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We study how the divide and conquer principle --- partition the availabledata into subsamples, compute an estimate from each subsample and combine theseappropriately to form the final estimator --- works in non-standard problemswhere rates of convergence are typically slower than $\sqrt{n}$ and limitdistributions are non-Gaussian, with a special emphasis on the least squaresestimator (and its inverse) of a monotone regression function. We find that thepooled estimator, obtained by averaging non-standard estimates across themutually exclusive subsamples, outperforms the non-standard estimator based onthe entire sample in the sense of pointwise inference. We also show that, underappropriate conditions, if the number of subsamples is allowed to increase atappropriate rates, the pooled estimator is asymptotically normally distributedwith a variance that is empirically estimable from the subsample-levelestimates. Further, in the context of monotone function estimation we show thatthis gain in pointwise efficiency comes at a price --- the pooled estimator'sperformance, in a uniform sense (maximal risk) over a class of models worsensas the number of subsamples increases, leading to a version of thesuper-efficiency phenomenon. In the process, we develop analytical results forthe order of the bias in isotonic regression, which are of independentinterest.

Divide and Conquer in Non-standard Problems and the Super-efficiency Phenomenon