least squares estimators of the mode of a unimodal regression function in this paper we consider nonparametric least squares estimators of the mode of an unknown unimodal regression function we establish almost sure convergence of these estimators with nearly optimal convergence rates under the assumption of the exponential tail for the error distributions

efficient estimation of population level summaries in general semiparametric regression models this article considers a wide class of semiparametric regression models in which interest focuses on population level quantities that combine both the parametric and the nonparametric parts of the model special cases in this approach include generalized partially linear models generalized partially linear single index models structural measurement error models and many others for estimating the parametric part of the model efficiently profile likelihood kernel estimation methods are well established in the literature here our focus is on estimating general population level quantities that combine the parametric and nonparametric parts of the model e g population mean probabilities etc we place this problem in a general context provide a general kernel based methodology and derive the asymptotic distributions of estimates of these population level quantities showing that in many cases the estimates are semiparametric efficient for estimating the population mean with no missing data we s

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

empirical evidence on the law of demand a sufficient condition for market demand to satisfy the law of demand is that the mean of all households income effect matrices be positive definite we show how this mean income effect matrix can be estimated from cross section data under metonymy an assumption about the distribution of households characteristics the estimation procedure uses the nonparametric method of average derivatives income effect matrices estimated this way from u k family expenditure data are in fact positive definite this result can be explained by a special form of heteroskedasticity in the data households demands are more dispersed at higher income levels

a semiparametric single index model with heterogeneous impacts on an unobserved variable this paper proposes a single index semiparametric model in which the unknown function has cross sectional unit specific weights the initial motivation comes from the search for a better measure of liquidity in stock trading which is captured by the unknown function here the model is estimated by semiparametric least squares developed by ichimura 1993 and ichimura and lee 1991 the proposed technique differs from theirs in at least two aspects first i show that the estimator has desirable asymptotic properties under less restrictive assumptions on data second the form of the unknown function is fixed however the coefficients are allowed to differ across the cross sectional units

nonparametric benefit risk assessment using marker process in the presence of a terminal event abstractbenefit risk assessment is a crucial step in medical decision process in many biomedical studies both longitudinal marker measurements and time to a terminal event serve as important endpoints for benefit risk assessment the effect of an intervention or a treatment on the longitudinal marker process however can be in conflict with its effect on the time to the terminal event thus questions arise on how to evaluate treatment effects based on the two endpoints for the purpose of deciding on which treatment is most likely to benefit the patients in this article we present a unified framework for benefit risk assessment using the observed longitudinal markers and time to event data we propose a cumulative weighted marker process to synthesize information from the two endpoints and use its mean function at a prespecified time point as a benefit risk summary measure we consider nonparametric estimation of the summary measure under two scenarios i the longitudinal marker is measured inter

optimal bandwidth choice for density weighted averages abstract this paper characterizes the optimal bandwidth value for estimating density weighted averages statistics that arise in semiparametric estimation methods for index models and models of selected samples based on nonparametric kernel estimators the optimal bandwidth is derived by minimizing the leading terms of mean squared error of the density weighted average the optimal bandwidth formulation is developed by comparison to the optimal pointwise bandwidth of a naturally associated nonparametric estimation problem highlighting the role of sample size and the structure of nonparametric estimation bias the methods are illustrated by estimators of average density density weighted average derivatives and conditional covariances and bandwidth values are calculated for normal designs a simple plug in estimator for the optimal bandwidth is proposed finally the optimal bandwidth for estimating ratios of density weighted averages is derived showing that the earlier optimal formulae can be implemented directly using naturally defined residual values

optimal two stage procedures for estimating location and size of the maximum of a multivariate regression function we propose a two stage procedure for estimating the location and size m of the maximum of a smooth d variate regression function f in the first stage a preliminary estimator of obtained from a standard nonparametric smoothing method is used at the second stage we zoom in near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity we fit an appropriate polynomial regression model to estimate the location and size of the maximum we establish that under suitable smoothness conditions and appropriate choice of the zooming the second stage estimators have better convergence rates than the corresponding first stage estimators of and m more specifically for smooth regression functions the optimal nonparametric rates n 1 2 d and n 2 d at the first stage can be improved to n 1 2 and n 1 2 respectively for 1 1 d 2 these rates are optimal in the class of all possible sequential estimators interestingly the two stage procedure resolves the curse of the dimensionality problem to some extent as the dimension d does not control the second stage convergence rates provided that the function class is sufficiently smooth we consider a multi stage generalization of our procedure that attains the optimal rate for any smoothness level 2 starting with a preliminary estimator with any power law rate at the first stage

quantile regression with censoring and endogeneity in this paper we develop a new censored quantile instrumental variable cqiv estimator and describe its properties and computation the cqiv estimator combines powell 1986 censored quantile regression cqr to deal with censoring with a control variable approach to incorporate endogenous regressors the cqiv estimator is obtained in two stages that are nonadditive in the unobservables the first stage estimates a nonadditive model with infinite dimensional parameters for the control variable such as a quantile or distribution regression model the second stage estimates a nonadditive censored quantile regression model for the response variable of interest including the estimated control variable to deal with endogeneity for computation we extend the algorithm for cqr developed by chernozhukov and hong 2002 to incorporate the estimation of the control variable we give generic regularity conditions for asymptotic normality of the cqiv estimator and for the validity of resampling methods to approximate its asymptotic distribution we verify these conditions for quantile and distribution regression estimation of the control variable our analysis covers two stage uncensored quantile regression with nonadditive first stage as an important special case we illustrate the computation and applicability of the cqiv estimator with a monte carlo numerical example and an empirical application on estimation of engel curves for alcohol

efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals this paper considers semiparametric efficient estimation ofconditional moment models with possibly nonsmooth residuals in unknown parametric components and unknown functions h of endogenous variables we show that 1 the penalized sieve minimum distance psmd estima tor can simultaneously achieve root n asymptotic normality ofand nonparametric optimal convergence rate of allowing for noncompact function parameter spaces 2 a simple weighted bootstrap procedure consistently estimates the limiting distribution of the psmd 3 the semi parametric efficiency bound formula of ai and chen 2003 remains valid for conditional models with nonsmooth residuals and the optimally weighted psmd estimator achieves the bound 4 the centered profiled optimally weighted psmd criterion is asymptotically chi square distributed we illustrate our theories using a partially linear quantile instrumental variables iv regression a monte carlo study and an empirical estimation of the shape invariant quantile iv engel curves

change point estimation under adaptive sampling we consider the problem of locating a jump discontinuity change point in a smooth parametric regression model with a bounded covariate it is assumed that one can sample the covariate at different values and measure the corresponding responses budget constraints dictate that a total of n such measurements can be obtained a multistage adaptive procedure is proposed where at each stage an estimate of the change point is obtained and new points are sampled from its appropriately chosen neighborhood it is shown that such procedures accelerate the rate of convergence of the least squares estimate of the change point further the asymptotic distribution of the estimate is derived using empirical processes techniques the latter result provides guidelines on how to choose the tuning parameters of the multistage procedure in practice the improved efficiency of the procedure is demonstrated using real and synthetic data this problem is primarily motivated by applications in engineering systems

twicing kernels and a small bias property of semiparametric estimators the purpose of this note is to show how semiparametric estimators with a small bias property can be constructed the small bias property sbp of a semiparametric estimator is that its bias converges to zero faster than the pointwise and integrated bias of the nonparametric estimator on which it is based we show that semiparametric estimators based on twicing kernels have the sbp we also show that semiparametric estimators where nonparametric kernel estimation does not affect the asymptotic variance have the sbp in addition we discuss an interpretation of series and sieve estimators as idempotent transformations of the empirical distribution that helps explain the known result that they lead to the sbp in monte carlo experiments we find that estimators with the sbp have mean square error that is smaller and less sensitive to bandwidth than those that do not have the sbp

nonparametric estimators which can be plugged in we consider nonparametric estimation of an object such as a probability density or a regression function can such an estimator achieve the ratewise minimax rate of convergence on suitable function spaces while at the same time when plugged in estimate efficiently at a rate of n 1 2 with the best constant many functionals of the object for example can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function we show that this is impossible for very large sets for example expectations of all functions bounded by m however we also show that it is possible for sets as large as indicators of all quadrants that is distribution functions we give appropriate constructions of such estimates

consistent estimation of limited dependent variable models despite misspecification of distribution abstract by exploiting conditions on the behavior of regressors estimators are developed which are consistent up to a factor of proportionality for the slope coefficients of the regression function of a imperfectly observed dependent variable limited dependent variables are a special case of interest the consistency is robust to misspecification of the distribution of the latent dependent variable the estimators are weighted pseudo maximum likelihood estimators which can be readily computed using popular methods and software

estimation of marginal effects in semiparametric selection models with binary outcomes this paper addresses the estimation of a semiparametric sample selection index model where both the selection rule and the outcome variable are binary since the marginal effects are often of primary interest and are difficult to recover in a semiparametric setting we focus on developing an estimator for the marginal effects this marginal effect estimator uses only observations where the selection probability is above a certain threshold a key innovation is that this high probability set is adaptive to the data we establish the large sample properties of the marginal effect estimator as well as those for an index estimator upon which it depends monte carlo studies show that these estimators perform well in finite samples

accelerated randomized stochastic optimization we propose a general class of randomized gradient estimates to be employed in a recursive search for the minimum of an unknown multivariate regression function here only two observations per iteration step are used special cases include random direction stochastic approximation kushner and clark simultaneous perturbation stochastic approximation spall and a special kernel based stochastic approximation method polyak and tsybakov if the unknown regression is p smooth p 2 at the point of minimum these methods achieve the optimal rate of convergence o n p 1 2p for both the classical stochastic approximation scheme kiefer and wolfowitz and the averaging scheme ruppert and polyak the related asymptotic distributions are computed

the asymptotic variance of semiparametric estimators this paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions the general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean square projections including conditional expectations regularity conditions for the validity of the formula are also given including primitive conditions for asymptotic normality when series estimators are present new examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative copyright 1994 by the econometric society

smoothing bias in density derivative estimation abstract this article discusses a generic feature of density estimation by local smoothing namely that estimated derivatives and location score vectors will display a systematic downward attenuation bias we study the behavior of kernel estimators indicating how the derivative bias arises and showing a simple result we then consider the estimation of score vectors negative log density derivatives which are motivated by the problem of estimating average derivatives and the adaptive estimation of regression models using fixed bandwidth limits we show how scores are proportionally downward biased for normal densities and argue from normal mixture densities that proportional bias can be a reasonable approximation we propose a simple diagnostic statistic for score bias

unstable solutions of the euler equations of certain variational problems soit cr n un domaine borne avec une frontiere appartenant a c 2 pour 0 1 soit f 0 f 0 x z p appartenant a c loc 2 r r n et soit f 0 x z r n r convexe pour tout x z r alors pour u c 1 on definit i 0 t u f 0 x u u dx on considere le probleme variationnel engendre en restreignant i 0 a l ensemble u c 1 u avec c 2

kernel based semiparametric estimators small bandwidth asymptotics and bootstrap consistency this paper develops asymptotic approximations for kernel based semiparametric estimators under assumptions accommodating slower than usual rates of convergence of their nonparametric ingredients our first main result is a distributional approximation for semiparametric estimators that differs from existing approximations by accounting for a bias this bias is nonnegligible in general and therefore poses a challenge for inference our second main result shows that some but not all nonparametric bootstrap distributional approximations provide an automatic method of correcting for the bias our general theory is illustrated by means of examples and its main finite sample implications are corroborated in a simulation study

a two stage hybrid procedure for estimating an inverse regression function we consider a two stage procedure tsp for estimating an inverse regression function at a given point where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two we establish that the convergence rate of the second stage estimate can attain the parametric n 1 2 rate furthermore a bootstrapped variant of tsp btsp is introduced and its consistency properties studied this variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity finally the finite sample performance of btsp is studied through simulations and the method is illustrated on a data set

on methods of sieves and penalization we develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates mle s in parametric semiparametric and nonparametric models we find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices the degree of smoothness of the functional and the local size of the underlying parameter space we show that when the local size of the parameter space is not very large the substitution standard nonsieve substitution sieve and substitution penalized mle s are asymptotically efficient in the fisher sense under certain stochastic equicontinuity conditions of the loglikelihood moreover when the convergence rate of the estimate is slow the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency when the size of the parameter space is very large the standard and penalized maximum likelihood procedures may be inefficient whereas the method of sieves may be able to overcome this difficulty this phenomenon is particularly manifested when the functional of interest is very smooth especially in the semiparametric case

estimating a class of triangular simultaneous equations models without exclusion restrictions this paper provides a control function estimator to adjust for endogeneity in the triangular simultaneous equations model where there are no available exclusion restrictions to generate suitable instruments our approach is to exploit the dependence of the errors on exogenous variables e g heteroscedasticity to adjust the conventional control function estimator the form of the error dependence on the exogenous variables is subject to restrictions but is not parametrically specified in addition to providing the estimator and deriving its large sample properties we present simulation evidence which indicates the estimator works well

robust data driven inference for density weighted average derivatives this paper presents a novel data driven bandwidth selector compatible with the small bandwidth asymptotics developed in cattaneo crump and jansson 2009 for density weighted average derivatives the new bandwidth selector is of the plug in variety and is obtained based on a mean squared error expansion of the estimator of interest an extensive monte carlo experiment shows a remarkable improvement in performance when the bandwidth dependent robust inference procedures proposed by cattaneo crump and jansson 2009 are coupled with this new data driven bandwidth selector the resulting robust data driven confidence intervals compare favorably to the alternative procedures available in the literature the online supplemental material to this paper contains further results from the simulation study