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This book has grown from the authors' investigations into the problem of finding consistent and efficient solutions to
estimating equations which admit multiple solutions. While there have been studies of multiple roots in likelihood
estimation, most notably by Barnett (1966), we felt that there was a need to incorporate more recent research by
studying the problem in other contexts such as semiparametrics and the construction of artificial likelihoods.
While the book began as the study of multiple roots in estimating functions, it soon expanded to become a study of
nonlinearity in estimating equations and iterative methods for the solution of these equations. Nonlinearity appears in
many different ways within a model, and in differing degrees of severity as an obstacle to statistical analysis. Some
simple forms of nonlinearity can be removed by a reparametrisation of the model. This type of nonlinearity is usually
no major obstacle to the construction of point estimators and the study of their properties. More entrenched forms of
nonlinearity cannot be removed by reparametrising the model, and may force the researcher to use more intensive
numerical methods to construct estimators. The use of root search algorithms or one-step estimators is standard for
these sorts of models. Perhaps the most severe type of nonlinearity in an estimating equation is that which affects the
monotonicity of the estimating function, making the function redescending, with the possibility of multiple roots.
When estimates are constructed by maximising an objective function, such as a likelihood or quasi-likelihood, an
analogous problem occurs when the objective function is not concave or not strongly concave. Hill climbing
algorithms which start at points of nonconcavity often have very poor convergence properties. |
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