  The family of rank estimators, including Han's maximum rank correlation (Han,
1987) as a notable example, has been widely exploited in studying regression
problems. For these estimators, although the linear index is introduced for
alleviating the impact of dimensionality, the effect of large dimension on
inference is rarely studied. This paper fills this gap via studying the
statistical properties of a larger family of M-estimators, whose objective
functions are formulated as U-processes and may be discontinuous in increasing
dimension set-up where the number of parameters, $p_{n}$, in the model is
allowed to increase with the sample size, $n$. First, we find that often in
estimation, as $p_{n}/n\rightarrow 0$, $(p_{n}/n)^{1/2}$ rate of convergence is
obtainable. Second, we establish Bahadur-type bounds and study the validity of
normal approximation, which we find often requires a much stronger scaling
requirement than $p_{n}^{2}/n\rightarrow 0.$ Third, we state conditions under
which the numerical derivative estimator of asymptotic covariance matrix is
consistent, and show that the step size in implementing the covariance
estimator has to be adjusted with respect to $p_{n}$. All theoretical results
are further backed up by simulation studies.
