local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

the zakai forms of the prediction and smoothing equations corresp zakai forms for the prediction and smoothing equations are obtained which like the zakai filtering equation are linear in the unnormalized conditional expectation

eigenschemes and the jordan canonical form abstract we study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix the two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the jordan canonical form of the matrix we also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space this interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

a nonparametric circular linear multivariate regression model with a rule of thumb bandwidth selector the purpose of this paper is to propose a nonparametric circular linear multivariate regression model using a kernel weighted local linear method the case of several linear regressors and one circular regressor is considered we extend results on the asymptotic bias and variance of the linear multivariate variable to the case of circular linear multivariate variable the rule of thumb selector is used to establish the optimal bandwidths for the nonparametric model the suitability of the model is judged from the coefficient of determination one simulation experiment and one real problem concerning wind energy are used to study the power performance of the nonparametric model

mismatched estimation and relative entropy a random variable with distribution p is observed in gaussian noise and is estimated by a mismatched minimum mean square estimator that assumes that the distribution is q instead of p this paper shows that the integral over all signal to noise ratios snrs of the excess mean square estimation error incurred by the mismatched estimator is twice the relative entropy d p q in nats this representation of relative entropy can be generalized to nonreal valued random variables and can be particularized to give new general representations of mutual information in terms of conditional means inspired by the new representation we also propose a definition of free relative entropy which fills a gap in and is consistent with the literature on free probability

on mutual information likelihood ratios and estimation error for the additive gaussian channel this paper considers the model of an arbitrarily distributed signal x observed through an added independent white gaussian noise w y x w new relations between the minimal mean square error of the noncausal estimator and the likelihood ratio between y and w are derived this is followed by an extended version of a recently derived relation between the mutual information i x y and the minimal mean square error these results are applied to derive infinite dimensional versions of the fisher information and the de bruijn identity a comparison between the causal and noncausal estimation errors yields a restricted form of the logarithmic sobolev inequality the derivation of the results is based on the malliavin calculus

semiparametric regression analysis for recurrent event interval counts this paper deals with analysis of data from longitudinal studies where the rate of a recurrent event characterizing morbidity is the primary criterion for treatment evaluation we consider clinical trials which require patients to visit their clinical center at successive scheduled times as part of follow up at each visit the patient reports the number of events that occurred since the previous visit or an examination reveals the number of accumulated events such as skin cancers the exact occurrence times of the events are unavailable and the actual patient visit times typically vary randomly about the scheduled follow up times each patient s record thus consists of a sequence of clinic visit dates event counts corresponding to the successive time intervals between clinic visits and baseline covariates we propose a semiparametric regression model extending the fully parametric model of thall 1988 biometrics 44 197 209 to estimate and test for covariate effects on the rate of events over time while also accounting for the possibly time varying nature of the underlying event rate covariate effects enter the model parametrically while the underlying time varying event rate is modelled nonparametrically the method of severini and wong 1992 annals of statistics 20 1768 1802 is used to construct asymptotically efficient estimators of the parametric component and to specify their asymptotic distribution a simulation study and application to a data set are provided

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

mutual information for stochastic signals and fractional brownian motion the mutual information between a stochastic signal and this signal plus a fractional brownian motion described as an additive fractional gaussian noise channel is expressed as the error of an estimation problem that can be naturally associated with this model if the stochastic signal with the additive fractional brownian motion occurs multiplied by a scalar parameter then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem these results generalize some results for a model where the fractional brownian motion is a brownian motion to a model with an arbitrary fractional brownian motion

monitoring profiles based on nonparametric regression methods the use statistical process control spc in monitoring and diagnosis of process and product quality profiles remains an important problem in various manufacturing industries the spc problem with a nonlinear profile is particularly challenging this article proposes a novel scheme to monitor changes in both the regression relationship and the variation of the profile online it integrates the multivariate exponentially weighted moving average procedure with the generalized likelihood ratio test based on nonparametric regression the proposed scheme not only provides an effective spc solution to handle nonlinear profiles which are common in industrial practice but it also resolves the latent problem in popular parametric monitoring methods of being unable to detect certain types of changes due to a misspecified out of control model our simulation results demonstrate the effectiveness and efficiency of the proposed monitoring scheme in addition a systematic diagnostic approach is provided to locate t

optimal convergence rates bahadur representation and asymptotic normality of partitioning estimators abstract this paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives mean square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions the uniform rate explicitly accounts for the effect of moment assumptions which is useful in semiparametric inference a general asymptotic integrated mean square error approximation is obtained and used to derive an optimal plug in tuning parameter selector a uniform bahadur representation is developed for linear functionals of the estimator using this representation asymptotic normality is established along with consistency of a standard error estimator the finite sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study

local linear regression on manifolds and its geometric interpretation high dimensional data analysis has been an active area and the main focus areas have been variable selection and dimension reduction in practice it occurs often that the variables are located on an unknown lower dimensional nonlinear manifold under this manifold assumption one purpose of this article is regression and gradient estimation on the manifold and another is developing a new tool for manifold learning as regards the first aim we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold and performing the popular local linear regression llr on a tangent plane estimate an immediate consequence is a dramatic reduction in the computational time when the ambient space dimension p d we provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature boundary and nonuniform sampling effects we propose a bandwidth selector that can handle heteroscedastic errors with re

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

comparison of smoothing parameterizations in bivariate kernel density estimation abstract the basic kernel density estimator in one dimension has a single smoothing parameter usually referred to as the bandwidth for higher dimensions however there are several options for smoothing parameterization of the kernel estimator for the bivariate case there can be between one and three independent smoothing parameters in the estimator which leads to a flexibility versus complexity trade off when using this estimator in practice in this article the performances of the different possible smoothing parameterizations are compared using both the asymptotic and exact mean integrated squared error our results show that it is important to have independent smoothing parameters for each of the coordinate directions although this is enough for many situations for densities with high amounts of curvature in directions different to those of the coordinate axes substantial gains can be made by allowing the kernel mass to have arbitrary orientations the sphering approaches to choosing this o

mutual information relative entropy and estimation in the poisson channel let be a nonnegative random variable and let the conditional distribution of a random variable given be poisson for a parameter we identify a natural loss function such that 1 the derivative of the mutual information between and with respect to is equal to the minimum mean loss in estimating based on regardless of the distribution of 2 when is estimated based on by a mismatched estimator that would have minimized the expected loss had the integral over all values of of the excess mean loss is equal to the relative entropy between and for a continuous time setting where is a nonnegative stochastic process and the conditional law of given is that of a non homogeneous poisson process with intensity function under the same loss function 1 the minimum mean loss in causal filtering when is equal to the expected value of the minimum mean loss in noncausal filtering smoothing achieved with a channel whose parameter is uniformly distributed between 0 and bridging the two quantities is the mutual information between and 2 this relationship between the mean losses in causal and noncausal filtering holds also in the case where the filters employed are mismatched i e optimized assuming a law on which is not the true one bridging the two quantities in this case is the sum of the mutual information and the relative entropy between the true and the mismatched distribution of thus relative entropy quantifies the excess estimation loss due to mismatch in this setting these results are parallel to those recently found for the gaussian channel the i mmse relationship of guo the relative entropy and mismatched estimation relationship of verdu and the relationship between causal and noncasual mismatched estimation of weissman

application of nonparametric quantile regression to body mass index percentile curves from survey data increasing rates of overweight among children in the u s stimulated interest in obtaining national percentile curves of body size to serve as a benchmark in assessing growth development in clinical and population settings in 2000 the u s centers for disease control and prevention cdc developed conditional percentile curves for body mass index bmi for ages 2 20 years the 2000 cdc bmi for age curves are partially parametric and only partially incorporated the survey sample weights in the curve estimation as a result they may not fully reflect the underlying pattern of bmi for age in the population this motivated us to develop a nonparametric double kernel based method and automatic bandwidth selection procedure we include sample weights in the bandwidth selection conduct median correction to reduce small sample smoothing bias and rescale the bandwidth to make it scale invariant using this procedure we re estimate the national percentile bmi for age curves and the prevalence of high bmi children in the u s

the resolution of the ideal of 2 2 minors of a 2 nmatrix of linear forms abstract let k be an algebraically closed field and let s k x 1 x m let m be a 2 n matrix of linear forms of s and let i 2 m denote the ideal generated by the determinants of the 2 2 minors of m we study in this paper the minimal finite free resolution of s i 2 m as an s module m corresponds to a certain 2 dimensional vector space l of m n matrices that is to a matrix pencil the kronecker weierstrass theory of such matrix pencils provides a normal form for l and we characterize the resolution of s i 2 m in terms of this normal form in particular if the general element of l is injective we explicitly construct the minimal resolution of s i 2 m by repeated application of the horseshoe lemma for any m we express the regularity of s i 2 m as a function of the invariants of l

nonparametric rotations for sphere sphere regression regression of data represented as points on a hypersphere has traditionally been treated using parametric families of transformations that include the simple rigid rotation as an important special

pointwise relations between information and estimation in gaussian noise many of the classical and recent relations between information and estimation in the presence of gaussian noise can be viewed as identities between expectations of random quantities these include the relationship between mutual information and minimum mean square error i mmse of guo the relative entropy and mismatched estimation relationship of verdu the relationship between causal estimation and mutual information of duncan and its extension to the presence of feedback by kadota the relationship between causal and non casual estimation of guo and its mismatched version of weissman we dispense with the expectations and explore the nature of the pointwise relations between the respective random quantities the pointwise relations that we find are as succinctly stated as and give considerable insight into the original expectation identities as an illustration of our results consider duncan s 1970 discovery that the mutual information is equal to the causal mmse in the additive white gaussian noise channel which can equivalently be expressed saying that the difference between the input output information density and half the causal estimation error is a zero mean random variable regardless of the distribution of the channel input we characterize this random variable explicitly rather than merely its expectation classical estimation and information theoretic quantities emerge with new and surprising roles for example the variance of this random variable turns out to be given by the causal mmse which in turn is equal to twice the mutual information by duncan s result

weighted local regression and kernel methods for nonparametric curve fitting abstract weighted local regression a popular technique for smoothing scatterplots is shown to be asymptotically equivalent to certain kernel smoothers since both methods are local weighted averages of the data it is proved that in the fixed design regression model given a weighted local regression procedure with any weight function there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity it is demonstrated by examples that in some instances the weights are nearly the same for both methods even for small samples the asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression further a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order number of vanishing moments of the kernel employed in the kernel smoot

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

a note on convergence rates and asymptotic normality for series estimators uniform convergence rates this paper establishes improved uniform convergence rates for series estimators series estimators are least squares fits of a regression function where the number of regressors depends on sample size i will specialize my results to the cases of polynomials and regression splines these results improves upon results obtained earlier by newey yet fail to attain the optimal rates of convergence

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

robust nonparametric confidence intervals for regression discontinuity designs in the regression discontinuity rd design units are assigned to treatment based on whether their value of an observed covariate exceeds a known cutoff in this design local polynomial estimators are now routinely employed to construct confidence intervals for treatment effects the performance of these confidence intervals in applications however may be seriously hampered by their sensitivity to the specific bandwidth employed available bandwidth selectors typically yield a large bandwidth leading to data driven confidence intervals that may be biased with empirical coverage well below their nominal target we propose new theory based more robust confidence interval estimators for average treatment effects at the cutoff in sharp rd sharp kink rd fuzzy rd and fuzzy kink rd designs our proposed confidence intervals are constructed using a bias corrected rd estimator together with a novel standard error estimator for practical implementation we discuss mean squared error optimal bandwidths which are by construction not valid for conventional confidence intervals but are valid with our robust approach and consistent standard error estimators based on our new variance formulas in a special case of practical interest our procedure amounts to running a quadratic instead of a linear local regression more generally our results give a formal justification to simple inference procedures based on increasing the order of the local polynomial estimator employed we find in a simulation study that our confidence intervals exhibit close to correct empirical coverage and good empirical interval length on average remarkably improving upon the alternatives available in the literature all results are readily available in r and stata using our companion software packages described in calonico cattaneo and titiunik 2014d 2014b

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

a method of estimating the average derivative abstract we derive a simple semi parametric estimator of the direct average derivative e d m x where m x is the regression function and s the support of the density of x is compact we partition s into disjoint bins and the local slope d m x within these bins is estimated by using ordinary least squares our average derivative estimate a is then obtained by taking the weighted average of these least squares slopes we show that this estimator is asymptotically normally distributed we also propose a consistent estimator of the variance of a using monte carlo simulation experiments based on a censored regression model with tobit model as a special case we produce small sample results comparing our estimator with the hardle stoker 1989 investigating smooth multiple regression by the method of average derivatives journal of american statistical association 84 408 986 995 method we conclude that a performs better that the hardle stoker estimator for bounded and discontinuous covariates

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

optimal data driven regression discontinuity plots exploratory data analysis plays a central role in applied statistics and econometrics in the popular regression discontinuity rd design the use of graphical analysis has been strongly advocated because it provides both easy presentation and transparent validation of the design rd plots are nowadays widely used in applications despite its formal properties being unknown these plots are typically presented employing ad hoc choices of tuning parameters which makes these procedures less automatic and more subjective in this article we formally study the most common rd plot based on an evenly spaced binning of the data and propose several optimal data driven choices for the number of bins depending on the goal of the researcher these rd plots are constructed either to approximate the underlying unknown regression functions without imposing smoothness in the estimator or to approximate the underlying variability of the raw data while smoothing out the otherwise uninformative scatterplot of the da

heteroscedasticity robust inference in linear regression models with many covariates we consider inference in linear regression models that is robust to heteroscedasticity and the presence of many control variables when the number of control variables increases at the same rate as

the mean shift algorithm and its relation to kernel regression we investigate the connection between the asymptotic bias of the well known nadaraya watson kernel regression and the mean shift ms vector with the gaussian kernel we first show that the asymptotic bias for the univariate nadaraya watson kernel regression can be estimated using the ms scalar then we investigate the general d dimensional case d 1 and derive a formula for the asymptotic bias as a function of the ms vector we show that when the regression function is a linear function of its entries then the asymptotic bias can be represented as a linear function of the ms vector the ms algorithm for the univariate and the linear cases can be used to find points where the nadaraya watson kernel regression gives an unbiased estimate furthermore this connection suggests that exploiting the theoretical properties of the asymptotic bias of the nadaraya watson kernel regression may be helpful to show the convergence of the ms algorithm through the simulations we show that how the given theoretical results can be used to estimate the bias of the estimated clean signal by just observing the noisy signal having access to the bias of the estimator enables us to evaluate the accuracy of the estimated clean data which later will be used to design a quantizer

the relationship between causal and noncausal mismatched estimation in continuous time awgn channels a continuous time finite power process with distribution p is observed through an awgn channel at a given signal to noise ratio snr and is estimated by an estimator that would have minimized the mean square error if the process had distribution q we show that the causal filtering mean square error mse achieved at snr level snr is equal to the average value of the noncausal smoothing mse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr emerging as the bridge for equating these two quantities are mutual information and relative entropy our result generalizes that of guo shamai and verdu 2005 from the nonmismatched case where p q to general p and q among our intermediate results is an extension of duncan s theorem that relates mutual information and causal mmse to the case of mismatched estimation some further extensions and implications are discussed key to our findings is the recent result of verdu on mismatched estimation and relative entropy

an improved estimator of the density function at the boundary abstract we propose a new method of boundary correction for kernel density estimation the technique is a kind of generalized reflection method involving reflecting a transformation of the data the transformation depends on a pilot estimate of the logarithmic derivative of the density at the boundary in simulations the new method is seen to clearly outperform an earlier generalized reflection idea it also has overall advantages over boundary kernel methods and a nonnegative adaptation thereof although the latter are competitive in some situations we also present the theory underlying the new methodology

information estimation and lookahead in the gaussian channel we consider mean squared estimation with lookahead of a continuous time signal corrupted by additive white gaussian noise we show that the mutual information rate function i e the mutual information rate as function of the signal to noise ratio snr does not in general determine the minimum mean squared error mmse with fixed finite lookahead in contrast to the special cases with 0 and infinite lookahead filtering and smoothing errors respectively which were previously established in the literature further we investigate the simple class of continuous time stationary gauss markov processes ornstein uhlenbeck processes as channel inputs and explicitly characterize the behavior of the minimum mean squared error mmse with finite lookahead and signal to noise ratio snr we extend our results to mixtures of ornstein uhlenbeck processes and use the insight gained to present lower and upper bounds on the mmse with lookahead for a class of stationary gaussian input processes whose spectrum can be expressed as a mixture of ornstein uhlenbeck spectra

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

mutual information for stochastic signals and levy processes in this paper some relations between estimation and mutual information are given by expressing two mutual information calculations in terms of two distinct estimation errors specifically the mutual information between a stochastic signal and a pure jump levy process whose rate function depends on the signal is expressed in terms of a filtering error and the rate of change of this mutual information with respect to a parameter multiplying the rate function of the levy process is expressed in terms of a smoothing error these results generalize the analogous mutual information results for some gaussian noise processes with additive stochastic signals

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation