The consistency of the BIC Markov order estimator

• 成果类型: Article
• 署名作者: Csiszár, I; Shields, PC
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Hungarian Academy of Sciences; University System of Ohio; University of Toledo
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2000
• 页码: 1601-1619
• 关键词: model

摘要

The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x(1), x(2), ..., x(n), as that value k = (k) over cap that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term \A\(k)(\A\-1)/2. We show that (k) over cap equals the correct order of the chain, eventually almost surely as it --> infinity, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong ratio-typicality result for Markov sample paths. We also show that the Bayesian estimator or minimum description length estimator, of which the BIC estimator is regarded as an approximation, fails to be consistent for the uniformly distributed i.i.d. process.