A cross-validation approach to bandwidth selection for a kernel-based estimate of the density of a conditional expectation

To estimate the density f of a conditional expectation μ(Z) = E[X|Z], Steckley and Henderson (2003) sample independent copies Z₁,...,Z_m; then, conditional on Z_i, they sample n independent samples of X, and their sample mean ̅X_i is an approximate sample of μ(Z_i). For a kernel density estimate ̂f of f based on such samples and a bandwidth (smoothing parameter) h, they consider the mean integrated squared error (MISE), ∫(̂f(x)-f(x))²dx, and find rates of convergence of m, n and h that optimize the rate of convergence of MISE to zero. Inspired by the cross-validation approach in classical density estimation, we develop an estimate of MISE (up to a constant) and select the h that minimizes this estimate. While a convergence analysis is lacking, numerical results suggest that our method is promising.