergence which encompasses many commonly used loss functions in the regression analysis classification procedures and machine learning literature we investigate new statistical properties of the resulting class of estimators with the number pn of parameters either diverging with the sample size n or even nearly comparable with n and develop statistical inference tools it is shown that the resulting penalized estimator combined with appropriate penalties achieves the same oracle property as the penalized likelihood estimator but asymptotically does not rely on the complete specification of the underlying distribution furthermore the choice of loss function in the penalized classifiers has an asymptotically relatively negligible impact on classification performance we illustrate the proposed method for quasilikelihood regression and binary classification with simulation evaluation and real data application

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

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

unbiasedness as the dual of being bayes in this note we define the notion of unbiasedness for a decision function for an arbitrary loss function this is a generalization of lehmann s 1951 definition we show that this notion of unbiasedness is a dual to the notion of being bayes that is if the role of the random variable and the parameter is interchanged then unbiasedness is equivalent to being bayes and vice versa some consequences of this fact are discussed

making and evaluating point forecasts typically point forecasting methods are compared and assessed by means of an error measure or scoring function with the absolute error and the squared error being key examples the individual scores are averaged over forecast cases to result in a summary measure of the predictive performance such as the mean absolute error or the mean squared error i demonstrate that this common practice can lead to grossly misguided inferences unless the scoring function and the forecasting task are carefully matched effective point forecasting requires that the scoring function be specified ex ante or that the forecaster receives a directive in the form of a statistical functional such as the mean or a quantile of the predictive distribution if the scoring function is specified ex ante the forecaster can issue the optimal point forecast namely the bayes rule if the forecaster receives a directive in the form of a functional it is critical that the scoring function be consistent for it in the sense that t

a fast high order finite difference algorithm for pricing american options we describe an improvement of han and wu s algorithm h han x wu a fast numerical method for the black scholes equation of american options siam j numer anal 41 6 2003 2081 2095 for american options a high order optimal compact scheme is used to discretise the transformed black scholes pde under a singularity separating framework a more accurate free boundary location based on the smooth pasting condition and the use of a non uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained comparisons with some recently proposed methods for the american options problem are carried out to show the advantage of our numerical method

flexible expectile regression in reproducing kernel hilbert spaces abstractexpectile first introduced by newey and powell in 1987 in the econometrics literature has recently become increasingly popular in risk management and capital allocation for financial institutions due to its desirable properties such as coherence and elicitability the current standard tool for expectile regression analysis is the multiple linear expectile regression proposed by newey and powell in 1987 the growing applications of expectile regression motivate us to develop a much more flexible nonparametric multiple expectile regression in a reproducing kernel hilbert space the resulting estimator is called kere which has multiple advantages over the classical multiple linear expectile regression by incorporating nonlinearity nonadditivity and complex interactions in the final estimator the kernel learning theory of kere is established we develop an efficient algorithm inspired by majorization minimization principle for solving the entire solution path of kere it is shown that the algori

conditional value at risk and average value at risk estimation and asymptotics we discuss linear regression approaches to the estimation of law invariant conditional risk measures two estimation procedures are considered and compared one is based on residual analysis of the standard least squares method and the other is in the spirit of the m estimation approach used in robust statistics in particular value at risk and average value at risk measures are discussed in detail large sample statistical inference of the estimators is derived furthermore finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive monte carlo study empirical results on the real data different financial asset classes are also provided to illustrate the performance of the estimators

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

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

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

superquantile regression with applications to buffered reliability uncertainty quantification and conditional value at risk abstract the paper presents a generalized regression technique centered on a superquantile also called conditional value at risk that is consistent with that coherent measure of risk and yields more conservatively fitted curves than classical least squares and quantile regression in contrast to other generalized regression techniques that approximate conditional superquantiles by various combinations of conditional quantiles we directly and in perfect analog to classical regression obtain superquantile regression functions as optimal solutions of certain error minimization problems we show the existence and possible uniqueness of regression functions discuss the stability of regression functions under perturbations and approximation of the underlying data and propose an extension of the coefficient of determination r squared for assessing the goodness of fit the paper presents two numerical methods for solving the error minimization problems and illustrates the methodology in several numerical examples in the areas of uncertainty quantification reliability engineering and financial risk management

volatility forecast comparison using imperfect volatility proxies the use of a conditionally unbiased but imperfect volatility proxy can lead to undesirable outcomes in standard methods for comparing conditional variance forecasts we motivate our study with analytical results on the distortions caused by some widely used loss functions when used with standard volatility proxies such as squared returns the intra daily range or realised volatility we then derive necessary and sufficient conditions on the functional form of the loss function for the ranking of competing volatility forecasts to be robust to the presence of noise in the volatility proxy and derive some useful special cases of this class of robust loss functions the methods are illustrated with an application to the volatility of returns on ibm over the period 1993 to 2003

on the measurement of economic tail risk this paper attempts to provide a decision theoretic foundation for the measurement of economic tail risk which is not only closely related to utility theory but also relevant to statistical model uncertainty the main result is that the only risk measures that satisfy a set of economic axioms for the choquet expected utility and the statistical property of general elicitability i e there exists an objective function such that minimizing the expected objective function yields the risk measure are the mean functional and value at risk var in particular the median shortfall which is the median of tail loss distribution and is also the var at a higher confidence level we also discuss various approaches of backtesting and their relations to elicitability and co elicitability in particular we show that the co elicitability of var and expected shortfall does not lead to a reliable backtesting method for expected shortfall and there have been only indirect backtesting methods for expected shortfall f

a fast numerical method for the black scholes equation of american options this paper introduces a fast numerical method for computing american option pricing problems governed by the black scholes equation the treatment of the free boundary is based on some properties of the solution of the black scholes equation an artificial boundary condition is also used at the other end of the domain the finite difference method is used to solve the resulting problem computational results are given for some american call option problems the results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method

elicitation complexity of statistical properties a property or statistical functional is said to be elicitable if it minimizes expected loss for some loss function the study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization while recent work asks which properties are elicitable we instead advocate for a more nuanced question how many dimensions are required to indirectly elicit a given property this number is called the elicitation complexity of the property we lay the foundation for a general theory of elicitation complexity including several basic results about how elicitation complexity behaves and the complexity of standard properties of interest building on this foundation our main result gives tight complexity bounds for the broad class of bayes risks we apply these results to several properties of interest including variance entropy norms and several classes of financial risk measures we conclude with discussion and open directions

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

the mode functional is not elicitable this article is concerned with point forecasting of a real valued random variable with a general lebesgue density answering a question of gneiting 2011 it is shown that the mode is not elicitable or in other words that it is impossible to find a loss or scoring function under which the mode is the bayes predictor

higher order elicitability and osband s principle a statistical functional such as the mean or the median is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score such scoring functions are called strictly consistent for the functional the elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores in this paper we explore the notion of elicitability for multi dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions we cover the case of functionals with elicitable components but we also show that one dimensional functionals that are not elicitable can be a component of a higher order elicitable functional in the case of the variance this is a known result however an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the correct quantiles a direct consequence of applied interest is that the pair value at risk expected shortfall is jointly elicitable under mild conditions that are usually fulfilled in risk management applications

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

asymptotic stability of empirical processes and related functionals abstract let e be a space of observables in a sequence of trials n and define m n to be the empirical distributions of the outcomes we discuss the almost sure convergence of the sequence m n in terms of the weak topology of measures when the sequence n is assumed to be stationary in this respect the limit variable is naturally described as a certain canonical conditional distribution then given some functional defined on a space of laws the consistency of the estimators m n is investigated moreover a result on the asymptotic stability of the distribution of m n w r t certain transformations of the data is given

bias corrected score decomposition for generalized quantiles summarydecompositions of the score of a forecast represent useful tools for assessing its performance we consider local score decompositions permitting detailed forecast assessments across a spectrum of conditions of interest we derive corrections to the bias of the decomposition components in the framework of point forecasts of quantile type functionals and illustrate their performance by simulation related bias corrections have thus far only been known for squared error criteria

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

interface problems for quasilinear elliptic equations in this paper we consider interface problems for quasilinear elliptic partial differential equations in two dimensional spaces the main result is that the bounded weak solution in the neighborhood of a singular point can be divided into two parts one of which is a function in h2 and the other one can be expressed as a finite linear combination of particular functions the most singular function is a particular solution to the homogeneous equation with constant coefficients

evaluating quantile assessments quantile assessments are commonly encountered in the elicitation of probability distributions in decision analysis forecasting and risk analysis scoring rules have been developed to provide ex ante incentives for careful and truthful assessments and ex post evaluation measures in the context of probability assessment we show that these scoring rules designed for probability assessment provide inappropriate incentives if used for quantile assessment we investigate the properties of a linear family of scoring rules that are intended specifically for quantile assessment including the assessment of multiple quantiles and can be related to a realistic decision making problem these rules provide proper incentives for quantile assessment and yield higher expected scores for distributions that are more informative in the sense of having less dispersion we discuss the special case of interval forecasts and a generalization involving transformations and we briefly mention other possible extensions

quasi maximum likelihood estimation for conditional quantiles in this paper we derive the asymptotic distribution of a new class of quasi maximum likelihood estimators qmle based on a tick exponential family of densities we show that the tick exponential assumption is a necessary and sufficient condition for a qmle to be consistent for the parameters of a correctly specified model of a given conditional quantile hence the role of this family of densities in the conditional quantile estimation is analog to the role of the linear exponential family in the conditional mean estimation the tick exponential qmles are shown to be asymptotically normal with an asymptotic covariance matrix that has a novel form not seen in earlier work and which accounts for possible model misspecification for practical purposes we show that the maximization of the tick exponential quasi log likelihood can conveniently be carried out by using standard gradient based optimization techniques more importantly we provide a consistent estimator for the asymptotic covariance matrix based on the scores of the log likelihood which allows us to compute the conditional quantile confidence intervals

m estimation convexity and quantiles the paper develops a class of extensions of the univariate quantile function to the multivariate case m quantiles related in a certain way to m parameters of a probability distribution and their m estimators the spatial geometric quantiles recently introduced by koltchinskii and dudley and by chaudhuri as well as the regression quantiles of koenker and basset are the examples of the m quantile function discussed in the paper we study the main properties of m quantiles and develop the asymptotic theory of empirical m quantiles we use m quantiles to extend l parameters and l estimators to the multivariate case to introduce a bootstrap test for spherical symmetry of a multivariate distribution and to extend the notion of regression quantiles to multiresponse linear regression models

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

asymmetric least squares estimation and testing this paper considers estimation and hypothesis tests for coefficients of linear regression models where the coefficient estimates are based on location measures defined by an asymmetric least squares criterion function these asymmetric least squares estimators have properties which are analogous to regression quantile estimators but are much simpler to calculate as are the corresponding test statistics the coefficient estimators can be used to construct test statistics for homoskedasticity and conditional symmetry of the error distribution and we find these tests compare quite favorably with other commonly used tests of these null hypotheses in terms of local relative efficiency consequently asymmetric least squares estimation provides a convenient and relatively efficient method of summarizing the conditional distribution of a dependent variable given the regressors and a means of testing whether a linear model is an adequate characterization of the typical value for this conditional distribution

bayesian estimation and prediction using asymmetric loss functions abstract estimators and predictors that are optimal relative to varian s asymmetric linex loss function are derived for a number of well known models their risk functions and bayes risks are derived and compared with those of usual estimators and predictors it is shown that some usual estimators for example a scalar sample mean or a scalar least squares regression coefficient estimator are inadmissible relative to asymmetric linex loss by providing alternative estimators that dominate them uniformly in terms of risk

building cost functions minimizing to some summary statistics a learning machine or a model is usually trained by minimizing a given criterion the expectation of the cost function measuring the discrepancy between the model output and the desired output as is already well known the choice of the cost function has a profound impact on the probabilistic interpretation of the output of the model after training in this work we use the calculus of variations in order to tackle this problem in particular we derive necessary and sufficient conditions on the cost function ensuring that the output of the trained model approximates 1 the conditional expectation of the desired output given the explanatory variables 2 the conditional median and more generally the q quantile 3 the conditional geometric mean and 4 the conditional variance the same method could be applied to the estimation of other summary statistics as well we also argue that the least absolute deviations criterion could in some cases act as an alternative to the ordinary least squares criterion for nonlinear regression in the same vein the concept of regression quantile is briefly discussed

least absolute relative error estimation multiplicative regression model or accelerated failure time model which becomes linear regression model after logarithmic transformation is useful in analyzing data with positive responses such as stock prices or life times that are particularly common in economic financial or biomedical studies least squares or least absolute deviation are among the most widely used criterions in statistical estimation for linear regression model however in many practical applications especially in treating for example stock price data the size of relative error rather than that of error itself is the central concern of the practitioners this paper offers an alternative to the traditional estimation methods by considering minimizing the least absolute relative errors for multiplicative regression models we prove consistency and asymptotic normality and provide an inference approach via random weighting we also specify the error distribution with which the proposed least absolute relative errors estimatio