De-Biasing Weighted MLE via Indirect Inference: The Case of Generalized Linear Latent Variable Models

In this paper we study bias-corrections to the weighted MLE (Dupuis and Morgenthaler, 2002), a robust estimator simply defined through a weighted score function. Indeed, although the WMLE is relatively simple to compute, for most models it is not consistent and hence not very helpful. For example, the model we consider in this paper is the generalized linear latent variable model (GLLVM) proposed in Moustaki and Knott (2000) (see also Moustaki, 1996, Sammel, Ryan, and Legler, 1997 and Bartholomew and Knott, 1999). The score functions of this model are very complicated. They contain integrals that need to be evaluated. Moreover, they are highly nonlinear in the parameters which makes the use of complicated robust estimator quite impossible in practice. Moustaki and Victoria-Feser (2006) propose to use a weighted MLE and develop indirect inference (Gouri´eroux, Monfort, and Renault, 1993, Gallant and Tauchen, 1996 and also Genton and de Luna, 2000, Genton and Ronchetti, 2003) to remove the bias. It can be computed in a simple iterative fashion. In this paper, we actually focus on indirect inference for bias correction in general. We rely heavily on the findings of Moustaki and Victoria-Feser (2006).