  The multiplicative censoring model introduced in Vardi [Biometrika 76 (1989)
751--761] is an incomplete data problem whereby two independent samples from
the lifetime distribution $G$, $\mathcal{X}_m=(X_1,...,X_m)$ and
$\mathcal{Z}_n=(Z_1,...,Z_n)$, are observed subject to a form of coarsening.
Specifically, sample $\mathcal{X}_m$ is fully observed while
$\mathcal{Y}_n=(Y_1,...,Y_n)$ is observed instead of $\mathcal{Z}_n$, where
$Y_i=U_iZ_i$ and $(U_1,...,U_n)$ is an independent sample from the standard
uniform distribution. Vardi [Biometrika 76 (1989) 751--761] showed that this
model unifies several important statistical problems, such as the deconvolution
of an exponential random variable, estimation under a decreasing density
constraint and an estimation problem in renewal processes. In this paper, we
establish the large-sample properties of kernel density estimators under the
multiplicative censoring model. We first construct a strong approximation for
the process $\sqrt{k}(\hat{G}-G)$, where $\hat{G}$ is a solution of the
nonparametric score equation based on $(\mathcal{X}_m,\mathcal{Y}_n)$, and
$k=m+n$ is the total sample size. Using this strong approximation and a result
on the global modulus of continuity, we establish conditions for the strong
uniform consistency of kernel density estimators. We also make use of this
strong approximation to study the weak convergence and integrated squared error
properties of these estimators. We conclude by extending our results to the
setting of length-biased sampling.
