galton edgeworth frisch and prospects for quantile regression in econometrics the work of three leading figures in the early history of econometrics is used to motivate some recent developments in the theory and application of quantile regression we stress not only the robustness advantages of this form of semiparametric statistical method but also the opportunity to recover a more complete description of the statistical relationship between variables a recent proposal for a more x robust form of quantile regression based on maximal depth ideas is described along with an interesting historical antecedent finally the notorious computational burden of median regression and quantile regression more generally is addressed it is argued that recent developments in interior point methods for linear programming together with some new preprocessing ideas make it possible to compute quantile regressions as quickly as least squares regressions throughout the entire range of problem sizes encountered in econometrics

nonparametric quantile estimations for dynamic smooth coefficient models in this article quantile regression methods are suggested for a class of smooth coefficient time series models we use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework we establish the asymptotic properties of both the local polynomial and local constant estimators for mixing time series also a bandwidth selector based on the nonparametric version of the akaike information criterion is suggested together with a consistent estimate of the asymptotic covariance matrix furthermore the asymptotic behaviors of the estimators at boundaries are examined a comparison of the local polynomial quantile estimator with the local constant estimator is presented a simulation study is carried out to illustrate the performance of estimates an empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures

a quantile regression approach to generating prediction intervals exponential smoothing methods do not involve a formal procedure for identifying the underlying data generating process the issue is then whether prediction intervals should be estimated by a theoretical approach with the assumption that the method is optimal in some sense or by an emp irical procedure in this paper we present an alternative hybrid approach which applies quantile regression to the empirical fit errors to produce forecast error quantile models these models are functions of the lead time as suggested by the theoretical variance expressions in addition to avoiding the optimality assumption the method is nonparametric so there is no need for the common normality assumption application of the new approach to simple holt s and damped holt s exponential smoothing using simulated and real data sets gave encouraging results

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

inference on the quantile regression process tests based on the quantile regression process can be formulated like the classical kolmogorov smirnov and cramer von mises tests of goodness of fit employing the theory of bessel processes as in kiefer 1959 however it is frequently desirable to formulate hypotheses involving unknown nuisance parameters thereby jeopardizing the distribution free character of these tests we characterize this situation as the durbin problem since it was posed in durbin 1973 for parametric empirical processes in this paper we consider an approach to the durbin problem involving a martingale transformation of the parametric empirical process suggested by khmaladze 1981 and show that it can be adapted to a wide variety of inference problems involving the quantile regression process in particular we suggest new tests of the location shift and location scale shift models that underlie much of classical econometric inference the methods are illustrated with a reanalysis of data on unemployment durations from the pennsylvania reemployment bonus experiments the pennsylvania experiments conducted in 1988 89 were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells

factorizations of ep operators abstract in this paper we characterize ep operators through the existence of different types of factorizations our results extend to ep operators existing characterizations for ep matrices and give new characterizations both for ep matrices and ep operators

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

once more on algebras generated by two projections we consider von neumann algebras generated by two arbitrary orthoprojections on a hilbert space a canonical decomposition is obtained for elements a of these algebras in terms of the operator angle between the ranges of the above mentioned projections this decomposition leads to explicit descriptions and formulas for kernels ranges spectra and essential spectra generalized inverses and other objects related to a

sparse and non negative bss for noisy data non negative blind source separation bss has raised interest in various fields of research as testified by the wide literature on the topic of non negative matrix factorization nmf in this context it is fundamental that the sources to be estimated present some diversity in order to be efficiently retrieved sparsity is known to enhance such contrast between the sources while producing very robust approaches especially to noise in this paper we introduce a new algorithm in order to tackle the blind separation of non negative sparse sources from noisy measurements we first show that sparsity and non negativity constraints have to be carefully applied on the sought after solution in fact improperly constrained solutions are unlikely to be stable and are therefore sub optimal the proposed algorithm named ngmca non negative generalized morphological component analysis makes use of proximal calculus techniques to provide properly constrained solutions the performance of ngmca compared to other state of the art algorithms is demonstrated by numerical experiments encompassing a wide variety of settings with negligible parameter tuning in particular ngmca is shown to provide robustness to noise and performs well on synthetic mixtures of real nmr spectra

additivity properties of operator ranges abstract in the present paper we study different conditions regarding the sum of operator ranges in particular we find conditions on operators a b which imply that r a b r a r b and derive some results concerning projections

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

quantile regression methods for reference growth charts estimation of reference growth curves for children s height and weight has traditionally relied on normal theory to construct families of quantile curves based on samples from the reference population age specific parametric transformation has been used to significantly broaden the applicability of these normal theory methods non parametric quantile regression methods offer a complementary strategy for estimating conditional quantile functions we compare estimated reference curves for height using the penalized likelihood approach of cole and green statistics in medicine 1992 11 1305 1319 with quantile regression curves based on data used for modern finnish reference charts an advantage of the quantile regression approach is that it is relatively easy to incorporate prior growth and other covariates into the analysis of longitudinal growth data quantile specific autoregressive models for unequally spaced measurements are introduced and their application to diagnostic screening is illustrated copyright 2005 john wiley sons ltd

on additive conditional quantiles with high dimensional covariates we investigate the estimation of the conditional quantile when many covariates are involved in particular we model the conditional quantile of a response as a nonlinear additive function of relevant covariates for this setup we propose a nonparametric smoother to estimate the unknown functions the estimator provides direct computation of the nonlinear functions because it does not require any iteration the estimator allows fast and routine data analysis on the theoretical front we also show asymptotic properties of the estimator including mean squared error and limiting distribution the theory confirms that for moderate dimension of the covariates the estimator escapes the curse of dimensionality problem both simulated and real data examples are provided to illustrate the methodology

banach algebras generated by idempotents the goal of this chapter is to provide a possible tool to study the invertibility of the local cosets which arise after localization of convolution type operators the basic observation is that the local algebras in some cases are generated by a finite number of elements p which are idempotent in the sense that p 2 p under some additional conditions it turns out that such algebras possess matrix valued symbols thus one can associate with every element of the algebra a matrix valued function such that the element is invertible if and only if the associated function is invertible at every point in this way one gets an effective criterion for the invertibility of elements in local algebras

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

on flat portions on the boundary of the numerical range abstract the paper is devoted to matrices with flat portions on the boundary of their numerical range a constructive criterion for such portions to exist is obtained in case of tridiagonal matrices and a particular case of continuant matrices is considered as an application the cases of arbitrary 3 3 and 4 4 matrices are treated it is shown in particular that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 4 matrices is four three if the matrices are assumed unitarily irreducible

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

on certain finite dimensional algebras generated by two idempotents abstract this paper is concerned with algebras generated by two idempotents p and q satisfying pq m qp m and pq m 1 qp m 1 the main result is the classification of all these algebras implying that for each m 2 there exist exactly eight nonisomorphic copies as an application it is shown that if an element of such an algebra has a nondegenerate leading term then it is group invertible and a formula for the explicit computation of the group inverse is given

interval forecasting an analysis based upon arch quantile estimators abstract in this paper we explore techniques for obtaining interval forecasts based on estimated time series models for processes which may exhibit autoregressive conditional heteroskedasticity arch to deal with the available variety of possible interval forecasts we propose a method for combining these forecasts based on quantile regression techniques our approach is practical rather than theoretical with attention focused directly on obtaining interval forecasts for two u s time series a measure of unemployment and a treasury bill rate we evaluate the performance of our procedures using a variety of diagnostics we find interval estimates which perform reasonably well judged by both in sample and out of sample criteria our experience suggests that a certain amount of care is required in order to obtain useful forecasts

local linear quantile regression abstract in this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting two such estimators are considered one is based on localizing the characterization of a regression quantile as the minimizer of e pp y a x x where p is the appropriate check function the other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters rather than one our aim is to present fully operational versions of both approaches and to show that each works quite well although either might be used in practice we have a particular preference for the second our automatic smoothing parameter selection method is novel the main regression quantile smoothing parameters are chosen by rule of thumb adaptations of state of the art methods for smoothing parameter selection for regression mean estimation the techniques are illustrated by application to two datasets and compared in simulations

direct use of regression quantiles to construct confidence sets in linear models direct use of the empirical quantile function provides a standard distribution free approach to constructing confidence intervals and confidence bands for population quantiles we apply this method to construct confidence intervals and confidence bands for regression quantiles and to construct prediction intervals based on sample regression quantiles comparison of the direct method with the studentization and the bootstrap methods are discussed simulation results show that the direct method has the advantage of robustness against departure from the normality assumption of the error terms

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

nonparametric estimation of an additive quantile regression model this article is concerned with estimating the additive components of a nonparametric additive quantile regression model we develop an estimator that is asymptotically normally distributed with a rate of convergence in probability of n r 2r 1 when the additive components are r times continuously differentiable for some r 2 this result holds regardless of the dimension of the covariates and thus the new estimator has no curse of dimensionality in addition the estimator has an oracle property and is easily extended to a generalized additive quantile regression model with a link function the numerical performance and usefulness of the estimator are illustrated by monte carlo experiments and an empirical example

on average derivative quantile regression for fixed 0 1 the quantile regression function gives the th quantile x in the conditional distribution of a response variable y given the value x x of a vector of covariates it can be used to measure the effect of covariates not only in the center of a population but also in the upper and lower tails a functional that summarizes key features of the quantile specific relationship between x and y is the vector of weighted expected values of the vector of partial derivatives of the quantile function x in a nonparametric setting can be regarded as a vector of quantile specific nonparametric regression coefficients in survival analysis models e g cox s proportional hazard model proportional odds rate model accelerated failure time model and in monotone transformation models used in regression analysis gives the direction of the parameter vector in the parametric part of the model can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics we show that under suitable regularity conditions the estimate of obtained by using the locally polynomial quantile estimate of chaudhuri 1991a is n 1 2 consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional we discuss how the estimate of can be used for model diagnostics and in the construction of a link function estimate in general single index models

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

on the algebra generated by two projections the von neumann algebra generated by a pair of projections on a separable hilbert space occurs naturally in the study of quantum hall systems by a systematic use of kato s dual pair of self adjoint operators corresponding to the two projections the argument referring to the structure of the algebra can be simplified and the existence of an action of the group su 2 on the algebra via inner automorphisms can be shown also the algebra can be given an r structure with respect to which it is r commutative and an extension of the classical commutative differential calculus becomes possible

drazin inversion in the von neumann algebra generated by two orthogonal projections abstract criteria for drazin and moore penrose invertibility of operators in the von neumann algebra generated by two orthogonal projections are established and explicit representations for the corresponding inverses are given the results are illustrated by several examples that have recently been considered in the literature

conditional growth charts growth charts are often more informative when they are customized per subject taking into account prior measurements and possibly other covariates of the subject we study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions the model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers and it is also flexible for including covariate information we propose a rank score test for large sample inference on covariates and develop a new model assessment tool for longitudinal growth data our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

bivariate quantile smoothing splines it has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample we introduce bivariate quantile smoothing splines which belong to the space of bilinear tensor product splines as nonparametric estimators for the conditional quantile functions in a two dimensional design space the estimators can be computed by using standard linear programming techniques and can further be used as building blocks for conditional quantile estimations in higher dimensions for moderately large data sets we recommend penalized bivariate b splines as approximate solutions we use real and simulated data to illustrate the methodology proposed

estimation of conditional densities and sensitivity measures in nonlinear dynamical systems using locally polynomial regression we develop nonparametric estimators for the conditional density function and its square root and their partial derivatives two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed one of which relates fisher information with initial value sensitivity in dynamical systems we propose estimators for these and show asymptotic normality for one of them we further propose a simple method for choosing the bandwidth the methods are illustrated by simulation of two well known models in dynamical systems

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to