Modeling temporally dependent ordinal processes.

Abstract

This research deals with some methods for modeling and analyzing temporally dependent ordered categorical data. It is motivated by applications where one is interested in the nature and extent of temporal dependence or in detecting any changes in a process, for example in statistical process control applications. In this study, ordered categorical variables {<italic>Y_t</italic>} are viewed as indicator variables obtained from latent continuous variables {<italic>X_t</italic>} which are temporally dependent. In other words, the ordinal data {<italic>Y_t</italic>} are assumed to be generated from {<italic>X_t</italic>} as follows: <italic>Y_t</italic> = <italic>j</italic> if qj-1<Xt&le;q j, where { qj } are unknown cut points. We consider an <italic>ARMA</italic>(<italic> p</italic>, <italic>q</italic>) model for the latent variables {<italic>X_t</italic>} and develop general methodology to make inference about temporal dependence and about process changes by estimating the relevant parameters based on {<italic>Y_t</italic>}. The details are developed for <italic> AR</italic>(1) and other common models. Since maximum likelihood estimation is computationally intractable, we consider two alternative estimation procedures: (a) pseudo-likelihood estimators and (b) Bayesian inference using data augmentation methods. Large sample properties of the pseudo-likelihood estimators are developed, and implementation issues associated with the data augmentation procedures are addressed. Also, test procedures are developed for testing hypotheses about parameters of an ordinal process as well as for the two-sample homogeneity problem. Comparisons based on asymptotic power functions are made among the different testing statistics. We also consider statistical process control (SPC) issues for processes based on temporally dependent ordinal data. Methods for constructing Shewhart and CUSUM monitoring procedures are proposed, and the effectiveness of these methods is investigated through simulation results.