A hidden Markov regime is a Markov process that governs the time or space dependent distributions of an observed stochastic process. Recursive algorithms can be used to estimate parameters in mixed distributions governed by a Markov regime. The authors derive a recursive algorithm for estimation of parameters in a Markov-modulated Poisson process also called a Cox point process. By this the authors mean a doubly stochastic Poisson process with a time dependent intensity that can take on a finite number of different values. The intensity switches randomly between the possible values according to a Markov process. The authors consider two different ways to observe the Markov-modulated Poisson process: in the first model the observations consist of the observed time intervals between events, and in the second model they use the total number of events in successive intervals of fixed length. They derive an algorithm for recursive estimation of the Poisson intensities and the switch intensities between the two states and illustrate the algorithm in a simulation study. The estimates of the switch intensities are based on the observed conditional switch probabilities.

Presents on-line procedures for estimating density functions and their derivatives. At each step, M terms are updated. By increasing M the efficiency compared to the traditional off-line kernel density estimator tends to one. Already for M=2, it exceeds 99.1% for kernel orders and derivatives of practical interest.>

A recursive algorithm is proposed for estimation of parameters in mixture models, where the observations are governed by a hidden Markov chain. The often badly conditioned information matrix is estimated, and its inverse is incorporated into the algorithm. The performance of the algorithm is studied by simulations of a symmetric normal mixture. The algorithm seems to be stable and produce approximately normally distributed estimates, provided the adaptive matrix is kept well conditioned. Some numerical examples are included.>