Brazilian Journal of Probability and Statistics

Asymptotic predictive inference with exchangeable data

Abstract

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.

Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 4 (2018), 815-833.

Dates
Accepted: May 2017
First available in Project Euclid: 17 August 2018

https://projecteuclid.org/euclid.bjps/1534492903

Digital Object Identifier
doi:10.1214/17-BJPS367

Mathematical Reviews number (MathSciNet)
MR3845031

Zentralblatt MATH identifier
06979602

Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Asymptotic predictive inference with exchangeable data. Braz. J. Probab. Stat. 32 (2018), no. 4, 815--833. doi:10.1214/17-BJPS367. https://projecteuclid.org/euclid.bjps/1534492903

References

• Aldous, D. J. (1985). Exchangeability and Related Topics, Ecole de Probabilites de Saint-Flour XIII. Lect. Notes in Math. 1117. Berlin: Springer.
• Berti, P., Crimaldi, I., Pratelli, L. and Rigo, P. (2009). Rate of convergence of predictive distributions for dependent data. Bernoulli 15, 1351–1367.
• Berti, P., Mattei, A. and Rigo, P. (2002). Uniform convergence of empirical and predictive measures. Atti Sem. Mat. Fis. Univ. Modena 50, 465–477.
• Berti, P., Pratelli, L. and Rigo, P. (2004). Limit theorems for a class of identically distributed random variables. Ann. Probab. 32, 2029–2052.
• Berti, P., Pratelli, L. and Rigo, P. (2012). Limit theorems for empirical processes based on dependent data. Electron. J. Probab. 17, 1–18.
• Berti, P., Pratelli, L. and Rigo, P. (2013). Exchangeable sequences driven by an absolutely continuous random measure. Ann. Probab. 41, 2090–2102.
• Berti, P. and Rigo, P. (1997). A Glivenko–Cantelli theorem for exchangeable random variables. Statist. Probab. Lett. 32, 385–391.
• Blackwell, D. and Dubins, L. E. (1962). Merging of opinions with increasing information. Ann. Math. Statist. 33, 882–886.
• Cifarelli, D. M., Dolera, E. and Regazzini, E. (2016). Frequentistic approximations to Bayesian prevision of exchangeable random elements. Internat. J. Approx. Reason. 78, 138–152.
• Cifarelli, D. M. and Regazzini, E. (1996). De Finetti’s contribution to probability and statistics. Statist. Sci. 11, 253–282.
• Crane, H. (2016). The ubiquitous Ewens sampling formula. Statist. Sci. 31, 1–19.
• De Blasi, P., Favaro, S., Lijoi, A., Mena, R. H., Prunster, I. and Ruggiero, M. (2015). Are Gibbs-type priors the most natural generalization of the Dirichlet process? IEEE Trans. Pattern Anal. Machine Intell. 37, 212–229.
• Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist. 14, 1–26.
• Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge: Cambridge University Press.
• Efron, B. (2003). Robbins, empirical Bayes and microarrays. Ann. Statist. 31, 366–378.
• Fortini, S., Ladelli, L. and Regazzini, E. (2000). Exchangeability, predictive distributions and parametric models. Sankhya A 62, 86–109.
• Gaenssler, P. and Stute, W. (1979). Empirical processes: A survey of results for independent and identically distributed random variables. Ann. Probab. 7, 193–243.
• Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge: Cambridge University Press.
• Hjort, N. L., Holmes, C., Muller, P. and Walker, S. G. (2010). Bayesian Nonparametrics. Cambridge: Cambridge University Press.
• Kingman, J. F. C. (1975). Random discrete distributions (with discussion). J. Royal Stat. Soc. B 37, 1–22.
• Kuelbs, J. and Dudley, R. M. (1980). Log log laws for empirical measures. Ann. Probab. 8, 405–418.
• Mijoule, G., Peccati, G. and Swan, Y. (2016). On the rate of convergence in de Finetti’s representation theorem. ALEA (Lat. Am. J. Probab. Math. Stat.) 13, 1165–1187.
• Phadia, E. G. (2016). Prior Processes and Their Applications, 2nd ed. Berlin: Springer.
• Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900.
• Robbins, H. (1964). The empirical Bayes approach to statistical decision problems. Ann. Math. Statist. 35, 1–20.
• Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Stat. Sinica 4, 639–650.
• van der Vaart, A. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Berlin: Springer.
• Zabell, S. L. (2005). Symmetry and Its Discontents (Essays on the History of Inductive Probability). Cambridge: Cambridge University Press.