Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 244

Proposition 9.3.1 Assume that there exist n 2 B n and c nj 2 R q n for j =
1; 2; ;d such that

kb 0 n n 0 k = O(k 1 and kb 0 n c n;j 0;j k = O(k 1

1n ) 2n ); j = 1; 2; ;d;

(9.12)

where k 1n and k 2n are two sequences of numbers satisfying k 1n ! 1 and
k 2n !1 as n !1. Suppose that the conditions of Theorem 9.1 are satised.
Then the observed information matrix O n defined in Equation (9.11) is a
consistent estimator of I( 0 ).

This proposition justifies the use of the inverse of the negative observed infor-

mation matrix as an estimator of the variance matrix of the semiparametric

MLE.

9.4 Examples

In this section, we illustrate the proposed method in two semiparametric re-

gression models, including the Cox model (Cox, 1972) for interval-censored

data studied in Huang and Wellner (1997) and the Poisson proportional mean

model for panel count data studied in Wellner and Zhang (2007). In these

examples, the parameter space will be the space of smooth functions de-

fined below. The sieve n is the space of polynomial splines. The polynomial
splines have been used in many fully nonparametric regression models, see for

example, Stone (1985) and Stone (1986).

Let a = d 1 = d 2 = = d m < d m+1 < < d m+K n < d m+K n +1 =
= d 2m+K n = b be a partition of [a;b] into K n sub-intervals I Kt =
[d m+t ;d m+t+1 );t = 0;:::;K, where K K n n v is a positive integer such
that max 1kK+1 jd m+k d m+k1 j = O(n v ). Denote the set of partition
points by D n = fd m ;:::;d m+K n +1 g. Let S n (D n ;K n ;m) be the space of poly-
nomial splines of order m 1 consisting of functions s satisfying (i) the