A Study of Bootstrap and Likelihood Based Methods in Non-Standard Problems.

Abstract

In this dissertation we investigate bootstrap and likelihood based methods for constructing confidence intervals in some non-standard problems. The non-standard problems studied include problems with non root-n convergence (e.g., cube-root convergence), estimation problems where the parameter is on the boundary and study of non-smooth/abrupt-change models.

We consider estimating a bounded parameter in presence of nuisance parameters and propose methods of constructing confidence intervals for the parameter of interest in some typical examples that arise in high energy physics and astronomy.

In epidemiological applications interest lies in constructing confidence sets for the distribution function of time to infection/illness (the failure time) with interval censored data. We use a pseudo-likelihood function based on the marginal likelihood of a Poisson process to construct a pseudo-likelihood ratio statistic for testing point null hypotheses for the distribution function and show that the test statistic converges to a pivotal quantity.

A major part of the thesis has been motivated by an astronomy application --estimation of dark matter distribution in dwarf galaxies. An essential component of the application involves estimation and inference on functions that obey shape restrictions, like monotonicity/convexity. We study the performance of bootstrap methods for inference in two non-parametric estimation problems – the estimation of a monotone density and the Wicksell’s problem. Our results show the inconsistency of conventional bootstrap methods in the monotone density estimation problem; in fact, we claim that the bootstrap estimate of the sampling distribution does not have any weak limit conditionally (given the data), in probability. We establish limit distributions of shape restricted estimators and the consistency of bootstrap methods in the Wicksell’s problem.

Whether a dwarf spheroidal galaxy is in equilibrium or being tidally disrupted by the Milky Way is an important question for the study of its dark matter content and distribution. We investigate the presence of such a streaming motion focusing our attention to the Leo I galaxy. Statistical tools include isotonic and change-point estimators, asymptotic theory and resampling methods. We find that although there is evidence for streaming, the effect is not alarming.