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Applications of Bernstein polynomials in Statistics

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Bernstein polynomials are studied in analysis as a probabilistic approximation scheme, and they are well known to provide a constructive proof of Weirstrass theorem. In fact, their simple probabilistic nature make them also attractive in many statistical problems. In this talk I will give an overview of applications of Bernstein polynomials in statistics, in particular in Bayesian nonparametric inference. I will briefly review their use to construct a nonparametric prior, which provides a smoothing of a Dirichlet process, and allows to fairly easily incorporate prior information. I will discuss theoretical properties and show some applications, in particular to shrinkage estimation, comparing with the clustering properties of the Dirichlet process. Bernstein polynomials also arise in quantile estimation from a Dirichlet process. I will finally discuss extensions to data on a general subset of R and to multivariate data.

This talk is part of the Statistics series.

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