Some stochastic shape applications in time series and Markov chains

This dissertation explores several shape ordering applications in statistics, pri- marily in time series and reversible Markov chains. The dissertation has two major goals. First, we introduce shape orderings for stationary time series and explore their convergence rate ramications. The shapes explored include increasing likeli- hood ratio, decreasing hazard rate and new better than used, structures reminiscent from stochastic processes settings. Examples of ARMA(p, q) time series having these shapes are presented. The shapes are then applied to obtain explicit geometric con- vergence rates of several one-step-ahead forecasting quantities. The second goal of this dissertation identies a monotonicity property in reversible Markov chains and examine consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate structure on the even time indices. Good and sometimes even optimal convergence rates of a time reversible Markov chain are deduced from this monotonicity.