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).

翻译：本文发展了若干有趣、重要且相互关联的非参数或半参数统计推断方法。参数模型下备受青睐的最大似然估计量（MLE）因其强一致性和最优性而闻名，这通常归功于Cramer。然而，当模型非正则或不完全准确时，这些性质便难以保持。此外，这些性质的适用性仅限于接近未知真实参数值的局部极大值。因此，必须确保在一般条件下似然函数的全局最大值具有强一致性（Wald，1949）。全局一致性也是经验似然（Owen，2001）背景下的一个关键研究问题。EL为非参数统计推断提供了开创性平台。随后，通过将估计函数纳入EL框架（Qin and Lawless，1994）实现了又一里程碑。由此得到的剖面经验似然函数具有参数似然的许多优良性质，但也存在同样缺陷。除非我们已知当前局部极大值接近于未知真实参数值，否则这些性质无法利用。为克服这一障碍，我们首先提出一组简洁条件，在此条件下全局最大值具有一致性。随后，我们发展了一种全局最大值检验方法，以判断当前局部极大值是否为全局最大值。此外，我们通过扩展EL下的估计函数集，发明了一种全局最大值修正方法以确保全局一致性。基于文献中众多例子的模拟实验充分证实，所提方法按预期有效。我们的方法还为DeHaan（1981）、Veall（1991）以及Gan和Jiang（1999）所研究的参数对应问题提供了更优解。