Estimation of the MISE and the optimal bandwidth vector of a product kernel density estimate

Given i.i.d. d-dimensional ( d > 1 ) data, a bootstrap estimate of the mean integrated squared error (MISE) of a product kernel density estimate is proposed. We propose to select the 1 × d bandwidth vector by minimizing the bootstrap estimate of the MISE. symptotic properties of the proposed estimators are obtained. For fixed sample size, the bootstrap MISE estimator by Sain et al. (1994) can fail to estimate the MISE, especially for large smoothing along any one of the d directions. Our MISE estimator overcomes this demerit. For a Gaussian kernel the exact formula of the bootstrap MISE estimator is obtained. The smoothed cross-validation method (SCV) is similar to the proposed bootstrap scheme. The proposed bootstrap estimate can be asymptotically more accurate than the SCV estimate of the MISE. We provide insight into the accuracy of the SCV and the bootstrap bandwidth vectors in terms of minimizing the MISE. oduct kernel density estimate, using the bootstrap bandwidths, for Old Faithful geyser data seem to compare well with the kernel density estimate using full bandwidth matrix chosen by plug-in method.