There are numerous situations in which observed data is generated by some (unknown) mechanism, where interest lies in estimating a function that is related to a model for the data. Using polynomial splines, an unknown function is modeled to be in a linear space. An algorithm employing stepwise addition and stepwise deletion of basis functions makes it possible to determine this space adaptively. The coefficients of the basis functions are estimated by the maximum likelihood method. The problems that are discussed in this proposal are hazard regression, including situations in which there are time-dependent covariates or interval censored data; bivariate survival estimation; bivariate survival regression; modeling of aftershocks of a major earthquake; polychotomous regression and classification; rotation invariant regression and density estimation. The primary aim of this project is to develop methodologies for the estimation of unknown functions using polynomial splines in a variety of problems. Polynomial splines are building stones that can be used to model functions without making assumptions of their form. The problems that are discussed in this proposal are: survival analysis, including situations in which there are time-dependent explanatory variables, situations in which the survival times are only known to be in an interval and situations in which more than one survival time is measured for each unit; modeling of aftershocks of a major earthquake; classification of observations based upon explanatory variables; rotation invariant regression and density estimation. For each of the problems that we propose, there are vast amounts of data whose improved analysis would be of much scientific interest.
