On the use of Cauchy integral formula for the embedding problem of discrete-time Markov chains
About: This article is published in Communications in Statistics-theory and Methods.The article was published on 2021-05-20. It has received 1 citations till now. The article focuses on the topics: Cauchy's integral formula & Markov chain.
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25 Jun 2021
TL;DR: This work proposes a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.
Abstract: We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.
7 citations
References
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Book•
01 Nov 2008
TL;DR: Finite Non-Negative Matrices as mentioned in this paper are a generalization of finite stochastic matrices, and finite non-negative matrices have been studied extensively in the literature.
Abstract: Finite Non-Negative Matrices.- Fundamental Concepts and Results in the Theory of Non-negative Matrices.- Some Secondary Theory with Emphasis on Irreducible Matrices, and Applications.- Inhomogeneous Products of Non-negative Matrices.- Markov Chains and Finite Stochastic Matrices.- Countable Non-Negative Matrices.- Countable Stochastic Matrices.- Countable Non-negative Matrices.- Truncations of Infinite Stochastic Matrices.
2,855 citations
TL;DR: Leslie's work, rather than that of his predecessors Bernardelli and Lewis, is most commonly cited in the widespread literature using matrices, largely for the reason that Leslie worked out the mathematics and the application with great thoroughness.
Abstract: Leslie’s work, rather than that of his predecessors Bernardelli and Lewis, is most commonly cited in the widespread literature using matrices, largely for the reason that Leslie worked out the mathematics and the application with great thoroughness. Some of his elaboration was designed to save arithmetic—for example his transformation of the projection matrix into an equivalent form with unity in the subdiagonal positions. Such devices, like a considerable part of classical numerical analysis, are unnecessary in a computer era.
2,340 citations
TL;DR: In this article, the authors consider a class of issues which are central to modeling social phenomena by continuous-time Markov structures and discuss how to select the specific structure from the list of alternatives which should be associated with the empirical process.
Abstract: In this paper we consider a class of issues which are central to modeling social phenomena by continuous-time Markov structures. In particular, we discuss (a) embeddability, or how to determine whether observations on an empirical process could have arisen via the evolution of a continuous-time Markov structure; and (b) identification, or what to do if the observations are consisten with more than one continuous-time Markov structure. With respect to the latter topic, we discuss how to select the specific structure from the list of alternatives which should be associated with the empirical process. We point out that the issues of embeddability and identification are especially pertinent to modeling empirical processes when one has available only fragmentary data and when the observations contain "noise" or other sources of error. These characteristics, of course, describe the typical work situation of sociologists. Finally, we note the type of situation in which a continuous-time model is the proper struc...
265 citations
TL;DR: In this paper, the authors identify conditions under which a true generator does or does not exist for an empirically observed Markov transition matrix and search for valid generators and choose the "correct" one that is the most compatible with bond rating behaviors.
Abstract: In this paper we identify conditions under which a true generator does or does not exist for an empirically observed Markov transition matrix. We show how to search for valid generators and choose the “correct” one that is the most compatible with bond rating behaviors. We also show how to obtain an approximate generator when a true generator does not exist. We give illustrations using credit rating transition matrices published by Moody's and by Standard and Poor's.
194 citations
98 citations