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 213

where Q 0 is the limit in probability of the targeted minimum loss-based esti-
mator Q n of Q 0 . In this case, the asymptotic variance 2

of p n( n 0 ) can

be estimated consistently by

X

1
n

^ n = D (Q n ;g n )(O i ) 2 :

i=1

If, however, g 0 is unknown but consistently estimated by g n , a maximum
likelihood estimator in some model G, then the influence curve of n is
IC 1 = D (Q 0 ;g 0 ) Y h

T(G) i

D (Q 0 ;g 0 )

where Q [ jT(G)] is the operator projecting onto the tangent space T(G)
functions in the Hilbert space L 0 (P 0 ) of square-integrable mean-zero functions
with inner product hf;gi = R f(o)g(o)dP 0 of Because computation of the
involved projection may sometimes be quite involved, ^ n may be used, in
practice, as a conservative estimate of 2 .

Asymptotic confidence intervals can easily be constructed using the asymp-
totic linearity of n as well as the Central Limit Theorem. Precisely, denoting,
for each 2 (0; 1), the -quantile of the standard normal distribution by z ,
the interval defined as

!

r ^ n r ^ n

n

n z 1=2

n ; n + z 1=2

will have asymptotic coverage no smaller than 1, with equality occurring,

in particular, when g n = g 0 . Similarly, given a xed 2R, a test of the null
hypothesis 0 = at asymptotic level no larger than is obtained by rejecting
the null hypothesis if and only if the test statistic

p n( n )
p ^ n T n =

is larger than z 1=2 . Once more, an exact asymptotic level of will be at-
tained when g n = g 0 .