Title: Semi-parametric Dynamic Models for Longitudinal Ordinal Categorical Data

Author(s): Brajendra C. Sutradhar
Issue: Volume 80 Series A Part 1 Year 2018
Pages: 80 -- 109
The over all regression function in a semi-parametric model involves a partly specified regression function in some primary covariates and a non-parametric function in some other secondary covariates. This type of semi-parametric models in a longitudinal setup has recently been discussed extensively both for repeated Poisson and negative binomial count data. However, when it is appropriate to interpret the longitudinal binary responses through a bi- nary dynamic logits model, the inferences for semi-parametric Poisson and negative binomial models cannot be applied to such binary models as these models unlike the count data models produce recursive means and variances containing the dynamic dependence or correlation parameters. In this paper, we consider a general multinomial dynamic logits model in a semi-parametric setup first to analyze nominal categorical data in a semi-parametric longi tudinal setup, and then modify this model to analyze ordinal categorical data. The ordinal responses are fitted by using a cumulative semi-parametric multinomial dynamic logits model. For the benefits of practitioners, a step by step estimation approach is developed for the non-parametric function, and for both regression and dynamic dependence parameters. In summary, a kernel-based semi-parametric weighted likelihood approach is used for the estimation of the non-parametric function. This weighted likelihood estimate for the non-parametric function is shown to be consistent. The regression and dynamic dependence parameters of the model are estimated by maximizing an approximate semi-parametric likelihood function for the parameters, which is constructed by replacing the non-parametric function with its consistent estimate. Asymptotic properties including the proofs for the consistency of the likelihood estimators of the regression and dynamic dependence parameters are discussed.
AMS (2000) subject classification . Primary 62F10, 62H20; Secondary 62F12, 62H12
Keywords and phrases: Binary mapping for cumulative multinomial responses, Consistency; Dynamic logits model for repeated nominal multinomial responses, Kernel-based weighted likelihood, Lag 1 transitional probabilities for cumulative responses, Non-parametric function in secondary covariates, Ordinal categories; Parametric linear predictor in primary covariates, Semi-parametric models and likelihood estimation.