Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 252

Lu et al. (2009) studied the sieve MLE for the above semiparametric Pois-

son model using monotone polynomial splines in M n (D n ;K n ;m) dened in
Equation (9.14). Replacing (t) by the exp( P q n

l=1 l b j (t)) in the likelihood
above, Lu et al. (2009) solved the constraint optimization problem over the

space B n . It turns out that, compared to Wellner and Zhang's method, the
B-splines sieve MLE is less computationally demanding with a better overall

convergence rate. In addition, the sieve MLE of is still semiparametrically

ecient with the same Fisher information matrix, I( 0 ). The bootstrap pro-
cedure was implemented to estimate I( 0 ) consistently by Lu et al. (2009)
due to the computation advantage of the B-splines sieve MLE method. Some

simple algebra shows that

X

z h Kj
Kj

` 1 (x; ) ` 2 (x; )(h) =

(N K;j exp( 0 z) Kj )

j=1

Under the regularity conditions given in Wellner and Zhang (2007), it can be

shown that (9.6), (A1), and (A2) are satisfied. The consistency of the least-

squares estimator of the information matrix follows from Proposition 9.3.1.

9.5 Numerical Results

9.5.1 Simulation Studies

In this section, we conduct simulation studies for the examples discussed in

the preceding section to evaluate the finite sample performance of the pro-

posed estimator. In each example, we estimate the unknown parameters using

the cubic B-splines sieve maximum likelihood estimation and estimate the

standard error of the regression parameter estimates using the proposed least-

squares method based on the cubic B-splines as well. For the B-splines sieve,
the number of knots is chosen as K n = [N 1=3 ], the largest integer smaller than
N 1=3 , where N is the number of distinct observation time-points in the data,