Nonparametric Knn estimation with monotone constraints

ABSTRACT

The K-nearest-neighbor (Knn) method is known to be more suitable in fitting nonparametrically specified curves than the kernel method (with a globally fixed smoothing parameter) when data sets are highly unevenly distributed. In this paper, we propose to estimate a nonparametric regression function subject to a monotonicity restriction using the Knn method. We also propose using a new convergence criterion to measure the closeness between an unconstrained and the (monotone) constrained Knn-estimated curves. This method is an alternative to the monotone kernel methods proposed by Hall and Huang (2001), and Du et al. (2013). We use a bootstrap procedure for testing the validity of the monotone restriction. We apply our method to the “Job Market Matching” data taken from Gan and Li (2016) and find that the unconstrained/constrained Knn estimators work better than kernel estimators for this type of highly unevenly distributed data.

KEYWORDS:

• Knn estimation
• monotone restriction
• nonparametric method
• testing monotonicity

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• C14

Acknowledgments

This paper was presented at the Emory University Econometrics Conference in honor of Professor Esfandiar Maasoumi in November, 2014. We thank two anonymous referees and the Emory Econometrics Conference participates for their helpful comments that greatly improved the paper. Li would also like to thank China National Science Foundation (project # 71133001) for partial research support. Liu’s research is supported by the Fundamental Research Funds for the Central Universities (project number 20720171061).

Notes

¹Although one can use adaptive bandwidth to avoid this problem, adaptive bandwidth method requires that one selects a different bandwidth at each different data evaluation point. It is computationally quite costly especially in large sample applications.

²We thank a referee pointing out that Du et al. (2013) do not require ${\sum }_{i=1}^n{p}_i=1$ and that this condition was listed in the paper is in fact a typo.

³The asymptotic analysis for the Knn method is more complex than that for the kernel method, see Mack and Rosenblatt (1979), Mack (1981), Ouyang et al. (2006), Fan and Liu (2015), among others.

⁴We would like to thank a referee for pointing out that local-polynomial estimates would serve better as the criterion in the curve method.

⁵In the simulation part, because the null hypothesis is a monotonic increasing function, we estimate all $\hat{g}\left(x\right|p)$ and $\hat{g}\left(x\right|p^{\ast })$ by selecting $p=\hat{p}$ or p* to minimize some distance functions D(p), while imposing that derivative function is always non-negative.

⁶We would like to thank Jeffrey Racine for his helpful suggestion about correcting bias in estimating test size.