Bernoulli

  • Bernoulli
  • Volume 10, Number 3 (2004), 549-564.

Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter

James P. Hobert, Dobrin Marchev, and Jason Schweinsberg

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Abstract

Let Z be a continuous random variable with a lower semicontinuous density f that is positive on (0,∞) and 0 elsewhere. Put G(x) = ∨x f(z)dz. We study the tail Markov chain generated by Z, defined as the Markov chain Ψ=(Ψn)n=0 with state space [0, ∞) and Markov transition density k(y|x) = f(y+x)/G(x). This chain is irreducible, aperiodic and reversible with respect to G. It follows that Ψ is positive recurrent if and only if Z has a finite expectation. We prove (under regularity conditions) that if E Z = ∞, then Ψ is null recurrent if and only if 1 1/[ z3 f(z) ] dz = ∞. Furthermore, we describe an interesting decision-theoretic application of this result. Specifically, suppose that X is an Exp(θ) random variable; that is, X has density θe- θx for x>0. Let ν be an improper prior density for θ that is positive on (0,∞). Assume that 0 θ ν(θ) dθ< ∞, which implies that the posterior density induced by ν is proper. Let mν denote the marginal density of X induced by ν; that is, mν(x) = ∨0 θe-θx ν(θ) dθ. We use our results, together with those of Eaton and of Hobert and Robert, to prove that ν is a \cal P-admissible prior if 1 1/ [x2 mν(x)]dx = ∞.

Article information

Source
Bernoulli, Volume 10, Number 3 (2004), 549-564.

Dates
First available in Project Euclid: 7 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1089206409

Digital Object Identifier
doi:10.3150/bj/1089206409

Mathematical Reviews number (MathSciNet)
MR2061443

Zentralblatt MATH identifier
1049.60068

Keywords
admissibility coupling hazard rate null recurrence reversibility stochastic comparison stochastically monotone Markov chain transience

Citation

Hobert, James P.; Marchev, Dobrin; Schweinsberg, Jason. Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter. Bernoulli 10 (2004), no. 3, 549--564. doi:10.3150/bj/1089206409. https://projecteuclid.org/euclid.bj/1089206409


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