We propose a method for kernel smoothing data that is either
interval-censored or has been aggregated into bins.
The method relies on weights computed using a proposed
kernel density estimate for such data. This estimate retains the
simplicity and intuitive appeal of the usual kernel density estimate for
complete data and is easy to compute. It results from an algorithm where
conditional expectations of a kernel are computed at each iteration. These
conditional expectations are computed with respect to the density estimate
from the previous iteration, allowing the estimator to extract more
information from the data at each step. The estimator is applied to HIV
data where interval censoring is common and to the Ontario health survey
where data has been aggregated into bins.
In terms of the cumulative distribution function the algorithm is shown to
coincide with the self-consistency algorithms of Efron (1967), Turnbull
(1976), and Li et al. (1997), as the window size of the kernel shrinks to
zero. It modifies these algorithms by introducing kernel smoothing at each
iteration. Viewing the iterative scheme as a generalized EM algorithm
permits a natural interpretation of the estimator as being close to the ideal
kernel density estimate where the data is not censored in any way. Simulation
results support the conjecture that kernel smoothing at every iteration does
not effect convergence. In addition, comparison to the standard kernel
density estimate, based on smoothing Turnbull's estimator, reflect favourably
on the proposed estimator for all criteria considered. Use of the estimator
for smoothing histograms, hazard estimation and scatterplot smoothing is also
considered.
This is joint work with Thierry Duchesne of the Department of Statistics
at University of Toronto.
