  We study how the divide and conquer principle --- partition the available
data into subsamples, compute an estimate from each subsample and combine these
appropriately to form the final estimator --- works in non-standard problems
where rates of convergence are typically slower than $\sqrt{n}$ and limit
distributions are non-Gaussian, with a special emphasis on the least squares
estimator (and its inverse) of a monotone regression function. We find that the
pooled estimator, obtained by averaging non-standard estimates across the
mutually exclusive subsamples, outperforms the non-standard estimator based on
the entire sample in the sense of pointwise inference. We also show that, under
appropriate conditions, if the number of subsamples is allowed to increase at
appropriate rates, the pooled estimator is asymptotically normally distributed
with a variance that is empirically estimable from the subsample-level
estimates. Further, in the context of monotone function estimation we show that
this gain in pointwise efficiency comes at a price --- the pooled estimator's
performance, in a uniform sense (maximal risk) over a class of models worsens
as the number of subsamples increases, leading to a version of the
super-efficiency phenomenon. In the process, we develop analytical results for
the order of the bias in isotonic regression, which are of independent
interest.
