On the Performance of Kernel Estimators for High-Dimensional, Sparse Binary Data

We develop mathematical models for high-dimensional binary distributions, and apply them to the study of smoothing methods for sparse binary data. Specifically, we treat the kernel-type estimator developed by Aitchison and Aitken (Biometrika63 (1976), 413-420). Our analysis is of an asymptotic nature. It permits a concise account of the way in which data dimension, data sparseness, and distribution smoothness interact to determine the over-all performance of smoothing methods. Previous work on this problem has been hampered by the requirement that the data dimension be fixed. Our approach allows dimension to increase with sample size, so that the theoretical model may accurately reflect the situations encountered in practice; e.g., approximately 20 dimensions and 40 data points. We compare the performance of kernel estimators with that of the cell frequency estimator, and describe the effectiveness of cross-validation.