An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.

Abstract—The discovery of networks is a fundamental problem

In this paper we investigate an efficient implementation of the Monte Carlo EM al-gorithm based on Quasi-Monte Carlo sampling. The Monte Carlo EM algorithm is a stochastic version of the deterministic EM (Expectation-Maximization) algorithm in which an intractable E-step is replaced by a Monte Carlo approximation. Quasi-Monte Carlo methods produce deterministic sequences of points that can significantly improve the accuracy of Monte Carlo approximations over purely random sampling. One draw-back to deterministic Quasi-Monte Carlo methods is that it is generally difficult to determine the magnitude of the approximation error. However, in order to implement the Monte Carlo EM algorithm in an automated way, the ability to measure this error is fundamental. Recent developments of randomized Quasi-Monte Carlo methods can overcome this drawback. We investigate the implementation of an automated, data-driven Monte Carlo EM algorithm based on randomized Quasi-Monte Carlo methods. We apply this algorithm to a geostatistical model of online purchases and find that it can significantly decrease the total simulation effort, thus showing great potential for improving upon the efficiency of the classical Monte Carlo EM algorithm. Key words and phrases: Monte Carlo error; low-discrepancy sequence; Halton sequence; EM algo-rithm; geostatistical model.

The EM algorithm is a popular tool for maximizing likelihood functions in the presence of missing data. Unfortunately, EM often requires the evaluation of analytically intractable and high-dimensional integrals. The Monte Carlo EM (MCEM) algorithm is the natural extension of EM that employs Monte Carlo methods to estimate the relevant integrals. Typically, a very large Monte Carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence. Even if this sample size were known at the onset of implementation of MCEM, its use throughout all iterations is wasteful, especially when accurate starting values are not available. We propose a data-driven strategy for controlling Monte Carlo resources in MCEM. The proposed algorithm improves on similar existing methods by: (i) recovering EM’s ascent (i.e., likelihood-increasing) property with high probability, (ii) being more robust to the impact of user defined inputs, and (iii) handling classical Monte Carlo and Markov chain Monte Carlo within a common framework. Because of (i) we refer to the algorithm as “Ascent-based MCEM”. We apply Ascent-based MCEM to a variety of examples, including one where it is used to dramatically accelerate the convergence of deterministic EM.

Data cloning method is a new computational tool for computing maximum likelihood estimates in complex statistical models such as mixed models. The data cloning method is synthesized with integrated nested Laplace approximation to compute maximum likelihood estimates efficiently via a fast implementation in generalized linear mixed models. Asymptotic normality of the hybrid data cloning based distribution is established aided by modification of Stein’s Identity. The results are illustrated through a series of well known examples. It is shown that the proposed method as well as normal approximation perform very well and justify the theory.

The Hidden semi-Markov models (HSMMs) have been introduced to overcome the constraint of a geometric sojourn time distribution for the different hidden states in the classical hidden Markov models. Several variations of HSMMs have been proposed that model the sojourn times by a parametric or a nonparametric family of distributions. In this article, we concentrate our interest on the nonparametric case where the duration distributions are attached to transitions and not to states as in most of the published papers in HSMMs. Therefore, it is worth noticing that here we treat the underlying hidden semi–Markov chain in its general probabilistic structure. In that case, Barbu and Limnios (2008) proposed an Expectation–Maximization (EM) algorithm in order to estimate the semi-Markov kernel and the emission probabilities that characterize the dynamics of the model. In this paper, we consider an improved version of Barbu and Limnios ’ EM algorithm which is faster than the original one. Moreover, we propose a stochastic version of the EM algorithm that achieves comparable estimates with the EM algorithm in less execution time. Some numerical examples are provided which illustrate the efficient performance of the proposed algorithms.