Targeted Minimum Loss-Based Estimation of a Causal E ect Using Interval-Censored Time-to-Event Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 206

to van der Laan and Rose (2011) for an extensive treatment. In addition to
constituting an automatic procedure for producing estimators that are locally
ecient and often robust to a certain degree of misspecification, targeted min-
imum loss-based estimation achieves maximal bias reduction in finite samples
by tailoring the estimation procedure to the specific target parameter of in-
terest. This is in contrast to the vast majority of other approaches, including
standard maximum likelihood, which often, either explicitly or implicitly, op-
timize quantities not directly of interest. The targeted minimum loss-based
estimation framework yields substitution estimators, that is, estimators that
can be expressed as an application of the parameter mapping on some proba-
bility measure in the statistical model. This characteristic is crucial because it
automatically ensures that the estimator will respect the natural constraints of
the parameter space and ascribes to it some robustness to outliers. For exam-
ple, targeted minimum loss-based estimators of probabilities will always yield
estimates that are themselves probabilities. This is true even in the presence
of outliers and despite serious violations of the positivity assumption, which
often cause conventional estimating equations-based estimators to yield pa-
rameter estimates outside the parameter space. Finally, because the iterative
process in targeted minimum loss-based estimation only requires minimization
of a loss over a low-dimensional space, common practical issues often encoun-
tered in other types of estimation approaches, such as seeking the solution of
equations with multiple or no roots, are, by construction, eliminated.
Suppose that the observed data consist of independent and identically
distributed random vectors O 1 ;O 2 ;:::;O n with probability distribution P 0 and
realizations inR m . The definition of targeted minimum loss-based estimation
requires three elements: (i) a well-defined statistical model and parameter,
(ii) an appropriate loss function, and (iii) a suitable class of fluctuation sub-
models. First, the statistical parameter of interest should be expressed as a
mapping from a statistical model M to a subset of the Euclidean spaceR q
for some q 2N. In general, (P) only depends on some portion of P 2 M;