a semiparametric single index risk score across populations abstractwe consider a problem motivated by issues in nutritional epidemiology across diseases and populations in this area it is becoming increasingly common for diseases to be modeled by a single diet score such as the healthy eating index the mediterranean diet score etc for each disease and for each population a partially linear single index model is fit the partially linear aspect of the problem is allowed to differ in each population and disease however and crucially the single index itself having to do with the diet score is common to all diseases and populations and the nonparametrically estimated functions of the single index are the same up to a scale parameter using b splines with an increasing number of knots we develop a method to solve the problem and display its asymptotic theory an application to the nih aarp study of diet and health is described where we show the advantages of using multiple diseases and populations simultaneously rather than one at a time in understand

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

convolutional autoregressive models for functional time series abstract functional data analysis has became an increasingly popular class of problems in statistical research however functional data observed over time with serial dependence remains a less studied area motivated by bosq 2000 who first introduced the functional autoregressive models we propose a convolutional functional autoregressive model where the function at time t is a result of the sum of convolutions of the past functions and a set of convolution functions plus a noise process mimicking the vector autoregressive process it provides an intuitive and direct interpretation of the dynamics of a stochastic process instead of principal component analysis commonly used in functional data analysis we adopt a sieve estimation procedure based on b spline approximation of the convolution functions we establish convergence rate of the proposed estimator and investigate its theoretical properties the model building model validation and prediction procedures are also developed both simulated and real data examples are presented

estimation of the mean function with panel count data using monotone polynomial splines we study nonparametric likelihood based estimators of the mean function of counting processes with panel count data using monotone polynomial splines the generalized rosen algorithm proposed by zhang jamshidian 2004 is used to compute the estimators we show that the proposed spline likelihood based estimators are consistent and that their rate of convergence can be faster than n 1 3 simulation studies with moderate samples show that the estimators have smaller variances and mean squared errors than their alternatives proposed by wellner zhang 2000 a real example from a bladder tumour clinical trial is used to illustrate this method

a unified framework for spline estimators this dissertation develops a uni ed framework to study the asymptotic properties of all periodic spline based estimators that is of regression penalized and smoothing splines the explicit form of the periodic demmler reinsch basis of general degree in terms of exponential splines allows to derive the exact expression for the equivalent kernel of all spline estimators simultaneously the corresponding bandwidth which drives the asymptotic behavior of spline estimators is shown to be a function of both the number of knots and the smoothing parameter a strategy for the optimal bandwidth selection is discussed

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible

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

sieve m inference on irregular parameters abstract this paper presents sieve inferences on possibly irregular i e slower than root n estimable functionals of semi nonparametric models with i i d data we provide a simple consistent variance estimator of the plug in sieve m estimator of a possibly irregular functional and the asymptotic standard normality of the sieve t statistic we show that for hypothesis testing of irregular functionals the sieve likelihood ratio statistic is asymptotically chi square distributed these results are useful in inference on structural parameters that may have singular semiparametric efficiency bounds a simulation study and an empirical application of heckman and singer 1984 duration model are presented

semiparametric estimation methods for panel count data using monotone b splines we study semiparametric likelihood based methods for panel count data with proportional mean model e t z 0 t exp 0tz where z is a vector of covariates and 0 t is the baseline mean function we propose to estimate 0 t and 0 jointly with 0 t approximated by monotone b splines and to compute the estimators using generalized rosen algorithm proposed by jamshidian 2004 we show that the proposed spline based likelihood estimators of 0 t are consistent with a possibly better than n1 3 convergence rate if 0 t is sufficiently smooth the normality of the estimators of 0 is also established comparisons between the proposed estimators and their alternatives studied in wellner and zhang 2007 are made through simulations studies regarding their finite sample performance and computational complexity a real example from a bladder tumor clinical trial is used to illustrate the methods

free knot polynomial splines with confidence intervals the estimates of knot locations and coefficients are obtained through a non linear least squares solution that corresponds to the maximum likelihood estimate confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator average coverage probabilities and the accuracy of the estimate are examined via simulation this includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a bayesian version of the variable knots method simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions it also performs well for fairly small sample sizes

breaking the curse of nonregularity with subagging inference of the mean outcome under optimal treatment regimes precision medicine is an emerging medical approach that allows physicians to select the treatment options based on individual patient information the goal of precision medicine is to identify the optimal treatment regime otr that yields the most favorable clinical outcome prior to adopting any otr in clinical practice it is crucial to know the impact of implementing such a policy although considerable research has been devoted to estimating the otr in the literature less attention has been paid to statistical inference of the otr challenges arise in the nonregular cases where the otr is not uniquely defined to deal with nonregularity we develop a novel inference method for the mean outcome under an otr the optimal value function based on subsample aggregating subagging the proposed method can be applied to multi stage studies where treatments are sequentially assigned over time bootstrap aggregating bagging and subagging have been recognized as effective variance reduction techniques to improve unstable estimators or classifiers buhlmann and yu 2002 however it remains unknown whether these approaches can yield valid inference results we show the proposed confidence interval ci for the optimal value function achieves nominal coverage in addition due to the variance reduction effect of subagging our method enjoys certain statistical optimality specifically we show that the mean squared error of the proposed value estimator is strictly smaller than that based on the simple sample splitting estimator in the nonregular cases moreover under certain conditions the length of our proposed ci is shown to be on average shorter than cis constructed based on the existing state of the art method luedtke and van der laan 2016 and the oracle method which works as well as if an otr were known extensive numerical studies are conducted to back up our theoretical findings

a density quantile function approach to optimal spacing selection in this paper design techniques for continuous parameter time series regression analysis are employed to develop a general approach to optimal spacing selection for the linear estimation of location and scale parameters by sample quantiles from uncensored or censored samples the spacings derived from this approach are asymptotically optimal in the sense that they result in near optimal asymptotic relative efficiencies for large values of k the number of spacing elements a comparison with the optimum efficiencies for several distribution types indicates that the asymptotically optimum spacings perform well for k 7 the regression framework is also utilized to develop sufficient conditions for optimal spacing unicity and to obtain asymptotically optimal spacings for quantile estimation

semi nonparametric estimation of the call option price surface under strike and time to expiry no arbitrage constraints abstract we suggest a semi nonparametric estimator for the call option price surface the estimator is a bivariate tensor product b spline to enforce no arbitrage constraints across strikes and expiry dates we establish sufficient no arbitrage conditions on the control net of the b spline surface the conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques the consistency of the estimator is proved by means of simulations we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state price densities under the full set of no arbitrage constraints we estimate a call option price surface families of first order strike derivatives and state price densities for s p 500 option data

spike and slab group lassos for grouped regression and sparse generalized additive models abstract we introduce the spike and slab group lasso ssgl for bayesian estimation and variable selection in linear regression with grouped variables we further extend the ssgl to sparse generali

hotelling s theorem on the volume of tubes some illustrations in simultaneous inference and data analysis we illustrate with contemporary examples hotelling s geometric approach to simultaneous probability calculations hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere and shows that this volume is often exactly given by length times cross sectional area we review hotelling s result together with some recent complements and then use the approach to set simultaneous prediction regions for some data from gait analysis to study andrews plots in multivariate data analysis and to construct significance tests for projection pursuit regression a by product is a numerical criterion for tube self overlap relevant for example to uniqueness of certain nonlinear least squares estimates

sparse additive ordinary differential equations for dynamic gene regulatory network modeling the gene regulation network grn is a high dimensional complex system which can be represented by various mathematical or statistical models the ordinary differential equation ode model is one of the popular dynamic grn models high dimensional linear ode models have been proposed to identify grns but with a limitation of the linear regulation effect assumption in this article we propose a sparse additive ode sa ode model coupled with ode estimation methods and adaptive group least absolute shrinkage and selection operator lasso techniques to model dynamic grns that could flexibly deal with nonlinear regulation effects the asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach an application example for identifying the nonlinear dynamic grn of t cell activation is used to illustrate the usefulness of the proposed method

on the asymptotics of marginal regression splines with longitudinal data summary there have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within cluster correlation when nonparametric regression models are used on longitudinal or cluster data in particular methods based on smoothing splines and local polynomial kernels exhibit different behaviour we show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property the asymptotic bias of the estimator does not depend on the working correlation matrix but the asymptotic variance and therefore the mean squared error is minimized when the true correlation structure is specified this property of the asymptotic bias distinguishes regression splines from smoothing splines

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

local asymptotics for quantile smoothing splines quantile smoothing splines were introduced by koenker ng and portnoy as natural and appealing estimates of conditional quantiles of response variables the natural setting for the problem considers minimization of a weighted combination of a fit penalty and a roughness penalty over the space of functions whose derivatives have bounded variation although this space is not traditional shen has shown recently that the quantile smoothing splines do indeed converge at the usual optimal rate n 2 5 in various norms here local asymptotic results are obtained by establishing bahadur representations for local parameters of the splines these are used to obtain local rates of convergence to establish uniform convergence rates to provide local distribution theory for quantile b splines and to expand the fit measure in order to analyze an information criterion for determining the smoothing parameter examples of using derivatives of the smoothing splines for estimating jump functions are also presented

asymptotic properties of penalized spline estimators we study the class of penalized spline estimators which enjoy similarities to both regression splines without penalty and with fewer knots than data points and smoothing splines with knots equal to the data points and a penalty controlling the roughness of the fit depending on the number of knots sample size and penalty we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines with a clear breakpoint distinguishing the cases we prove that using fewer knots results in better asymptotic rates than when using a large number of knots we obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter

simultaneous confidence bands for penalized spline estimators in this article we construct simultaneous confidence bands for a smooth curve using penalized spline estimators we consider three types of estimation methods a as a standard fixed effect nonparametric model b using the mixed model framework with the spline coefficients as random effects and c a full bayesian approach the volume of tube formula is applied for the first two methods and compared with bayesian simultaneous confidence bands from a frequentist perspective we show that the mixed model formulation of penalized splines can help obtain at least approximately confidence bands with either bayesian or frequentist properties simulations and data analysis support the proposed methods the r package confbands accompanies the article

uniform confidence bands for functions estimated nonparametrically with instrumental variables abstract this paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables we show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed the asymptotic normality result holds in both mildly and severely ill posed cases we present methods to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values monte carlo experiments illustrate the finite sample performance of the uniform confidence band

efficient estimation and computation of parameters and nonparametric functions in generalized semi non parametric regression models abstract the efficiency of estimation for the parameters in semiparametric models has been widely studied in the literature bickel et al 1993 in this paper we study efficient estimators for both parameters and nonparametric functions in a class of generalized semi non parametric regression models which cover commonly used semiparametric models such as partially linear models partially linear single index models and two sample semiparametric models we propose a maximum likelihood principle combined with the local linear technique for estimating the parameters and nonparametric functions the proposed estimators of the parameters and a linear functional of the nonparametric functions are consistent and asymptotically normal and are further shown to be semiparametrically efficient an efficient computational algorithm to achieve the maximization is proposed extensive simulation experiments show the superiority of the proposed methods three real data examples are analyzed and presented as an illustration

optimally weighted l2 distance for functional data many techniques of functional data analysis require choosing a measure of distance between functions with the most common choice being l2 distance in this article we show that using a weighted l2 distance with a judiciously chosen weight function can improve the performance of various statistical methods for functional data including k medoids clustering nonparametric classification and permutation testing assuming a quadratically penalized e g spline basis representation for the functional data we consider three nontrivial weight functions design density weights inverse variance weights and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure the benefits of weighting in particular with the proposed weight function are demonstrated both in simulation studies and in applications to the berkeley growth data and a functional magnetic resonance imaging data set

local asymptotics for regression splines and confidence regions in this paper we study the local behavior of regression splines in particular explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained in addition asymptotic normality for regression splines is established leading to the construction of approximate confidence intervals and confidence bands for the regression function

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

error variance estimation in ultrahigh dimensional additive models abstracterror variance estimation plays an important role in statistical inference for high dimensional regression models this article concerns with error variance estimation in high dimensional sparse additive model we study the asymptotic behavior of the traditional mean squared errors the naive estimate of error variance and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models we further propose an accurate estimate for error variance in ultrahigh dimensional sparse additive model by effectively integrating sure independence screening and refitted cross validation techniques the root n consistency and the asymptotic normality of the resulting estimate are established we conduct monte carlo simulation study to examine the finite sample performance of the newly proposed estimate a real data example is used to illustrate the proposed methodology supplementary materials for this article are avai

preincavity and fuzzy decision making since almost all practical problems are fuzzy and approximate fuzzy decision making becomes one of the most important practical approaches however the resulting problems are frequently complicated and difficult to solve one effective way to overcome these difficulties is to explore the concavity or generalized concavity properties of the resulting problems in this paper we introduce and study the concept of supp preincave and supp prequasiincave fuzzy sets we give characterizations for a supp preincave fuzzy set in terms of its fuzzy hypograph and a supp prequasiincave fuzzy set in terms of its level sets furthermore we also prove that any local maximizer of a supp preincave fuzzy set is also a global maximizer and that any strictly local maximizer of a supp prequasiincave fuzzy set is also a global maximizer finally some aggregation rules for supp preincave and supp prequasiincave fuzzy sets are given and some applications to fuzzy decision making are discussed

on quantile regression in reproducing kernel hilbert spaces with the data sparsity constraint for spline regressions it is well known that the choice of knots is crucial for the performance of the estimator as a general learning framework covering the smoothing splines learning in a reproducing kernel hilbert space rkhs has a similar issue however the selection of training data points for kernel functions in the rkhs representation has not been carefully studied in the literature in this paper we study quantile regression as an example of learning in a rkhs in this case the regular squared norm penalty does not perform training data selection we propose a data sparsity constraint that imposes thresholding on the kernel function coefficients to achieve a sparse kernel function representation we demonstrate that the proposed data sparsity method can have competitive prediction performance for certain situations and have comparable performance in other cases compared to that of the traditional squared norm penalty therefore the data sparsity method can serve as a competitive alternative to the squared norm penalty method some theoretical properties of our proposed method using the data sparsity constraint are obtained both simulated and real data sets are used to demonstrate the usefulness of our data sparsity constraint