Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 235

the mesh size. In addition, for moderate to high-dimensional parameters, im-
plementation of the profile likelihood approach is computationally demanding
and dicult, as the differentiability of the profile likelihood with respect to
the finite-dimensional parameter is hard to verify in general. To overcome
the computational diculty, Lee et al. (2005) proposed a Bayesian profile
sampler approach using MCMC methods. Higher order asymptotic and fre-
quentist properties of this approach were investigated by Cheng and Kosorok
(2008a,b). Klaassen (1987) has shown by construction that the influence func-
tion of locally asymptotically linear estimators can be estimated consistently
under mild regularity conditions. For semiparametic models, this result implies
that the semiparametrically ecient influence function and hence the Fisher
information can be estimated consistently. However, Klaassen's construction
requires the use of data splitting, which is not easy to use in problems with
only moderate sample sizes like the examples we consider in this chapter.
We propose a least-squares approach to consistent information estimation
in a class of semiparametric models, which naturally leads to consistent vari-
ance estimation for the semiparametric MLE. The proposed method is based
on the geometric interpretation of the ecient score function, which is the
residual of the projection of the score function onto the tangent space for
the infinite-dimensional parameter; see for example, van der Vaart (1991) and
Chapter 3 in Bickel et al. (1993). Thus the theoretical information calculation
is a least-squares problem in a Hilbert space. When a sample from the model
is available, this theoretical information can be estimated by its empirical
version. It turns out that this empirical version is essentially a least-squares
nonparametric regression problem, due to the fact that the score function
for the infinite-dimensional parameter is a linear operator. In this nonpara-
metric regression problem, the \response" is the score function for the nite-
dimensional parameter, the \covariate" is the linear score operator for the
innite-dimensional parameter, and the \regression parameter" is the least fa-
vorable direction that is used to define the ecient score. Computationally,