arXiv subjects

Alois Kneip

Publications and source records attributed to Alois Kneip.

11 recordsLinked to original sources

arXiv2026

Combining Concurrent and Historical Functional Linear Regression

We study a function-on-function linear regression model in which the response at time $t$ depends on both the past trajectory of a predictor and its concurrent value. The model combines an $L^2$-historical effect with a point-evaluation effect, and these two coefficient functions are not automatically identifiable. We characterize the resulting non-identifiability and show that the concurrent and historical effects are separately identifiable whenever the covariance eigenfunctions of the covariance operator of the predictor are not pointwise square-summable. This mild and novel condition prevents the concurrent point evaluation from being represented by an $L^2$-historical effect. Building on an orthogonalized representation of the predictor process, we propose a smoothing-spline estimator for both coefficient functions and establish consistency rates. The rates reveal an interesting trade-off between path regularity and eigenvalue decay: smoother predictor trajectories lead to faster convergence of the historical-effect estimator, whereas rougher trajectories lead to faster convergence of the concurrent-effect estimator. Simulation studies and two real-data applications demonstrate the practical importance of disentangling concurrent and historical effects.

math.ST↗

arXiv2025

Nonparametric local polynomial regression for functional covariates

We consider nonparametric regression with functional covariates, that is, they are elements of an infinite-dimensional Hilbert space. A locally polynomial estimator is constructed, where an orthonormal basis and various tuning parameters remain to be selected. We provide a general asymptotic upper bound on the estimation error and show that this procedure achieves polynomial convergence rates under appropriate tuning and supersmoothness of the regression function. Such polynomial convergence rates have usually been considered to be non-attainable in nonparametric functional regression without any additional strong structural constraints such as linearity of the regression function.

math.ST↗

arXiv2021

A Wavelet Method for Panel Models with Jump Discontinuities in the Parameters

While a substantial literature on structural break change point analysis exists for univariate time series, research on large panel data models has not been as extensive. In this paper, a novel method for estimating panel models with multiple structural changes is proposed. The breaks are allowed to occur at unknown points in time and may affect the multivariate slope parameters individually. Our method adapts Haar wavelets to the structure of the observed variables in order to detect the change points of the parameters consistently. We also develop methods to address endogenous regressors within our modeling framework. The asymptotic property of our estimator is established. In our application, we examine the impact of algorithmic trading on standard measures of market quality such as liquidity and volatility over a time period that covers the financial meltdown that began in 2007. We are able to detect jumps in regression slope parameters automatically without using ad-hoc subsample selection criteria.

econ.EM↗

arXiv2021

Semiparametric inference for partially linear regressions with Box-Cox transformation

In this paper, a semiparametric partially linear model in the spirit of Robinson (1988) with Box- Cox transformed dependent variable is studied. Transformation regression models are widely used in applied econometrics to avoid misspecification. In addition, a partially linear semiparametric model is an intermediate strategy that tries to balance advantages and disadvantages of a fully parametric model and nonparametric models. A combination of transformation and partially linear semiparametric model is, thus, a natural strategy. The model parameters are estimated by a semiparametric extension of the so called smooth minimum distance (SmoothMD) approach proposed by Lavergne and Patilea (2013). SmoothMD is suitable for models defined by conditional moment conditions and allows the variance of the error terms to depend on the covariates. In addition, here we allow for infinite-dimension nuisance parameters. The asymptotic behavior of the new SmoothMD estimator is studied under general conditions and new inference methods are proposed. A simulation experiment illustrates the performance of the methods for finite samples.

econ.EM↗

arXiv2020

Cross-Component Registration for Multivariate Functional Data, With Application to Growth Curves

Multivariate functional data are becoming ubiquitous with advances in modern technology and are substantially more complex than univariate functional data. We propose and study a novel model for multivariate functional data where the component processes are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. To address this previously unconsidered mode of warping, we propose new registration methodology which is based on a shift-warping model. Our method differs from all existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual-specific registration, we focus on shift registration across the components of a multivariate functional data vector on a population-wide level. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence and often have intuitive physical interpretations, all in contrast to traditional curve-specific registration approaches. We demonstrate the implementation and interpretation of the proposed method by applying our methodology to the Zürich Longitudinal Growth data and study its finite sample properties in simulations.

stat.ME↗

arXiv2020

Super-Consistent Estimation of Points of Impact in Nonparametric Regression with Functional Predictors

Predicting scalar outcomes using functional predictors is a classic problem in functional data analysis. In many applications, however, only specific locations or time-points of the functional predictors have an impact on the outcome. Such ``points of impact'' are typically unknown and have to be estimated in addition to estimating the usual model components. We show that our points of impact estimator enjoys a super-consistent convergence rate and does not require knowledge or pre-estimates of the unknown model components. This remarkable result facilitates the subsequent estimation of the remaining model components as shown in the theoretical part, where we consider the case of nonparametric models and the practically relevant case of generalized linear models. The finite sample properties of our estimators are assessed by means of a simulation study. Our methodology is motivated by data from a psychological experiment in which the participants were asked to continuously rate their emotional state while watching an affective video eliciting a varying intensity of emotional reactions.

math.ST↗

arXiv2019

On the Optimal Reconstruction of Partially Observed Functional Data

We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed at $m_i$, $i=1,\dots,n$, discretization points. We derive rates of consistency for our nonparametric estimation procedures using a double asymptotic. For data situations, as in our real data application where $m_i$ is considerably smaller than $n$, we show that our functional principal components based estimator can provide better rates of convergence than conventional nonparametric smoothing methods.

math.ST↗

arXiv2016

Functional linear regression with points of impact

The paper considers functional linear regression, where scalar responses $Y_1,\ldots,Y_n$ are modeled in dependence of i.i.d. random functions $X_1,\ldots,X_n$. We study a generalization of the classical functional linear regression model. It is assumed that there exists an unknown number of "points of impact," that is, discrete observation times where the corresponding functional values possess significant influences on the response variable. In addition to estimating a functional slope parameter, the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients. Identifiability of the generalized model is considered in detail. It is shown that points of impact are identifiable if the underlying process generating $X_1,\ldots,X_n$ possesses "specific local variation." Examples are well-known processes like the Brownian motion, fractional Brownian motion or the Ornstein-Uhlenbeck process. The paper then proposes an easily implementable method for estimating the number and locations of points of impact. It is shown that this number can be estimated consistently. Furthermore, rates of convergence for location estimates, regression coefficients and the slope parameter are derived. Finally, some simulation results as well as a real data application are presented.

math.ST↗

arXiv2012

Factor models and variable selection in high-dimensional regression analysis

The paper considers linear regression problems where the number of predictor variables is possibly larger than the sample size. The basic motivation of the study is to combine the points of view of model selection and functional regression by using a factor approach: it is assumed that the predictor vector can be decomposed into a sum of two uncorrelated random components reflecting common factors and specific variabilities of the explanatory variables. It is shown that the traditional assumption of a sparse vector of parameters is restrictive in this context. Common factors may possess a significant influence on the response variable which cannot be captured by the specific effects of a small number of individual variables. We therefore propose to include principal components as additional explanatory variables in an augmented regression model. We give finite sample inequalities for estimates of these components. It is then shown that model selection procedures can be used to estimate the parameters of the augmented model, and we derive theoretical properties of the estimators. Finite sample performance is illustrated by a simulation study.

math.ST↗

arXiv2009

Smoothing splines estimators for functional linear regression

The paper considers functional linear regression, where scalar responses $Y_1,...,Y_n$ are modeled in dependence of random functions $X_1,...,X_n$. We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in an out-of-sample prediction of the response for a new random function $X_{n+1}$. It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.

math.ST↗

arXiv2009

Common functional principal components

Functional principal component analysis (FPCA) based on the Karhunen--Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation, but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the log-returns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. First we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop the two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.

math.ST↗