Statistical analysis and inference in shape restricted problems with applications to astronomy.

This dissertation is based on the development of methods for statistical problems with inherent shape-restrictions and the inferential properties related to them. We discuss scientific phenomena which naturally give rise to monotonicity or concavity/convexity. Astronomical examples include the mass distribution of galaxies having a spherically symmetric and decreasing density. In the ubiquitous applications of function estimation in non-parametric scenarios, the shape-restriction improves the finite sample as well as asymptotic behavior when the true function itself has that restriction. Both density estimation and regression methodology can be developed along this line. Algorithms are devised to find estimators for Polya frequency functions; smooth monotone link functions in generalized exponential families. The limit theorems developed in this non-regular class of problems are fairly different from the classical regular problems. In case of monotone regression estimation, the NPMLE has some undesirable properties, including a spiking problem at the end-points giving rise to inconsistency. We incorporate a smoothness penalty to deal with that and achieve consistency as well as asymptotic normality. For density estimation, the spiking problem has been countered with a similar penalty by Woodroofe and Sun. We develop a penalized likelihood ratio based confidence interval for f(0+) in this problem using constrained and unconstrained optimization of the penalized log-likelihood. Finally, it should be emphasized that the main focus of this entire thesis is to investigate physical problems and the statistical methodology has been developed keeping that in mind. So, all the algorithms are actually implemented in some of the relevant data sets and shown to work fairly well compared to the existing methods. Simulation studies are presented to show their relative performances. A vast amount of literature study has been done to form the background work of the research and has been mentioned in relevant sections.PhDPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126252/2/3238049.pd