Convergence rates for density estimation with Bernstein polynomials

By Subhashis Ghosal. 2001

157 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/480c5a1abfba07fd33a7ffe7d3d5976cfaf92150

Citations in the corpus

A Unified Framework for Fitting Bayesian Semiparametric Models to Arbitrarily Censored Survival Data, Including Spatially Referenced Data (2017, unknown): “Smooth densities with support (0, 1) can be well approximated by a Bernstein polynomial (Ghosal 2001): if f (x) is any continuously differentiable density with support (0, 1) and bounded second derivative, wJ can be chosen such that sup 0 x 1 | f (x) − d(x|J,wJ )| = O(J−1).”
A variable selection approach to monotonic regression with Bernstein polynomials (2011, unknown): “Ghosal 27 has proved theoretical convergence rates of Bernstein polynomials in the related problem of non-parametric density estimation.”
Adaptive Bayesian density estimation with location-scale mixtures (2010, unknown): “…with rate n = n − β/ (1+2 β ) (log n ) t , where r and b are as in (11) and (14) , The proof is based on Theorem 5 of Ghosal and van der Vaart [13], which is included here in appendix A. Condition (11) is usual in ﬁnite mixture models, see for instance [10], [20] and [26] for beta-mixtures.”
On Bayesian supremum norm contraction rates (2013, unknown): “eve the set of dyadic density histograms HLn, so that Π[HcL n ]=0. Entropy and prior mass conditions can then be veriﬁed with the same rate (logn/n)α/(2α+1); see also the remark before Theorem 2.4 in [12].ON BAYESIAN SUP-NORM RATES 33 Lemma 11. Let ∆(ζ) be the Jacobian of the change of variables ω→ζ”
Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparamatric estimation of the density (2010, unknown): “The papers [4, 5] and [15] study the concentration rate of the posterior distribution under Dirichlet mixtures of Gaussian priors, and Ghosal [3] considers the Bernstein polynomial’s case, that is, the mixture of Beta distribution with fixed parameters.”
Bernstein polynomial probability density estimation (2004, unknown): “For smooth estimate of a density function with a finite known support, Vitale’s method (1975) based on the Bernstein polynomials, illustrated below, also has been investigated in recent literature (e.g. Ghosal, 2001; Babu et al., 2002).”
On rates of convergence for posterior distributions in infinite-dimensional models (2007, unknown): “3 of [4], it is shown that the prior concentration rate is (logn)1/3/n1/3 and that the entropy rate is (logn)5/6/n1/3, thus leading to an overall convergence rate of (logn)5/6/n1/3.”
A bias-reduced approach to density estimation using Bernstein polynomials (2010, unknown): “This estimator relies on the family of the so-called Bernstein–Dirichlet priors introduced by Petrone (1999b) and has been further studied by Ghosal (2001) and Petrone and Wasserman (2002) .”
Convergence rates of posterior distributions for non-i.i.d. observations (2007, unknown): “Therefore, following Ghosal [12], (‖f −f0‖∞ < ε) e−cε log ε−1 .”
Flexible semiparametric generalized Pareto modeling of the entire range of rainfall amount (2019, unknown): “More recently, Leblanc (2010, 2012a,b) studied the boundary properties of both density and distribution estimators, (see also Petrone, 1999; Ghosal, 2001; Kakizawa, 2004; Bouezmarni and Rolin, 2007, for extensions in a Bayesian and/or multivariate context).”
Adaptive Bayesian Procedures Using Random Series Priors (2014, unknown): “Hence f satisfies conditions of Lemma 2 of Ghosal and van der Vaart [2007b] with respect to ‖ · ‖2,n. This implies that we can work with ‖ · ‖2,n instead of the Hellinger distance. Using the arguments in Section 7.2 of Ghosal and van der Vaart [2007b], we get max {”