Empirical likelihood, which was pioneered by Thomas and Grunkemeier (1975)
and Owen (1988), is a powerful nonparametric method of statistical inference that
has been widely used in the statistical literature. In this thesis, we investigate the
merits of empirical likelihood for various problems arising in ratio estimation. First,
motivated by the smooth empirical likelihood (SEL) approach proposed by Zhou &
Jing (2003), we develop empirical likelihood estimators for diagnostic test likelihood
ratios (DLRs), and derive the asymptotic distributions for suitable likelihood ratio
statistics under certain regularity conditions. To skirt the bandwidth selection problem
that arises in smooth estimation, we propose an empirical likelihood estimator
for the same DLRs that is based on non-smooth estimating equations (NEL). Via
simulation studies, we compare the statistical properties of these empirical likelihood
estimators (SEL, NEL) to certain natural competitors, and identify situations
in which SEL and NEL provide superior estimation capabilities.
Next, we focus on deriving an empirical likelihood estimator of a baseline cumulative
hazard ratio with respect to covariate adjustments under two nonproportional
hazard model assumptions. Under typical regularity conditions, we show
that suitable empirical likelihood ratio statistics each converge in distribution to a
2 random variable. Through simulation studies, we investigate the advantages of
this empirical likelihood approach compared to use of the usual normal approximation.
Two examples from previously published clinical studies illustrate the use of
the empirical likelihood methods we have described.
Empirical likelihood has obvious appeal in deriving point and interval estimators
for time-to-event data. However, when we use this method and its asymptotic
critical value to construct simultaneous confidence bands for survival or cumulative
hazard functions, it typically necessitates very large sample sizes to achieve reliable
coverage accuracy. We propose using a bootstrap method to recalibrate the critical
value of the sampling distribution of the sample log-likelihood ratios. Via simulation
studies, we compare our EL-based bootstrap estimator for the survival function
with EL-HW and EL-EP bands proposed by Hollander et al. (1997) and apply this
method to obtain a simultaneous confidence band for the cumulative hazard ratios
in the two clinical studies that we mentioned above.
While copulas have been a popular statistical tool for modeling dependent data
in recent years, selecting a parametric copula is a nontrivial task that may lead to
model misspecification because different copula families involve different correlation
structures. This observation motivates us to use empirical likelihood to estimate
a copula nonparametrically. With this EL-based estimator of a copula, we derive
a goodness-of-fit test for assessing a specific parametric copula model. By means
of simulations, we demonstrate the merits of our EL-based testing procedure. We
demonstrate this method using the data from Wieand et al. (1989).
In the final chapter of the thesis, we provide a brief introduction to several areas
for future research involving the empirical likelihood approach.
| Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/5817 |
| Date | 22 February 2011 |
| Creators | Dong, Bin |
| Source Sets | University of Waterloo Electronic Theses Repository |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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