Nonparametric and dimension reduction method for longitudinal and survival data

This dissertation consists of two parts. In the first part, we study a new dimension reduction method for longitudinal data. An extension of inverse regression for multivariate data is proposed for intermittently measured longitudinal data. Compared to previous work in dimension reduction with functional covariates where the whole sample trajectories of random functional covariates are assumed to be observed, we consider longitudinal covariate data that are recorded discretely, intermittently, and maybe even sparsely. This sampling plan violates the basic assumptions of previous methods such that previous approaches are not applicable under this sampling plan. We circumvent the methodology difficulties through nonparametric smoothing techniques and develop asymptotic theory for our procedure. Under some regularity conditions, the new dimension reduction approach converges at an optimal rate normally attained for a one-dimension smoother. Simulation studies and data analysis are also provided to demonstrate the performance of our method.;The second part of the dissertation involves a nonstandard survival analysis problem in which we develop a new method to study the age distribution of wild olive flies. Information about the age distribution and survival of wild populations is of much interest in ecology and biodemography, but is hard to obtain. Established schemes such as capture-recapture often are not feasible. In the residual demography paradigm, individuals are randomly sampled from the wild population at unknown ages and the resulting captive cohort is reared out in the laboratory until death. Under mild assumptions one obtains a demographic convolution equation that involves the unknown age distribution of the wild population, the observed survival function of the captive cohort, and the observed survival function of a reference cohort that is independently raised in the laboratory from birth. We adopt a statistical penalized least squares method for deconvolving this equation, aiming at extracting the age distribution of the wild population under suitable constraints. Under stationarity of the population, the age density is proportional to the survival function of the wild population and can thus be inferred. Several extensions are discussed. Residual demography is demonstrated for data on fruit flies {Bactrocera oleae}.