156 research outputs found
Large-sample study of the kernel density estimators under multiplicative censoring
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 , and
, are observed subject to a form of coarsening.
Specifically, sample is fully observed while
is observed instead of , where
and 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 , where is a solution of the
nonparametric score equation based on , and
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.Comment: Published in at http://dx.doi.org/10.1214/11-AOS954 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Second-Order Inference for the Mean of a Variable Missing at Random
We present a second-order estimator of the mean of a variable subject to
missingness, under the missing at random assumption. The estimator improves
upon existing methods by using an approximate second-order expansion of the
parameter functional, in addition to the first-order expansion employed by
standard doubly robust methods. This results in weaker assumptions about the
convergence rates necessary to establish consistency, local efficiency, and
asymptotic linearity. The general estimation strategy is developed under the
targeted minimum loss-based estimation (TMLE) framework. We present a
simulation comparing the sensitivity of the first and second order estimators
to the convergence rate of the initial estimators of the outcome regression and
missingness score. In our simulation, the second-order TMLE improved the
coverage probability of a confidence interval by up to 85%. In addition, we
present a first-order estimator inspired by a second-order expansion of the
parameter functional. This estimator only requires one-dimensional smoothing,
whereas implementation of the second-order TMLE generally requires kernel
smoothing on the covariate space. The first-order estimator proposed is
expected to have improved finite sample performance compared to existing
first-order estimators. In our simulations, the proposed first-order estimator
improved the coverage probability by up to 90%. We provide an illustration of
our methods using a publicly available dataset to determine the effect of an
anticoagulant on health outcomes of patients undergoing percutaneous coronary
intervention. We provide R code implementing the proposed estimator
Individualized treatment rules under stochastic treatment cost constraints
Estimation and evaluation of individualized treatment rules have been studied
extensively, but real-world treatment resource constraints have received
limited attention in existing methods. We investigate a setting in which
treatment is intervened upon based on covariates to optimize the mean
counterfactual outcome under treatment cost constraints when the treatment cost
is random. In a particularly interesting special case, an instrumental variable
corresponding to encouragement to treatment is intervened upon with constraints
on the proportion receiving treatment. For such settings, we first develop a
method to estimate optimal individualized treatment rules. We further construct
an asymptotically efficient plug-in estimator of the corresponding average
treatment effect relative to a given reference rule
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