Nonlinear factor analysis

Factor analysis and more general structural equation modeling have been popular statistical tools in social and behavioral sciences, and are useful statistical methods in a wide range of applications. Traditional models in these types of analyses are linear in underlying factors or latent variables. The linear model assumes that the observed variables are linear functions of the underlying factors or latent variables. This assumption may not be valid or realistic in many applications. In behavioral sciences, it is often the case that the linear model fits the data well but with a larger number of factors than expected. In such cases, the subject matter theory usually fails to explain the nature of the unexpected additional factors. Another problem pointed out in application is that the linear model with the maximum allowable number of factors sometimes fails to fit the data. These problems could be due to the existence of nonlinear relationships between observed variables and underlying factors. Models nonlinear in factors may fit the data well and may produce more interpretable results based on the subject matter theory. Nonlinear models can also provide a large class of useful exploratory data analysis tools.;This dissertation consists of two papers. In the first paper, we discuss the additive nonlinear factor analysis model which is nonlinear in underlying factors but is linear in parameters. Based on a certain conditional distribution, an estimation procedure for model parameters and a test procedure for checking the model fit are introduced. Asymptotic properties of the estimator and the test procedure are derived using the small-...