The constrained Fisher scoring method for maximum likelihood computation of a nonparametric mixing distribution

This paper proposes a new method for computing the nonparametric maximum likelihood estimate of a mixing distribution. It uses the Fisher scoring quadratic approximation to the log-likelihood function of the mixing proportions. At each iteration, new candidate support points are found and included, as guided by the gradient function, and bad support points are discarded, after being found redundant by optimizing the quadratic approximation. Numerical studies show that the CFS method is generally competitive with the fast and stable constrained Newton method; it may even have an advantage over the latter when the initial estimate is badly chosen.     

Estimating Equations with Nuisance Parameters: Theory and Applications

In a variety of statistical problems the estimate _n of a parameter is defined as the root of a generalized estimating equation G_n(_nn)=0 where _n is an estimate of a nuisance parameter . We give sufficient conditions for the asymptotic normality of #x0398;_n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;_n will not be influenced by that of _n) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.