Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 257

estimate the asymptotic standard error using the least-squares approach. The
semiparametric sieve MLE of 0 is b n = (0:2076;0:0356; 0:0647;0:7949)
with the asymptotic standard errors given by (0:0066; 0:0101; 0:0338; 0:0534)

and the corresponding p-values = (0.0000, 0.0004, 0.0553, 0.0000) based on

the asymptotic theorem developed in Wellner and Zhang (2007). We note that

the standard error estimates are different from those of Lu et al. (2009) using

the bootstrap method. The difference indirectly indicates that the working

assumption of the Poisson process model to form the likelihood may not be

valid as Lu{Zhang{Huang's inference procedure is shown to be robust against

misspecification of the underlying counting process.

9.6 Discussion

When the infinite-dimensional parameter as nuisance parameter cannot be

eliminated in estimating the finite-dimensional parameter, a general semi-

parametric maximum likelihood estimation is often a challenging task. The

sieve MLE method, proposed originally by Geman and Hwang (1982), renders

a practical approach for alleviating the diculty in the semiparametric esti-

mation problem. In particular, the spline-based sieve semiparametric method,

as exemplified by Zhang et al. (2010) and Lu et al. (2009), has many attrac-

tive properties in practice. Not only does it reduce the numerical diculty

in computing the semiparametric MLE, but it also achieves the semipara-
metric asymptotic estimation eciency for the finite-dimensional parameter.

However, the estimation of the information matrix remains a dicult task

in general. In this chapter, we propose an easy-to-implement least-squares

approach to estimate the semiparametric information matrix. We show that

the estimate is asymptotically consistent, and it is also a by-product of the

establishment of asymptotic normality for sieve semiparametric MLE. Inter-