The quest for a formula that satisfactorily measures the effective degrees of freedom in kernel density estimation (KDE) is a long standing problem with few solutions. Starting from the orthogonal polynomial sequence (OPS) expansion for the ratio of the empirical to the oracle density, we show how convolution with the kernel leads to a new OPS with respect to which one may express the resulting KDE. The expansion coefficients of the two OPS systems can then be related via a kernel sensitivity matrix, and this then naturally leads to a definition of effective parameters by taking the trace of a symmetrized positive semi-definite normalized version. The resulting effective degrees of freedom (EDoF) formula is an oracle-based quantity; the first ever proposed in the literature. Asymptotic properties of the empirical EDoF are worked out through influence functions. Numerical investigations confirm the theoretical insights.

翻译：在核密度估计（KDE）中，寻求一个能令人满意地衡量有效自由度的公式是一个长期存在且鲜有解决方案的问题。从经验密度与理想密度之比的正交多项式序列（OPS）展开出发，我们展示了与核的卷积如何导出一个新的OPS，并可以基于此表达所得的KDE。两个OPS系统的展开系数可通过一个核敏感度矩阵联系起来，这自然引出了通过取对称化半正定归一化版本的迹来定义有效参数的方法。由此得到的有效自由度（EDoF）公式是一个基于理想量的度量，为文献中首次提出。通过影响函数，我们推导了经验EDoF的渐近性质。数值研究证实了理论见解。