Limiting bias-reduced Amoroso kernel density estimators for non-negative data

ABSTRACT

The Amoroso kernel density estimator (Igarashi and Kakizawa 2017) for non-negative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n^{− 4/5}), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n^{− 8/9}, if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.

KEYWORDS:

• Amoroso kernel
• asymmetric kernel
• bias reduction
• boundary bias problem
• non-parametric density estimation.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G07
• 62G20

Acknowledgments

The first author has been supported by the Japan Society for the Promotion of Science (JSPS); Grant-in-Aid for Research Activity Start-up [grant number 15H06068] and Grant-in-Aid for Young Scientists (B) [grant number 17K13714]. The second author has been partially supported by the JSPS; Grant-in-Aid for Scientific Research (C) [grant number 26330030/17K00041].