SEMIPARAMETRIC METHODS FOR DATA FROM AN OUTCOME-DEPENDENT SAMPLING SCHEME - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 96

Empirical likelihood (EL) in its simplest form is just a nonparametric
likelihood. Let x 1 ;:::;x n be i.i.d. observations from an unknown d-variate
distribution F. The nonparametric likelihood is in fact maximized by the
empirical CDF, F n (x) =

P

n
i=1 I(X i

x). Owen 3;4 introduced an em-
pirical likelihood ratio statistic for nonparametric parameters. He showed
that the statistic has a limiting chi-square distribution and how to obtain
tests and condence intervals for a parameter, expressed as functional (F).
Many extensions and applications of empirical likelihood have been devel-
oped for biased sampling and censored data. See [5] for a comprehensive
review. Zhou, et al. 15 , Wang and Zhou 12 have applied the empirical like-
lihood to the problem of biased sampling (outcome dependent sampling).
Vardi 10;11 and Qin 6 have discussed the biased sampling problem in the
case of a completely known the weight function. When the weight function
involves unknown parameters, one usually needs methods to combine the
empirical likelihood with the parametric likelihood.

1
n

2.1. Data Structure and Likelihood
Zhou, et al. 15 proposed a semiparametric empirical likelihood to deal with
the two component outcome dependent data set, in which there is an overall
SRS sample and several supplemental samples. These supplemental sam-
ples are selected dependent on the outcome. The proposed semiparametric
empirical likelihood can deal with the continuous outcome and does not
make any assumption for the distribution of covariates.
Let Y denote the continuous outcome variable and X denote the vector
of covariates. Assume that the domain of Y is a union of K mutually
exclusive intervals: C k = (a k1 ;a k ]; k = 1;:::;K with a k ;k = 0; 1;:::;K
being known constants satisfying a 0 =1< a 1 < a 2 < ::: < a K =1:
The structure of the two component ODS sample consists an overall simple
random sample (the SRS sample) and a simple sample from each of the K
intervals of Y (the supplement samples). Let k be the index for intervals of Y
and i be the individuals. Then the observed data structure is as follows: one
observes the supplement samplefY ki ;X ki jY ki 2C k gwhere k = 1; 2;:::;K
and i = 1; 2;:::;n k . The overall SRS sample is denoted byfY 0i ;X 0i gwhere
i = 1; 2;:::;n 0 . For CPP data, in the sampling notation, we have that
a 1 = 82;a 2 = 110;n 0 = 849;n 1 = 81;n 2 = 0 and n 3 = 108:
For the ease of the presentation, let G X and g X denote the cumulative
distribution and density function of X, respectively. The joint likelihood