  The semi-parametric Cox proportional hazards regression model has been widely
used for many years in several applied sciences. However, a fully parametric
proportional hazards model, if appropriately assumed, can often lead to more
efficient inference. To tackle the extreme non-robustness of the traditional
maximum likelihood estimator in the presence of outliers in the data under such
fully parametric proportional hazard models, a robust estimation procedure has
recently been proposed extending the concept of the minimum density power
divergence estimator (MDPDE) under this set-up. In this paper, we consider the
problem of statistical inference under the parametric proportional hazards
model and develop robust Wald-type hypothesis testing and model selection
procedures using the MDPDEs. We have also derived the necessary asymptotic
results which are used to construct the testing procedure for general composite
hypothesis and study its asymptotic powers. The claimed robustness properties
are studied theoretically via appropriate influence function analyses. We have
studied the finite sample level and power of the proposed MDPDE based Wald type
test through extensive simulations where comparisons are also made with the
existing semi-parametric methods. The important issue of the selection of
appropriate robustness tuning parameter is also discussed. The practical
usefulness of the proposed robust testing and model selection procedures is
finally illustrated through three interesting real data examples.
