Gaku Igarashi

In regression discontinuity design (RDD), the continuity of the density of a running variable is required. Hence, a discontinuity test of density is used for RDD. In previous studies, tests using difference estimators between the left- and right-hand limits of a density at a (potential) discontinuity point were suggested. In the present paper, a new discontinuity test based on direct density ratio estimation using a beta kernel is proposed. By using the ratio estimator in the proposed test statistic, rather than a difference estimator, the characteristic form of the asymptotic variance of the test statistic is obtained. Consequently, the power of the proposed test is shown to increase when used as a one-tailed test. Simulation studies illustrate the larger power of the proposed test when used as a one-tailed test.

A new nonparametric density ratio estimator using the beta kernel is proposed. It is shown that the beta kernel density ratio estimator (KDRE) is free of boundary or tail bias, and the asymptotic properties of the beta KDRE are derived. Simulation studies are conducted to illustrate the finite sample performance of the beta KDRE.

Multiplicative bias correction technique is revisited for asymmetric kernel density estimators (KDEs) when the data is nonnegative or bounded. It is crucial to classify the recently developed asymmetric KDEs into two types. The multiplicative bias correction applied to the non two-regime type is shown to effectively reduce the order of the bias, at the expense of a constant-factor inflation of the variance. However, it is revealed that, in common with other bias corrections, the multiplicative bias correction applied to the two-regime type fails in reducing the bias near the boundary, unless the density to be estimated satisfies the shoulder condition. (C) 2019 Elsevier B.V. All rights reserved.

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.