Mathematical Sciences: Likelihood Functions for Estimating Equations

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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.