The focus of this paper is on the derivation of confidence and credibility intervals for Markov chains when discrete-time, continuous-time or discretely observed continuous-time data are available. Thereby, our contribution is threefold: First, we discuss and compare multinomial confidence regions for the rows of discrete-time Markov transition matrices in the light of empirical characteristics of credit rating migrations. Second, we derive an analytical expression for the expected Fisher information matrix of a continuous-time Markov chain which is used to construct credibility intervals using a non-informative Jeffreys prior distribution and a Metropolis-Hastings Algorithm. Third, we concretize profile and estimated/pseudo likelihood based confidence intervals in the continuous-time data settings, which in contrast to asymptotic normality based intervals explicitly consider non-negativity constraints for the parameters. Furthermore, we illustrate the described methods by Moody’s corporate ratings data with exact continuous-time transitions.

中文翻译：

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马尔可夫链的统计推断及其在信用风险中的应用

本文的重点是当离散时间，连续时间或离散观察到的连续时间数据可用时，马尔可夫链的置信度和可信区间的推导。因此，我们的贡献是三方面的：首先，根据信用评级迁移的经验特征，我们讨论并比较了离散时间马尔可夫转移矩阵行的多项式置信区域。其次，我们导出了连续时间马尔可夫链的预期Fisher信息矩阵的解析表达式，该表达式用于使用非信息性Jeffreys先验分布和Metropolis-Hastings算法构造可信区间。第三，我们在连续时间数据设置中具体化基于配置文件和估计/伪似然的置信区间，与基于渐近正态性的区间相反，它明确考虑了参数的非负约束。此外，我们以准确的连续时间转换来说明穆迪公司评级数据所描述的方法。