# Jason Schweinsberg

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## Publications (3)3.46 Total impact

• Source
##### Article: Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter
James P. Hobert, Dobrin Marchev, 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) = ∨<sub>x</sub><sup>∞</sup> f(z)dz. We study the tail Markov chain generated by Z, defined as the Markov chain Ψ=(Ψ<sub>n</sub>)<sub>n=0</sub><sup>∞</sup> 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 ∨<sub>1</sub><sup>∞</sup> 1/[ z<sup>3</sup> 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<sup>- θx</sup> for x>0. Let ν be an improper prior density for θ that is positive on (0,∞). Assume that ∨<sub>0</sub><sup>∞</sup> θ ν(θ) dθ< ∞, which implies that the posterior density induced by ν is proper. Let m<sub>ν</sub> denote the marginal density of X induced by ν; that is, m<sub>ν</sub>(x) = ∨<sub>0</sub><sup>∞</sup> θe<sup>-θx</sup> 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 ∨<sub>1</sub><sup>∞</sup> 1/ [x<sup>2</sup> m<sub>ν</sub>(x)]dx = ∞.
Bernoulli 01/2004; · 0.94 Impact Factor
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##### Article: Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability
James P. Hobert, Jason Schweinsberg
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ABSTRACT: Let $Z$ be a discrete random variable with support $\Z^+ = \{0,1,2,\dots\}$. We consider a Markov chain $Y=(Y_n)_{n=0}^\infty$ with state space $\Z^+$ and transition probabilities given by $P(Y_{n+1} = j|Y_n = i) = P(Z = i+j)/P(Z \geq i)$. We prove that convergence of $\sum_{n=1}^\infty 1/[n^3 P (Z=n)]$ is sufficient for transience of $Y$ while divergence of $\sum_{n=1}^\infty 1/[n^2 P (Z \geq n)]$ is sufficient for recurrence. Let $X$ be a $\mbox{Geometric}(p)$ random variable; that is, $P(X=x)=p(1-p)^x$ for $x \in \Z^+$. 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 $\mathscr{P}$-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 $\mathscr{P}$-admissible.
The Annals of Statistics 01/2002; · 2.53 Impact Factor
• ##### Article: Conditions for Recurrence and Transience of a Markov Chain on Z+ and Estimation of a Geometric Success Probability
James P. Hobert, Jason Schweinsberg
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ABSTRACT: Let Z be a discrete random variable with support Z + = f0; 1; 2; : : : g. We consider a Markov chain Y = (Yn ) 1 n=0 with state space Z + and transition probabilities given by P (Yn+1 = jjY n = i) = P (Z = i + j)=P (Z i). We prove that convergence of P 1 n=1 1=[n 3 P (Z = n)] is sucient for transience of Y while divergence of P 1 n=1 1=[n 2 P (Z n)] is sucient for recurrence. We use these results in conjunction with those of Eaton (1992) and Hobert and Robert (1999) to show that when p is the success probability of a Geometric random variable, all prior densities of the form p 1 (1 p) b 1 for b > 0 are P-admissible. We also use our results about Y to establish sucient conditions for recurrence and transience of the continuous analogue of Y that lives on R + . This is done via a coupling argument. This research partially supported by NSF Grant DMS-00-72827. AMS 2000 subject classications. Primary 60J10; secondary 62C15 Key words and phrases. Admissibility, Electrical network, Geometric distribution, Null recurrence, Reversibility, Weighted random walk 1 1
05/2001;