Random Bernstein Polynomials

@article{Petrone1999RandomBP,
  title={Random Bernstein Polynomials},
  author={Sonia Petrone},
  journal={Scandinavian Journal of Statistics},
  year={1999},
  volume={26}
}
  • Sonia Petrone
  • Published 1 September 1999
  • Mathematics
  • Scandinavian Journal of Statistics
Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so defined provides a novel prior on the space of distribution functions on [0, 1] which has full support and can easily select absolutely continuous distribution functions with a continuous and smooth derivative. In particular, the Bernstein polynomial which approximates a Dirichlet process is studied. This may be of interest in… 

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References

SHOWING 1-10 OF 52 REFERENCES

Bayesian density estimation using bernstein polynomials

TLDR
A Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval, say [0,1], using a prior based on Bemstein polynomials to express the density as a mixture of given beta densities, with random weights and a random number of components.

Two-dimensional Bernstein polynomial density estimators

A Bernstein polynomial estimator for fnN(x, y) for an unknown probability density functionf(x, y) concentrated on the triangle Δ={(x, y): 0≤x, y<1,x+y<1} or on the square ⊡=(x, y):0 ≤ x, y ≤ 1 is

BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS

Randomly generated distributions

A new scheme for randomly generating probability distributions on the interval [0, 1] is introduced. The scheme can also be viewed as a way to generate homeomorphisms at random. Conditions are given

Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems

process. This paper extends Ferguson's result to cases where the random measure is a mixing distribution for a parameter which determines the distribution from which observations are made. The

Bayesian Nonparametric Estimation for Incomplete Data Via Successive Substitution Sampling

In the problem of estimating an unknown distribution function F in the presence of censoring, one can use a nonparametric estimator such as the Kaplan-Meier estimator, or one can consider parametric

Generalized linear models with unknown link functions

function by incorporating it as an unknown in the model. Since the link function is usually taken to be strictly increasing, by a strictly increasing transformation of its range to the unit interval

Bayesian Approaches to Non- and Semiparametric Density Estimation

This paper proposes and discusses several Bayesian attempts at nonparametric and semiparametric density estimation. The main categories of these ideas are as follows: ( 1) Build a non parametric
...