  For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues
$\lambda_1, \ldots, \lambda_n$, the seminal work of Rider and Silverstein
asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n
f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test
functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$,
where $K$ randomly chosen eigenvalues have been removed from the sum. In this
case, we identify the limiting distribution and show that it need not be
Gaussian. Our results hold for the case when $K$ is fixed as well as the case
when $K$ tends to infinity with $n$.
  The proof utilizes the predicted locations of the eigenvalues introduced by
E. Meckes and M. Meckes. As a consequence of our methods, we obtain a rate of
convergence for the empirical spectral distribution of $X_n$ to the circular
law in Wasserstein distance, which may be of independent interest.
