Some problems of statistical inference in absorbing Markov chains
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吸收马尔可夫链中的若干统计推断问题
- 研究问题
- 真正具有吸收状态的马尔可夫链仅观察一次实现时，如何研究最大似然估计的渐近性质，增加非吸收状态数能否提供可用框架？
- 方法与思路
- 作者先用两个简单例子考察大量非吸收状态下的渐近可能性，再分析 Moran 群体遗传模型，并以模拟支持相关渐近猜想。
- 材料陈述的贡献
- 摘要指出固定状态数的单次实现最终吸收，不能靠延长观察持续获得信息；论文探索状态数增大的路径，但复杂模型结论仍是猜想。
- 阅读关注点
- 应核对状态数增长的具体安排、简单例子的已证结论与遗传模型猜想的界限，以及模拟证据和相关马尔可夫链理论的不足。
- 阅读边界
- OpenAlex 索引摘要未注明稿件版本；作者明确未证明遗传模型参数估计的渐近猜想，不能将模拟支持表述为一般吸收链的已证定理。
原文摘要
The estimation of parameters in an absorbing Markov chain has been discussed by Gani (1956), Bhat & Gani (1960), Bhat (1961) and possibly by other authors. Asymptotic theory for the distribution of maximum-likelihood estimators applies if a large number of independent replicates of the chain is available. These replicates, however, could be considered as occurring sequentially in time, so that the chain has implicitly been altered in such a way that the absorbing state has been replaced by an ‘instantaneous return’ state; once it is reached, a new replicate is commenced starting from the original initial state. In this paper, we concern ourselves with the asymptotic theory of maximum-likelihood estimators from a single realization of truly absorbing chains. It is clear that if the number of states is kept fixed there can be no asymptotic theory, since with probability one, an absorbing state will be reached after a finite number of transitions, and no further information can be obtained by continuing observation. In § 2, we show by means of two simple examples that asymptotic theory may, at least in some cases, be available if the number of non-absorbing states is large. In the remaining sections, forming the main part of the paper, we discuss a more complex example, a population genetic model of Moran (1958a). We do not prove that the conjectured asymptotic theory holds for the estimation of the parameter, but produce numerical evidence from simulation studies to support this conjecture. Further research is needed to clarify the general problem of inference in absorbing Markov chains. The problem can be made to depend on the theory of positively regular chains but the latter is itself incomplete for this purpose.