University of Washington - Department of Statistics
Advisor: Jon Wellner
The method of maximum likelihood continues to be a very popular general method of estimation. However, even though MLEs have many nice theoretical properties, there has been particular difficulty with the likelihood approach to estimation in the presence of nuisance parameters. In this talk, we will focus on what Gong and Samaniego (1981) called the "pseudo-likelihood" approach, which is of particular interest in problems when the likelihood surface is ill-behaved in higher dimensions but well-behaved in lower dimensions.
Let X1,..., Xn be i.i.d. random variables with common unknown probability measure indexed by k real parameters of interest and m nuisance parameters. The framework in this research allows some nuisance parameters in an "infinite-dimensional" space. The main idea is to replace the nuisance parameters by their ad-hoc square-root n consistent estimators, which are easily computable and explicitly defined, into the likelihood function. We then treat this resulting "pseudo-likelihood," which is now a function of k real parameters only, as an usual likelihood function. We call any estimator that nearly solves a system of (pseudo) score equations a "pseudo Z-estimator," and any estimator that nearly maximizes the pseudo- likelihood a "pseudo M-estimator." We will present our general theorem concerning asymptotic normality of pseudo Z-estimators. The general theorems established here not only successfully extend the classical results of Gong and Samaniego (1981) from parametric to semiparametric models, but are also applicable to some less smooth models, such as the double exponential distribution, that classical results do not cover.
We choose copula models to illustrate the applications of our main theorem, because there has been an interest in families of multivariate distributions with the association between the components as a parameter of interest and the marginal distributions as nuisance parameters. For the bivariate normal copula model, we have shown that even the pseudo Z-estimator achieves efficiency asymptotically, which agrees with the main finding of Klaassen and Wellner (1995).