Mathematical Sciences: Likelihood Functions for Estimating Equations

我的研究重点是构建用于估计方程的似然函数。当估计方程不与势函数积分时，就需要这样的函数。似然函数可以由真实分数函数在一类保守估计方程上的投影得到，也可以由一对中心似然比的投影得到。给出了拟似然方法的投影、广义估计方程的投影以及其他一些估计方程的投影。它们既可以应用于独立的情况，也可以应用于依赖的情况。这些函数将用于构造不变置信区间，以区分估计方程的一致和不一致的解决方案，并检验统计假设。我还介绍了测试函数中包含的信息的概念，并将使用它来研究一些常用的测试函数。由此得到了一阶辅助性的一个性质。统计学的关键问题是建立真理和数据之间的关系，并利用这种关系在数据的基础上对真理进行推断。这通常是通过我们所说的可能性方法来实现的。有时，传统的似然方法会面临一些困难：它可能需要做出难以满足的假设；这可能在计算上难以实现；它可能不够丰富，无法容纳某些应用程序。为了应对这些挑战，近年来，拟似然方程和估计方程的理论和方法得到了发展和广泛应用。新方法对似然方法进行了扩展和丰富，可以在更现实的假设下、更方便地应用于更广泛的情况。我的研究方向是准似然和估计方程的某些方面，特别是估计方程的似然函数的构造及其在统计推断中的应用。

My research focuses on the construction of likelihood functions for estimating equations. The need for such functions arises when the estimating equations do not integrate to potential functions. The likelihood functions can be either obtained from the projection of the true score function onto a class of conservative estimating equations or from the projection of a pair of centered likelihood ratios. The projections are developed for the quasi-likelihood method, and for the generalized estimating equations, as well as some other classes of estimating equations. They can be applied to both independent and dependent situations. These functions will be used to construct invariant confidence intervals, to distinguish between consistent and inconsistent solutions to estimating equations, and to test statistical hypotheses. I also have introduced a notion of the information contained in a testing function, and will use it to study some commonly used testing functions. In relation to this, a property of the first order ancillarity is obtained. The key issue in statistics is to establish a relation between the truth and the data, and to use the relation to make inference about the truth based on the data. This is often achieved by what we call the likelihood methods. Sometimes the traditional likelihood method faces difficulties: it may need to make assumptions that are difficult to satisfy; it may be computationally difficult to realize; it may not be rich enough to accommodate certain applications. Meeting these challenges, the theories and methods of quasi-likelihood and estimating equations have been developed and widely used during recent years. Extending and enriching the likelihood method, the new methods can be applied under more realistic assumptions, more conveniently, and in wider situations. My research is on certain aspects of quasi-likelihood and estimating equations, and in particular the construction of likelihood functions for estimating equations and their use in statistical inference.