Determining the Number of Effective Parameters in Kernel Density Estimation

The hat matrix maps the vector of response values in a regression to its predicted counterpart. The trace of this hat matrix is the workhorse for calculating the effective number of parameters in both parametric and nonparametric regression settings. Drawing on the regression literature, the standard kernel density estimate is transformed to mimic a regression estimate thus allowing extraction of a usable hat matrix for calculating the effective number of parameters of the kernel density estimate. Asymptotic expressions for the trace of this hat matrix are derived under standard regularity conditions for mixed, continuous, and discrete densities. Simulations validate the theoretical contributions. Several empirical examples demonstrate the usefulness of the method suggesting that calculating the effective number of parameters of a kernel density estimator may be useful in interpreting differences across estimators