Asymptotic predictive inference with exchangeable data
| Autor: | Pietro Rigo, Luca Pratelli, Patrizia Berti |
|---|---|
| Přispěvatelé: | Berti, Patrizia, Pratelli, Luca, RIGO, PIETRO |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
High Energy Physics::Phenomenology Mathematics::Analysis of PDEs 02 engineering and technology conditional identity in distribution exchangeability random probability measure Bayesian consistency Computer Science::Numerical Analysis 01 natural sciences Combinatorics 010104 statistics & probability predictive measure 0202 electrical engineering electronic engineering information engineering Filtration (mathematics) empirical measure 020201 artificial intelligence & image processing Almost surely 0101 mathematics Bayesian consistency -- Conditional identity in distribution -- Empirical measure -- Exchangeability -- Predictive measure -- Random probability measure Random variable Mathematics |
| Zdroj: | Braz. J. Probab. Stat. 32, no. 4 (2018), 815-833 |
| ISSN: | 0103-0752 |
| DOI: | 10.1214/17-bjps367 |
| Popis: | Let $(X_{n})$ be a sequence of random variables, adapted to a filtration $(\mathcal{G}_{n})$, and let $\mu_{n}=(1/n)\sum_{i=1}^{n}\delta_{X_{i}}$ and $a_{n}(\cdot)=P(X_{n+1}\in\cdot|\mathcal{G}_{n})$ be the empirical and the predictive measures. We focus on \begin{equation*}\Vert \mu_{n}-a_{n}\Vert =\mathop{\mathrm{sup}}_{B\in\mathcal{D}}\vert\mu_{n}(B)-a_{n}(B)\vert,\end{equation*} where $\mathcal{D}$ is a class of measurable sets. Conditions for $\Vert \mu_{n}-a_{n}\Vert \rightarrow0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_{n}\Vert \mu_{n}-a_{n}\Vert $ is investigated for suitable constants $r_{n}$. Special attention is paid to $r_{n}=\sqrt{n}$ and $r_{n}=\sqrt{\frac{n}{\log\log n}}$. The sequence $(X_{n})$ is exchangeable or, more generally, conditionally identically distributed. |
| Databáze: | OpenAIRE |
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