Recursions for the MMPP Score Vector and Observed Information Matrix

Abstract

Exact forward recursions for the score vector and observed information matrix of the Markov-modulated Poisson process (MMPP) are developed. The recursions are motivated by similar recursions developed for hidden Markov models by Lystig and Hughes who extended earlier work by LeGland and Mèvel. Explicit expressions for the first derivative and Hessian of the MMPP transition density matrix are developed and coupled with the recursions. The recursions are implemented and applied to confidence interval estimation in a simulation study.

Collaborative Spectrum Sensing via Online Estimation of Hidden Bivariate Markov Models

“…The online HBMM parameter estimation algorithm developed in [10] is based on an algorithm for the HMM developed by Rydén [9], and a recursion for computing the score function for an HMM from Lystig and Hughes [22] (see also Willy et al [23]). The parameter estimation algorithm operates on a block of m observations at a time.…”

Bivariate Markov Processes and Their Estimation

“…An explicit expression for the derivative of the matrix exponential in (7.7), and its efficient implementation using Van Loan's result [110], were also derived in [117]. In [118], recursions for the score function (7.2) and for the observed information matrix, for vectors of observations from an MMPP, were developed. A vector version of (7.1) may be applied to a bivariate Markov chain, when vectors of consecutive observations are assumed iid, while the random variables within each vector maintain their dependency.…”

“…Asymptotic properties of this recursion, and a numerical study comparing its performance with that of the maximum split-time likelihood estimator [94], for estimating the parameter of a hidden Markov model, were provided in [97]. A recursion for the score function ψ(y m ; φ) = D φ log p φ (y m ), where y m is a vector of observations from a continuous-time bivariate Markov chain, can be derived using an approach similar to that of [118]. We demonstrate this recursion for m = 1 without loss of generality.…”