Estimating α-frontierα-frontier technical efficiency with shape-restricted kernel quantile regression

In frontier analysis, most of nonparametric approaches produce a full frontier that envelopes all observations. Its sensitivity to extreme values and outliers can be overcome by α-frontierα-frontier, which is defined as the α-quantileα-quantile of the output conditional on a given input. The α-frontierα-frontier can be regarded as the benchmark output whether a specified firm achieves top αα efficiency. This paper proposes a nonparametric smooth multivariate estimation for α-frontierα-frontier based on shape-restricted kernel quantile regression. This method explicitly appends the classical kernel quantile regression with two shape restrictions: nondecreasing and concave, which are necessary conditions for production functions. Its training is a semi-infinite programming and can be discretized to a semidefinite programming problem, which is computationally tractable. Theoretical analysis shows that the rate of exceedance in the samples will converge to αα as the size of training data increases. Experimental results on two toy data sets clearly show that this exploitation of these prior shape knowledge can greatly improve learning performance. Experimental results on a data set from the NBER-CES Manufacturing Industry Database clearly show that the shaped restricted kernel quantile regression can achieve better out-of-sample performance than those of two benchmark methods.