A Change-point Problem and Preliminary Test Estimation in Circular Statistics
This thesis investigates two different problems relating to circular data. One relates to change-point problems. Tests in this context are meant to detect the point in time at which a sample of observations changes the probability distribution from which they came. Suppose one has a set of independent vectors of measurements, observed in a time-ordered or space-ordered sequence. In our set-up, these observations are circular data and we are interested as to which point in time does the distribution change from having one mode to having more than one mode. In this work we model unimodality or bimodality with a mixture of two Circular Normal distributions, which admits both possibilities, albeit for different parameter values. Tests for detecting the change-point are derived using the generalized likelihood ratio method. We obtain simulated distri- butions and critical values for the appropriate test statistics in finite samples, as well as provide the asymptotic distributions, under some regularity conditions. We also tackle this problem from a Bayesian perspective. In the second part, the goal is to estimate the concentration parameter of a Circular Normal distribution when the mean direction is unknown. We present two alternate approaches that incorporate prior knowledge on the mean direction (i) via a preliminary test on the mean direction, the so-called “preliminary test estimators” and (ii) through an assumed prior distribution on the mean direction as one does in Bayes procedures. We compare such alternate estimators with the standard maximum likelihood estimator and explore when one method is superior to the other.