  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.
