  Traditionally, assessing the accuracy of inference based on regression
quantiles has relied on the Bahadur representation. This provides an error of
order $n^{-1/4}$ in normal approximations, and suggests that inference based on
regression quantiles may not be as reliable as that based on other (smoother)
approaches, whose errors are generally of order $n^{-1/2}$ (or better in
special symmetric cases). Fortunately, extensive simulations and empirical
applications show that inference for regression quantiles shares the smaller
error rates of other procedures. In fact, the "Hungarian" construction of
Koml\'{o}s, Major and Tusn\'{a}dy [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131,
Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] provides an alternative expansion
for the one-sample quantile process with nearly the root-$n$ error rate
(specifically, to within a factor of $\log n$). Such an expansion is developed
here to provide a theoretical foundation for more accurate approximations for
inference in regression quantile models. One specific application of
independent interest is a result establishing that for conditional inference,
the error rate for coverage probabilities using the Hall and Sheather [J. R.
Stat. Soc. Ser. B Stat. Methodol. 50 (1988) 381-391] method of sparsity
estimation matches their one-sample rate.
