  This paper develops several interesting, significant, and interconnected
approaches to nonparametric or semi-parametric statistical inferences. The
overwhelmingly favoured maximum likelihood estimator (MLE) under parametric
model is renowned for its strong consistency and optimality generally credited
to Cramer. These properties, however, falter when the model is not regular or
not completely accurate. In addition, their applicability is limited to local
maxima close to the unknown true parameter value. One must therefore ascertain
that the global maximum of the likelihood is strongly consistent under generic
conditions (Wald, 1949). Global consistency is also a vital research problem in
the context of empirical likelihood (Owen, 2001). The EL is a ground-breaking
platform for nonparametric statistical inference. A subsequent milestone is
achieved by placing estimating functions under the EL umbrella (Qin and
Lawless, 1994). The resulting profile EL function possesses many nice
properties of parametric likelihood but also shares the same shortcomings.
These properties cannot be utilized unless we know the local maximum at hand is
close to the unknown true parameter value. To overcome this obstacle, we first
put forward a clean set of conditions under which the global maximum is
consistent. We then develop a global maximum test to ascertain if the local
maximum at hand is in fact a global maximum. Furthermore, we invent a global
maximum remedy to ensure global consistency by expanding the set of estimating
functions under EL. Our simulation experiments on many examples from the
literature firmly establish that the proposed approaches work as predicted. Our
approaches also provide superior solutions to problems of their parametric
counterparts investigated by DeHaan (1981), Veall (1991), and Gan and Jiang
(1999).
