TITLE:
Nonparametric Estimation in Linear Mixed Models with Uncorrelated Homoscedastic Errors
AUTHORS:
Eugène-Patrice Ndong Nguéma, Betrand Fesuh Nono, Henri Gwét
KEYWORDS:
Clustered Data, Linear Mixed Model, Fixed Effect, Uncorrelated Homoscedastic Error, Random Effects Predictor
JOURNAL NAME:
Open Journal of Statistics,
Vol.11 No.4,
August
30,
2021
ABSTRACT: Today, Linear Mixed Models (LMMs) are fitted, mostly,
by assuming that random effects and errors have Gaussian distributions,
therefore using Maximum Likelihood (ML) or REML estimation. However, for many
data sets, that double assumption is unlikely to hold, particularly for the
random effects, a crucial component in which assessment of magnitude is key in such
modeling. Alternative fitting methods not relying on that assumption (as ANOVA
ones and Rao’s MINQUE) apply, quite often, only to the very
constrained class of variance components models. In this paper, a new
computationally feasible estimation methodology is designed, first for the
widely used class of 2-level (or longitudinal) LMMs with only assumption
(beyond the usual basic ones) that residual errors are uncorrelated and
homoscedastic, with no distributional assumption imposed on the random effects.
A major asset of this new approach is that it yields nonnegative variance
estimates and covariance matrices estimates which are symmetric and, at least,
positive semi-definite. Furthermore, it is shown that when the LMM is, indeed,
Gaussian, this new methodology differs from ML just through a slight variation
in the denominator of the residual variance estimate. The new methodology
actually generalizes to LMMs a well known nonparametric fitting procedure for
standard Linear Models. Finally, the methodology is also extended to ANOVA
LMMs, generalizing an old method by Henderson for ML estimation in such models
under normality.