Nonparametric Density Estimation for Stratified Samples

In this paper, we consider the non-parametric, kernel estimate of the density, f(x), for data drawn from stratified samples. Much of the data used by social scientists is gathered in some type of complex survey violating the usual assumptions of independently and identically distributed data. Such effects induced by the survey structure are rarely considered in the literature on non-parametric density estimation, yet they may have serious consequences for our analysis, as shown in this paper. A weighted estimator is developed which provides asymptotically unbiased density estimation for stratified samples. A data-based method for choosing the optimal bandwidth is suggested, using information on withinstratum variances and means. The weighted estimator and proposed bandwidth are shown to give smaller mean squared error for stratified samples than an un-weighted estimator and a commonly used method of choosing the bandwidth. Surprisingly, the single bandwidth outperforms optimally choosing stratum-specific bandwidths in some cases. Several illustrations from simulation are provided. We also show that the optimal sampling scheme in this case is always stratified sampling proportional to size, irrespective of the stratum-specific densities