The Effect of Discretization Error on Bandwidth Selection for Kernel Density Estimation
解释了离散化数据导致交叉验证评分函数在带宽趋近零时趋于负无穷的原因，并提出一种简单修正方法，使所选带宽与原始未舍入数据的交叉验证结果渐近等价。
It is well known that the cross-validation score function of discretized data often tends to -∞ as the bandwidth tends to zero. This disturbing property causes some difficulty in applying cross-validation to discretized data. Since all data are rounded to some degree, the phenomenon limits the application of cross-validation in practice. Based on characteristic functions, the cause of the difficulty is explained, and a simple modification is suggested. Under some conditions, it is shown that the bandwidth selected by the proposed procedure is asymptotically equivalent to the bandwidth selected by cross-validation of the original unrounded data. The advantages of the proposed procedure are demonstrated in two examples.