  Influential diagnosis is an integral part of data analysis, of which most
existing methodological frameworks presume a deterministic submodel and are
designed for low-dimensional data (i.e., the number of predictors p smaller
than the sample size n). However, the stochastic selection of a submodel from
high-dimensional data where p exceeds n has become ubiquitous. Thus, methods
for identifying observations that could exert undue influence on the choice of
a submodel can play an important role in this setting. To date, discussion of
this topic has been limited, falling short in two domains: (i) constrained
ability to detect multiple influential points, and (ii) applicability only in
restrictive settings. After describing the problem, we characterize and
formalize the concept of influential observations on variable selection. Then,
we propose a generalized diagnostic measure, extended from an available metric
accommodating different model selectors and multiple influential observations,
the asymptotic distribution of which is subsequently establish large p, thus
providing guidelines to ascertain influential observations. A high-dimensional
clustering procedure is further incorporated into our proposed scheme to detect
multiple influential points. Simulation is conducted to assess the performances
of various diagnostic approaches. The proposed procedure further demonstrates
its value in improving predictive power when analyzing thermal-stimulated pain
based on fMRI data.
