The Annals of Probability

Contributions to the Theory of Dirichlet Processes

Ramesh M. Korwar and Myles Hollander

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Abstract

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 $\sum^{D(n)}_{i=1} Y_i/D(n)$.

Article information

Source
Ann. Probab., Volume 1, Number 4 (1973), 705-711.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996898

Digital Object Identifier
doi:10.1214/aop/1176996898

Mathematical Reviews number (MathSciNet)
MR350950

Zentralblatt MATH identifier
0264.60084

JSTOR
links.jstor.org

Subjects
Primary: 60K99: None of the above, but in this section
Secondary: 62G05: Estimation

Keywords
Dirichlet process consistent estimation strong law of large numbers distribution theory

Citation

Korwar, Ramesh M.; Hollander, Myles. Contributions to the Theory of Dirichlet Processes. Ann. Probab. 1 (1973), no. 4, 705--711. doi:10.1214/aop/1176996898. https://projecteuclid.org/euclid.aop/1176996898


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