349,852 research outputs found
Small sets and Markov transition densities
Full text linkThe theory of general state-space Markov chains can be strongly related to the case of discrete state-space by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated sigma-algebras, though the minorization provided by the theory concerns small sets of order n and n-step transition kernels for some unspecified n. Partly motivated by the growing importance of small sets for Markov chain Monte Carlo and Coupling from the Past, we show that in general there need be no small sets of order n = 1 even if the kernel is assumed to have a density function (though of course one can take n = 1 if the kernel density is continuous). However, n = 2 will suffice for kernels with densities (integral kernels), and in fact small sets of order 2 abound in the technical sense that the 2-step kernel density can be expressed as a countable sum of non-negative separable summands based on small sets, This can be exploited to produce a representation using a latent discrete Markov chain; indeed one might say, inside every Markov chain with measurable transition density there is a discrete state-space Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harris-recurrent Markov chains and we relate the counterexample to Turan problems for bipartite graphs
Asymptotic Expansions for Moments of Hitting Times for Nonlinearly Perturbed Semi-Markov Processes
No full textIn Chapter 3, we introduce a model of perturbed semi-Markov processes, formulate basic perturbation conditions, describe a one-step time-space screening procedure of phase space reduction for perturbed semi-Markov processes, introduce hitting times, and prove an invariant property of them with respect to the procedure of phase space reduction. We, also, present algorithms for re-calculation of asymptotic expansions for transition characteristics of nonlinearly perturbed semi-Markov processes with reduced phase spaces and algorithms for sequential reduction of phase space for semi-Markov processes and construction of Laurent asymptotic expansions, without and with explicit upper bounds for remainders, for power moment of hitting times. © 2017, The Author(s).</p
Nonlinearly Perturbed Birth-Death-Type Semi-Markov Processes
No full textIn Chapter 5, we present asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed birth-death-type semi-Markov processes, which play an important role in many applications. In this case, the corresponding expansions can be given in a more explicit form. © 2017, The Author(s).</p
Markov or not Markov - this should be a question
Full text linkAlthough it is well known that Markov process theory, frequently applied in the literature on income convergence, imposes some very restrictive assumptions upon the data generating process, these assumptions have generally been taken for granted so far. The present paper proposes, resp. recalls chi-square tests of the Markov property, of spatial independence, and of homogeneity across time and space to assess the reliability of estimated Markov transition matrices. As an illustration we show that the evolution of the income distribution across the 48 coterminous U.S. states from 1929 to 2000 clearly has not followed a Markov process.
Beveridge-Nelson Decomposition with Markov Switching
Full text linkThis paper considers Beveridge-Nelson decomposition in a context where the permanent and transitory components both follow a Markov switching process. Our approach incorporates Markov switching into a single source of error state-space framework, allowing business cycle asymmetries and regime switches in the long run multiplier.Beveridge-Nelson decomposition, Markov switching, Single source of error state space models
Markov-Switching Model Selection Using Kullback-Leibler Divergence
Full text linkIn Markov-switching regression models, we use Kullback-Leibler (KL) divergence between the true and candidate models to select the number of states and variables simultaneously. In applying Akaike information criterion (AIC), which is an estimate of KL divergence, we find that AIC retains too many states and variables in the model. Hence, we derive a new information criterion, Markov switching criterion (MSC), which yields a marked improvement in state determination and variable selection because it imposes an appropriate penalty to mitigate the over-retention of states in the Markov chain. MSC performs well in Monte Carlo studies with single and multiple states, small and large samples, and low and high noise. Furthermore, it not only applies to Markov-switching regression models, but also performs well in Markov- switching autoregression models. Finally, the usefulness of MSC is illustrated via applications to the U.S. business cycle and the effectiveness of media advertising.Research Methods/ Statistical Methods,
Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions
Full text linkRecent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this, geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed
Theory and inference for a Markov switching GARCH model
Full text linkWe develop a Markov-switching GARCH model (MS-GARCH) wherein the conditional mean and variance switch in time from one GARCH process to another. The switching is governed by a hidden Markov chain. We provide sufficient conditions for geometric ergodicity and existene of moments of the process. Because of path dependence, maximum likelihood estimation is not feasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We illustrate the model on SP500 daily returns.GARCH, Markov-switching, Bayesian inference
Theory and Inference for a Markov-Switching GARCH Model
Full text linkWe develop a Markov-switching GARCH model (MS-GARCH) wherein the conditional mean and variance switch in time from one GARCH process to another. The switching is governed by a hidden Markov chain. We provide sufficient conditions for geometric ergodicity and existence of moments of the process. Because of path dependence, maximum likelihood estimation is not feasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We illustrate the model on SP500 daily returns.GARCH, Markov-switching, Bayesian inference
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