  The ordinary Bayes estimator based on the posterior density suffers from the
potential problems of non-robustness under data contamination or outliers. In
this paper, we consider the general set-up of independent but non-homogeneous
(INH) observations and study a robustified pseudo-posterior based estimation
for such parametric INH models. In particular, we focus on the
$R^{(\alpha)}$-posterior developed by Ghosh and Basu (2016) for IID data and
later extended by Ghosh and Basu (2017) for INH set-up, where its usefulness
and desirable properties have been numerically illustrated. In this paper, we
investigate the detailed theoretical properties of this robust pseudo Bayes
$R^{(\alpha)}$-posterior and associated $R^{(\alpha)}$-Bayes estimate under the
general INH set-up with applications to fixed-design regressions. We derive a
Bernstein von-Mises types asymptotic normality results and Laplace type
asymptotic expansion of the $R^{(\alpha)}$-posterior as well as the asymptotic
distributions of the expected $R^{(\alpha)}$-posterior estimators. The required
conditions and the asymptotic results are simplified for linear regressions
with known or unknown error variance and logistic regression models with fixed
covariates. The robustness of the $R^{(\alpha)}$-posterior and associated
estimators are theoretically examined through appropriate influence function
analyses under general INH set-up; illustrations are provided for the case of
linear regression. A high breakdown point result is derived for the expected
$R^{(\alpha)}$-posterior estimators of the location parameter under a
location-scale type model. Some interesting real life data examples illustrate
possible applications.
