544,358 research outputs found
Functional linear regression analysis for longitudinal data
We propose nonparametric methods for functional linear regression which are
designed for sparse longitudinal data, where both the predictor and response
are functions of a covariate such as time. Predictor and response processes
have smooth random trajectories, and the data consist of a small number of
noisy repeated measurements made at irregular times for a sample of subjects.
In longitudinal studies, the number of repeated measurements per subject is
often small and may be modeled as a discrete random number and, accordingly,
only a finite and asymptotically nonincreasing number of measurements are
available for each subject or experimental unit. We propose a functional
regression approach for this situation, using functional principal component
analysis, where we estimate the functional principal component scores through
conditional expectations. This allows the prediction of an unobserved response
trajectory from sparse measurements of a predictor trajectory. The resulting
technique is flexible and allows for different patterns regarding the timing of
the measurements obtained for predictor and response trajectories. Asymptotic
properties for a sample of subjects are investigated under mild conditions,
as , and we obtain consistent estimation for the regression
function. Besides convergence results for the components of functional linear
regression, such as the regression parameter function, we construct asymptotic
pointwise confidence bands for the predicted trajectories. A functional
coefficient of determination as a measure of the variance explained by the
functional regression model is introduced, extending the standard to the
functional case. The proposed methods are illustrated with a simulation study,
longitudinal primary biliary liver cirrhosis data and an analysis of the
longitudinal relationship between blood pressure and body mass index.Comment: Published at http://dx.doi.org/10.1214/009053605000000660 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Varying-coefficient functional linear regression
Functional linear regression analysis aims to model regression relations
which include a functional predictor. The analog of the regression parameter
vector or matrix in conventional multivariate or multiple-response linear
regression models is a regression parameter function in one or two arguments.
If, in addition, one has scalar predictors, as is often the case in
applications to longitudinal studies, the question arises how to incorporate
these into a functional regression model. We study a varying-coefficient
approach where the scalar covariates are modeled as additional arguments of the
regression parameter function. This extension of the functional linear
regression model is analogous to the extension of conventional linear
regression models to varying-coefficient models and shares its advantages, such
as increased flexibility; however, the details of this extension are more
challenging in the functional case. Our methodology combines smoothing methods
with regularization by truncation at a finite number of functional principal
components. A practical version is developed and is shown to perform better
than functional linear regression for longitudinal data. We investigate the
asymptotic properties of varying-coefficient functional linear regression and
establish consistency properties.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ231 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The Application of Robust Regression to a Production Function Comparison – the Example of Swiss Corn
The adequate representation of crop response functions is crucial for agri-environmental modeling and analysis. So far, the evaluation of such functions focused on the comparison of different functional forms. The perspective is expanded in this article by considering an alternative regression method. This is motivated by the fact that exceptional crop yield observations (outliers) can cause misleading results if least squares regression is applied. We show that such outliers are adequately treated if robust regression is used instead. The example of simulated Swiss corn yields shows that the use of robust regression narrows the range of optimal input levels across different functional forms and reduces potential costs of misspecification.production function estimation, production function comparison, robust regression, crop response
Local linear regression for functional predictor and scalar response
The aim of this work is to introduce a new nonparametric regression technique in the context of
functional covariate and scalar response. We propose a local linear regression estimator and study
its asymptotic behaviour. Its finite-sample performance is compared with a Nadayara-Watson type
kernel regression estimator via a Monte Carlo study and the analysis of two real data sets. In all
the scenarios considered, the local linear regression estimator performs better than the kernel one,
in the sense that the mean squared prediction error and its standard deviation are lower
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
Nonparametric and Varying Coefficient Modal Regression
In this article, we propose a new nonparametric data analysis tool, which we
call nonparametric modal regression, to investigate the relationship among
interested variables based on estimating the mode of the conditional density of
a response variable Y given predictors X. The nonparametric modal regression is
distinguished from the conventional nonparametric regression in that, instead
of the conditional average or median, it uses the "most likely" conditional
values to measures the center. Better prediction performance and robustness are
two important characteristics of nonparametric modal regression compared to
traditional nonparametric mean regression and nonparametric median regression.
We propose to use local polynomial regression to estimate the nonparametric
modal regression. The asymptotic properties of the resulting estimator are
investigated. To broaden the applicability of the nonparametric modal
regression to high dimensional data or functional/longitudinal data, we further
develop a nonparametric varying coefficient modal regression. A Monte Carlo
simulation study and an analysis of health care expenditure data demonstrate
some superior performance of the proposed nonparametric modal regression model
to the traditional nonparametric mean regression and nonparametric median
regression in terms of the prediction performance.Comment: 33 page
Using Quantile Regression for Duration Analysis
Quantile regression methods are emerging as a popular technique in econometrics and biometrics for exploring the distribution of duration data. This paper discusses quantile regression for duration analysis allowing for a flexible specification of the functional relationship and of the error distribution. Censored quantile regression address the issue of right censoring of the response variable which is common in duration analysis. We compare quantile regression to standard duration models. Quantile regression do not impose a proportional effect of the covariates on the hazard over the duration time. However, the method can not take account of time{varying covariates and it has not been extended so far to allow for unobserved heterogeneity and competing risks. We also discuss how hazard rates can be estimated using quantile regression methods. A small application with German register data on unemployment duration for younger workers demonstrates the applicability and the usefulness of quantile regression for empirical duration analysis. --censored quantile regression,unemployment duration,unobserved heterogeneity,hazard rate
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