Conditions for recurrence and transience of a Markov chain on Z+ and estimation of a geometric success probability

• 成果类型: Article
• 署名作者: Hobert, JP; Schweinsberg, J
• 署名单位: State University System of Florida; University of Florida; Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1214-1223
• 关键词: Admissibility

摘要

Let Z be a discrete random variable with support Z(+) = {0, 1, 2,...}. We consider a Markov chain Y = (Y-n)(infinity)(n=0) with state space Z(+) and transition probabilities given by P(Yn+1 = j\Yn = i) = P(Z = i + j)/P(Z greater than or equal to i). We prove that convergence of Sigma(n=1)(infinity) 1/[n(3) P(Z = n)] is sufficient for transience of Y while divergence of Sigma(n=1)(infinity) 1/[n(2) P (Z greater than or equal to n)] is sufficient for recurrence. Let X be a Geometric (p) random variable; that is, P (X = x) = P (1 - P)(x) for x is an element of Z(+). We use our results in conjunction with those of M. L. Eaton [Ann. Statist. 20 (1992) 1147-1179] and J. P. Hobert and C. P. Robert [Ann. Statist. 27 (1999) 361-373] to establish a sufficient condition for. P-admissibility of improper priors on p. As an illustration of this result, we prove that all prior densities of the form p(-1) (1 - p)(b-1) with b > 0 are P-admissible.