Periodic Little's law

In this dissertation, we develop the theory of the periodic Little's law (PLL) as well as discussing one of its applications. As extensions of the famous Little's law, the PLL applies to the queueing systems where the underlying processes are strictly or asymptotically periodic. We give a sample-path version, a steady-state stochastic version and a central-limit-theorem version of the PLL in the first part. We also discuss closely related issues such as sufficient conditions for the central-limit-theorem version of the PLL and the weak convergence in countably infinite dimensional vector space which is unconventional in queueing theory. The PLL provides a way to estimate the occupancy level indirectly. We show how to construct a real-time predictor for the occupancy level inspired by the PLL as an example of its applications, which has better forecasting performance than the direct estimators

Relative fixed-width stopping rules for Markov chain Monte Carlo simulations

Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that is conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in a wide variety of MCMC estimation settings, such as expectation, quantile, or simultaneous multivariate estimation. Finally, we investigate the finite sample properties through a variety of examples and provide some recommendations to practitioners.Comment: 24 page

Statistical theory and robust methodology for nonlinear models with application to toxicology

Nonlinear regression models are commonly used in dose-response studies, especially when researchers are interested in determining various toxicity characteristics of a chemical or a drug. There are several issues one needs to pay attention to when fitting nonlinear models for toxicology data, such as structure for the error variance in the model and the presence of potential influential and outlying observations. In this dissertation I developed robust statistical methods for analyzing nonlinear regression models, which are based on robust M-estimation and preliminary test estimation (PTE) procedures. In the first part of this research the M-estimation methods in heteroscedastic nonlinear models are considered for two cases. In one case, the error variance is proportional to some known function of mean response, while in the other case the error variance is modeled as a polynomial function of dose. The asymptotic properties of the proposed M-procedures and the asymptotic efficiency of the proposed M-estimators are provided. In the second part I consider PTE-based methodology using M-methods for estimating the regression parameters. Based on the outcome of the preliminary test, the proposed methodology determines the appropriate error variance structure for the data and accordingly chooses the suitable estimation procedure. Since the resulting methodology uses M-estimators, it is expected to be robust to outliers and influential observations, although such issues have not been explored in this dissertation. Consequently, one does not have to pre-specify the error structure for the variances not does the user have to perform model diagnostics to choose a method of estimation. Some asymptotic results will be given to obtain the asymptotic covariance matrix of the PTE. Finally numerical studies are presented to illustrate the methodology. The results of the numerical studies suggest that the PTE using M-methods performs well and is robust to the error variance structure

Convex hulls of random walks

We study the convex hulls of random walks establishing both law of large numbers and weak convergence statements for the perimeter length, diameter and shape of the hull. It should come as no surprise that the case where the random walk has drift, and the zero-drift case behave differently. We make use of several different methods to gain a better insight into each case. Classical results such as Cauchy’s surface area formula, the law of large numbers and the central limit theorem give some preliminary law of large number results. Considering the convergence of the random walk and then using the continuous mapping theorem leads to intuitive results in the case with drift where, under the appropriate scaling, non-zero, deterministic limits exist. In the zero-drift case the random limiting process, Brownian motion, provides insight into the behaviour of such a walk. We add to the literature in this area by establishing tighter bounds on the expected diameter of planar Brownian motion. The Brownian motion process is also useful for proving that the convex hull of the zero-drift random walk has no limiting shape. In the case with drift, a martingale difference method was used by Wade and Xu to prove a central limit theorem for the perimeter length. We use this framework to establish similar results for the diameter of the convex hull. Time-space processes give degenerate results here, so we use some geometric properties to further what is known about the variance of the functionals in this case and to prove a weak convergence statement for the diameter. During the study of the geometrical properties, we show that, only finitely often is there a single face in the convex minorant (or concave majorant) of such a walk