Multinomial sequence based estimation using contiguous subsequences of length three

Abstract

The Maximum Likelihood (ML) and Bayesian estimation paradigms work within the model that the data, from which the parameters are to be estimated, is treated as a set rather than as a sequence. The pioneering paper that dealt with the field of sequence-based estimation [2] involved utilizing both the information in the observations and in their sequence of appearance. The results of [2] introduced the concepts of Sequence Based Estimation (SBE) for the Binomial distribution, where the authors derived the corresponding MLE results when the samples are taken two-at-a-time, and then extended these for the cases when they are processed three-at-a-time, four-at-a-time etc. These results were generalized for the multinomial “two-at-a-time” scenario in [3]. This paper (This paper is dedicated to the memory of Dr. Mohamed Kamel, who was a close friend of the first author.) now further generalizes the results found in [3] for the multinomial case and for subsequences of length 3. The strategy used in [3] (and also here) involves a novel phenomenon called “Occlusion” that has not been reported in the field of estimation. The phenomenon can be described as follows: By occluding (hiding or concealing) certain observations, we map the estimation problem onto a lower-dimensional space, i.e., onto a binomial space. Once these occluded SBEs have been computed, the overall Multinomial SBE (MSBE) can be obtained by combining these lower-dimensional estimates. In each case, we formally prove and experimentally demonstrate the convergence of the corresponding estimates.