  In this paper, we use the empirical likelihood method to construct the
confidence regions for the difference between the parameters of a two-phases
nonlinear model with random design. We show that the empirical likelihood ratio
has an asymptotic chi-squared distribution. The result is a nonparametric
version of Wilk's theorem. Empirical likelihood method is also used to
construct the confidence regions for the difference between the parameters of a
two-phases nonlinear model with response variables missing at randoms (MAR). In
order to construct the confidence regions of the parameter in question, we
propose three empirical likelihood statistics : Empirical likelihood based on
complete-case data, weighted empiri- cal likelihood and empirical likelihood
with imputed values. We prove that all three empirical likelihood ratios have
asymptotically chi-squared distributions. The effectiveness of the proposed
approaches in aspects of coverage probability and interval length is
demonstrated by a Monte-Carlo simulations.
