  Generalized linear mixed models are powerful tools for analyzing clustered
data, where the unknown parameters are classically (and most commonly)
estimated by the maximum likelihood and restricted maximum likelihood
procedures. However, since the likelihood based procedures are known to be
highly sensitive to outliers, M-estimators have become popular as a means to
obtain robust estimates under possible data contamination. In this paper, we
prove that, for sufficiently smooth general loss functions defining the
M-estimators in generalized linear mixed models, the tail probability of the
deviation between the estimated and the true regression coefficients have an
exponential bound. This implies an exponential rate of consistency of these
M-estimators under appropriate assumptions, generalizing the existing
exponential consistency results from univariate to multivariate responses. We
have illustrated this theoretical result further for the special examples of
the maximum likelihood estimator and the robust minimum density power
divergence estimator, a popular example of model-based M-estimators, in the
settings of linear and logistic mixed models, comparing it with the empirical
rate of convergence through simulation studies.
