Developing the maximum likelihood estimation for component analysis: A state space approach

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Traditionally, scientists have used component analysis (CA) as descriptive tools in exploratory data analysis. Mathematically, a component is defined as a linear combination of the observed variables. The prototype of CA is principal component analysis. Other basic methods of CA include canonical correlation analysis and redundancy analysis. Over the past several decades, various generalizations of these basic component models were developed, including simultaneous component analysis, three-way component analysis, multiple-set canonical correlation analysis, and extended redundancy analysis, just to name a few. In the statistical literature, most component models are estimated by the least square method, which only gives limited information for parameter estimation. In contrast, the very popular and most commonly used estimation procedure—the maximum likelihood (ML) method—is somewhat under-developed for CA. Despite the fact that the ML method requires the assumption of normal distribution on the variables, the ML method is often preferred in many instances, because 1) the ML estimator is consistent, 2) is more efficient than the LS estimator, and 3) provides more information that can be used for statistical inferences, including the standard errors for individual parameters, the likelihood ratio test statistic, and several information criteria for model selection purpose. Moreover, the ML estimator and the likelihood ratio test statistic derived under the normal distribution retain their robustness to some violations of the normality assumption. Finally, the ML method constructs the likelihood function that can be used to calculate the robust standard errors in case of assumption violations and/or model misspecifications. Given the many advantages of the ML method, I have two major objectives for my future research including 1) developing the ML method for some existing component models and 2) proposing new component models suitable for psychological and behavioral research. The first objective, in many situations, can be achieved through a proper specification of a state space model (SSM) for a particular component model, which brings a practical convenience to applied researchers that software programs designed for SSM can be used to estimate various component models, whereas the implementation of many component models developed in the modern literature requires a specialized program. For the first three years, six projects are planned for the two objectives, and each project is expected to produce a peer-reviewed journal article. For the last two years and beyond, more topics related to the ML method and CA can be done, such as the penalized ML method for both linear and nonlinear component models and the development of better algorithms for numeric optimization. In sum, this line of research will make important contributions to multivariate statistical analysis.