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August, 1973 Contributions to the Theory of Dirichlet Processes
Ramesh M. Korwar, Myles Hollander
Ann. Probab. 1(4): 705-711 (August, 1973). DOI: 10.1214/aop/1176996898


Consider a sample $X_1, \cdots, X_n$ from a Dirichlet process $P$ on an uncountable standard Borel space $(\mathscr{X}, \mathscr{A})$ where the parameter $\alpha$ of the process is assumed to be non-atomic and $\sigma$-additive. Let $D(n)$ be the number of distinct observations in the sample and denote these distinct observations by $Y_1, \cdots, Y_{D(n)}$. Our main results are (1) $D(n)/\log n \rightarrow_{\operatorname{a.s.}} \alpha(\mathscr{X}), n \rightarrow \infty$, and (2) given $D(n), Y_1, \cdots, Y_{D(n)}$ are independent and identically distributed according to $\alpha(\bullet)/\alpha(\mathscr{X})$. Result (1) shows that $\alpha(\mathscr{X})$ can be consistently estimated from the sample, and result (2) leads to a strong law for $\sum^{D(n)}_{i=1} Y_i/D(n)$.


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Ramesh M. Korwar. Myles Hollander. "Contributions to the Theory of Dirichlet Processes." Ann. Probab. 1 (4) 705 - 711, August, 1973.


Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0264.60084
MathSciNet: MR350950
Digital Object Identifier: 10.1214/aop/1176996898

Primary: 60K99
Secondary: 62G05

Keywords: consistent estimation , Dirichlet process , distribution theory , Strong law of large numbers

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 4 • August, 1973
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