Variance of Maximum Likelihood Estimates for the Hidden Markov Model with Multipartite Graph Structure Transition

Abstract

This study was conducted with the aim to derive an expression for variance of the maximum likelihood estimators of the hidden Markov model having multipartite graph structure transition. To obtain the estimates of variance, observed information matrix was derived using the Louise (1982) method. This study derived information matrix for the m1 and m2 partition of states which were observed at time 2t 􀀀 1 and 2t respectively. Also, in this study, lower bound for variance of maximum likelihood estimates was derived. The study also defines a parametric bootstrap procedure for computation of variance. To check the validity of derived matrix for maximum likelihood estimates, a numerical example was used to estimate the variance using derived information matrix and compared with the results of parametric bootstrap. For this purpose, a real world data, named, as ”faithful” considered, which is freely available in statistical software R. The data-set have 272 observations on each of two variables i.e. eruption time te and waiting time tw, both measured in minutes. In this study, variable te was considered in one partition of states, which observed at time 2t􀀀1 and variable tw was considered in second partition of states, which observed at time 2t. The study compared estimated variances by observed matrix and parametric bootstrap procedure for different combination of states and sample sizes. The comparison showed a smaller variation in values of maximum likelihood estimates obtained from observed matrix than by bootstrap procedure. In combination of states, both approaches showed almost similar variances. The overall comparison indicates that estimated variance of maximum likelihood estimators by observed matrix seems meaningful i.e. explaining less variation than that obtained from the bootstrap procedure. To study the empirical performance of the derived observed matrix for variance of maximum likelihood estimators, an extensive simulation study of various sample size was conducted. Simulated data were generated for different sizes and variance was calculated by observed matrix.