{"title":"Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees","authors":"Julien WeibelIDP, CERMICS","doi":"arxiv-2409.06295","DOIUrl":null,"url":null,"abstract":"We consider hidden Markov models indexed by a binary tree where the hidden\nstate space is a general metric space. We study the maximum likelihood\nestimator (MLE) of the model parameters based only on the observed variables.\nIn both stationary and non-stationary regimes, we prove strong consistency and\nasymptotic normality of the MLE under standard assumptions. Those standard\nassumptions imply uniform exponential memorylessness properties of the initial\ndistribution conditional on the observations. The proofs rely on ergodic\ntheorems for Markov chain indexed by trees with neighborhood-dependent\nfunctions.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"75 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - STAT - Statistics Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06295","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We consider hidden Markov models indexed by a binary tree where the hidden
state space is a general metric space. We study the maximum likelihood
estimator (MLE) of the model parameters based only on the observed variables.
In both stationary and non-stationary regimes, we prove strong consistency and
asymptotic normality of the MLE under standard assumptions. Those standard
assumptions imply uniform exponential memorylessness properties of the initial
distribution conditional on the observations. The proofs rely on ergodic
theorems for Markov chain indexed by trees with neighborhood-dependent
functions.