387 research outputs found
Multivariate varying coefficient model for functional responses
Motivated by recent work studying massive imaging data in the neuroimaging
literature, we propose multivariate varying coefficient models (MVCM) for
modeling the relation between multiple functional responses and a set of
covariates. We develop several statistical inference procedures for MVCM and
systematically study their theoretical properties. We first establish the weak
convergence of the local linear estimate of coefficient functions, as well as
its asymptotic bias and variance, and then we derive asymptotic bias and mean
integrated squared error of smoothed individual functions and their uniform
convergence rate. We establish the uniform convergence rate of the estimated
covariance function of the individual functions and its associated eigenvalue
and eigenfunctions. We propose a global test for linear hypotheses of varying
coefficient functions, and derive its asymptotic distribution under the null
hypothesis. We also propose a simultaneous confidence band for each individual
effect curve. We conduct Monte Carlo simulation to examine the finite-sample
performance of the proposed procedures. We apply MVCM to investigate the
development of white matter diffusivities along the genu tract of the corpus
callosum in a clinical study of neurodevelopment.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1045 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Perturbation and scaled Cook's distance
Cook's distance [Technometrics 19 (1977) 15-18] is one of the most important
diagnostic tools for detecting influential individual or subsets of
observations in linear regression for cross-sectional data. However, for many
complex data structures (e.g., longitudinal data), no rigorous approach has
been developed to address a fundamental issue: deleting subsets with different
numbers of observations introduces different degrees of perturbation to the
current model fitted to the data, and the magnitude of Cook's distance is
associated with the degree of the perturbation. The aim of this paper is to
address this issue in general parametric models with complex data structures.
We propose a new quantity for measuring the degree of the perturbation
introduced by deleting a subset. We use stochastic ordering to quantify the
stochastic relationship between the degree of the perturbation and the
magnitude of Cook's distance. We develop several scaled Cook's distances to
resolve the comparison of Cook's distance for different subset deletions.
Theoretical and numerical examples are examined to highlight the broad spectrum
of applications of these scaled Cook's distances in a formal influence
analysis.Comment: Published in at http://dx.doi.org/10.1214/12-AOS978 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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