On metrizing vague convergence of random measures with applications on Bayesian nonparametric models. Issue 2 (4th March 2018)

• Record Type: Journal Article
• Authors: Al-Labadi, Luai
• Is Part Of: Statistics. Volume 52:Issue 2(2018)
• Page Start: 445
• Page End: 457
• Publication Date: 2018-03-04
• Subjects: Beta process -- nonparametric Bayesian statistics -- random measures -- vague convergence; Primary: 62F15 -- 60G57 -- Secondary: 28A33; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2018.1425866
• Languages: English
• ISSNs: 0233-1888

Abstract

ABSTRACT: This paper deals with studying vague convergence of random measures of the formμ n = ∑ i = 1 n p i, n δ θ i, where( θ i ) 1 ≤ i ≤ n is a sequence of independent and identically distributed random variables with common distribution Π, δ θ i denotes the Dirac measure atθ i and( p i, n ) 1 ≤ i ≤ n are random variables, independent of( θ i ) i ≥ 1, chosen according to certain procedures such thatp i, n → p i almost surely, asn → ∞, for fixed i . We show that, asn → ∞, μ n converges vaguely almost surely toμ = ∑ i = 1 ∞ p i δ θ i if and only ifμ n ( k ) = ∑ i = 1 k p i, n δ θ i converges vaguely almost surely toμ ( k ) = ∑ i = 1 k p i δ θ i for all k fixed. The limiting process μ plays a central role in many areas in statistics, including Bayesian nonparametric models. A finite approximation of the beta process is derived from the application of this result. A simulated example is incorporated, in which the proposed approach exhibits an excellent performance over several existing algorithms.