Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 249

O p (n p=(1+2p) ). This is the optimal rate of convergence in nonparametric
regression with comparable smoothness assumptions.
(ii) n 1=2 ( b n 0 ) = n 1=2 I 1 ( 0 ) P i=1 `(X i ; 0 ; 0 )+o p (1) ! d N(0;I 1 ( 0 )).
Thus, b n is asymptotically normal and ecient.

(iii) The inverse observed information matrix is a consistent estimator of
I 1 ( 0 ), the asymptotic variance of n 1=2 ( b n 0 ).

The proof of this theorem, Parts (i) and (ii), is given in Zhang et al. (2010)

with some technical flaws later found for Part (i). The remedy of the proof for

Part (i) is provided in the appendix at the end of this chapter.

Under conditions (C1) through (C5), it can be shown that (9.6), (A1),

and (A2) are satisfied. Moreover, using monotone cubic B-splines in the sieve
estimation automatically gives kb 0 n n 0 k = O(n p ) for a 2 B n and
kb 0 n c n 0 k = O(n p ) for a c n 2 R q n due to Corollary 6.20 of Schumaker
(1981). Hence the Fisher information I( 0 ) in the Cox model with interval-
censored data can be consistently estimated using the proposed least-squares

approach according to Proposition 9.3.1.

The following proposition provides justification of the existence of positive

definite Fisher information matrix for the Cox model with interval-censored

data.

Proposition 9.4.1 For 2 R, under conditions (C1) through (C5), Equation

(9.15) has a unique solution h (t). Moreover, h (t) has bounded derivative. In
general, for 2 R d , h (t) is a d-dimensional vector and each component has
a bounded derivative. The ecient score for is

` (x) = ` (x) ` h (x):

The information bound for is

I() = E[` (X)] 2 ;

where a 2 = aa 0 for any column vector a 2 R d . Under conditions (C1) through

(C5), I() is a positive denite matrix with nite entries.