Exact mean and mean squared error of the smoothed bootstrap mean integrated squared error estimator (Q5943408)

Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed with density \(f\), and set \(X=\) \(\{ X_1, \ldots, X_n \}.\) \(\phi\) denotes the standard normal density and for \(\sigma >0\) let \(\phi(x, \sigma^2) = \sigma^{-1}\phi(x\sigma^{-1}).\) The authors consider kernel estimators for \(f\): the Gaussian kernel estimator with bandwidth \(b\), \(\hat f_b (x, X_n)\), that is \(\hat f_b (x, X_n) = n^{-1} \sum_{i=1}^n \phi (x - X_i, b^2)\), and the smoothed bootstrap mean integrated squared error estimator with pilot bandwidth \(b_p\), that is \[ \hat \Psi (b, b_p) = E \bigl[ \int (\hat f(x, Y_n) - \hat f_{b_p} (x, X_n))^2 dx |X_n], \] where \(Y = (Y_1, \ldots, Y_n)\) is a random sample of size \(n\) with density \(\hat f_{b_p}.\) They consider the case when \(f\) is a normal \(N\)-mixture density. In this case the expectation of \(\hat \Psi (b, b_p)\) and the mean squared error of \(\hat \Psi (b, b_p)\) are found. These results are easily computable. They are used to study the exact behaviour of the estimator for selected unimodal and bimodal densities. A noteworthy observation is that while asymptotics call for oversmoothing the estimator, the undersmoothing way in fact is more appropriate for small samples.