Nonparametric Models for Longitudinal Data Using Bernstein Polynomial Sieve

We develop a new nonparametric approach to the analysis of irregularly observed longitudinal data using a sieve of Bernstein polynomials within a Gaussian process framework. The proposed methodology has a number of novel features: (i) both the mean function and the covariance function can be estimated simultaneously under a set of mild regularity conditions; (ii) the derivative of the response process can be analyzed without additional modeling assumptions; (iii) shape constraint of the mean and covariance functions (e.g. nonnegativity, monotonicity and convexity) can be handled in a straightforward way; and (iv) the Lp approximation of the Gaussian process using the Bernstein polynomial sieve is established rigorously under mild regularities conditions. Further, in order to choose the appropriate order of the Bernstein sieve, a new Bayesian model selection criterion is proposed based on a predictive cross-validation criterion. Superior performance of the proposed nonparametric model and model selection criterion is demonstrated using both synthetic and real data examples.