Time series of random convex bodies

Date of Completion

January 2000






This thesis focuses on extending classical areas of dependent random variable theory to dependent random compact convex bodies (dependent random bodies). Much of this work is in the context of time series, with special reference to the convex body AR1 series Xt=gXt-1+Zt, where g is a non-negative scalar, Zt are independent, identically distributed random bodies called innovation bodies, and Xt is the convex body time series. ^ Three types of estimators of the parameter g are developed. The suitability of each depends on the kind of information available for the innovation body and its measurements. If no information is available, then a Yule-Walker style estimator is appropriate. The second type of estimator is similar to the Yule-Walker estimator but attempts to take advantage of the special structure of the AR1 model. The third estimator exploits the positive nature of many measurements. If it is known that the positive measurement is regularly varying, then the associated index of variation can be used to select among measurements in the experimental design. ^ The second area of traditional time series analysis that will be treated for random bodies is forecasting. Given the observations X0,&ldots;,Xn from the AR1 body series, a predictor of the random body Xn+1 is developed. As with the traditional AR1 series, the predictor is shown to be asymptotically unbiased. It is also shown to be efficient in the sense that it minimizes an asymptotic mean square error. ^ The third extension of classical results is in the area of empirical distribution theory. The distribution function and the empirical distribution function for random bodies are defined. These definitions are followed by a proof of the convergence of the empirical distribution of the AR1 body series to the underlying population distribution. Finally, we verify that the empirical distribution calculated from estimates of the innovation process for the AR1 model converges to the underlying distribution of the innovation process. ^