Nonlinear structural equation modeling as a statistical method

Structural equation modeling originated (Jöreskog (1973), Bentler (1980), Bollen (1989)) as a method for modeling linear relations among observed and hypothesized latent variables. Despite limitation inherent in the linearity assumption of traditional structural equation modeling, it has indeed provided a revolutionary and popular framework for addressing research questions in the social, psychological and behavioral sciences where latent variable are quite common. In order to expand the flexibility and thus applicability of this already useful statistical modeling method, a natural extension is to include the possibility of modeling nonlinear relations among the latent variables in addition to linear relations. There has a been growing literature (some of which described later in this paper) developing different kinds nonlinear structural equation models and estimation methods for them. Generally, the estimation methods in this literature can be described as either making and relying on distributional assumptions for the underlying latent variables or instead leaving the distribution unspecified. Furthermore, the methods can be described as being either tailor-made to a specific sort of nonlinear structural model, i.e. polynomial or specifically low dimensional polynomial, or else being applicable to a more general nonlinear structural model. In this paper we present a general nonlinear structural equation model and estimation methods for it. Section 2 presents the general nonlinear structural equation model as an extension of the linear structural equation model. Section 2 also describes special cases of