Alexandre Leblanc

A bias-reduced approach to density estimation using Bernstein polynomials

Mixtures of Beta densities have led to different methods of density estimation for univariate data assumed to have compact support. One such method relies on Bernstein polynomials and leads to good approximation properties for the resulting estimator of the underlying density f. In particular, if f is twice continuously differentiable, this estimator can be shown to attain the optimal nonparametric convergence rate of n −4/5 in terms of mean integrated squared error (MISE). However, this rate cannot be improved upon directly when relying on the usual Bernstein polynomials, no matter what other assumptions are made on the smoothness of f. In this note, we show how a simple method of bias reduction can lead to a Bernstein-based estimator that does achieve a higher rate of convergence. Precisely, we exhibit a bias-corrected estimator that achieves the optimal nonparametric MISE rate of n −8/9 when the underlying density f is four times continuously differentiable on its support.

Chung–Smirnov property for Bernstein estimators of distribution functions

In this article, we show that the Chung–Smirnov property holds for Bernstein estimators of distribution functions under different conditions on the underlying distribution to be estimated. In doing so, we obtain general results that characterise the closeness between these Bernstein estimators and the empirical distribution function F n .

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Raos score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the tests statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.

Smooth conditional distribution estimators using Bernstein polynomials

In a variety of statistical problems, estimation of the conditional distribution function remains a challenge. To this end, a two-stage Bernstein estimator for conditional distribution functions is introduced. The method consists in smoothing a first-stage NadarayaWatson or local linear estimator by constructing its Bernstein polynomial. Some asymptotic properties of the proposed estimator are derived, such as its asymptotic bias, variance and mean squared error. The asymptotic normality of the estimator is also established under appropriate conditions of regularity. Lastly, the performance of the proposed estimator is briefly studied through a few examples.

Robust Nonparametric Data Approximation of Point Sets via Data Reduction

In this paper we present a novel nonparametric method for simplifying piecewise linear curves and we apply this method as a statistical approximation of structure within sequential data in the plane. We consider the problem of minimizing the average length of sequences of consecutive input points that lie on any one side of the simplified curve. Specifically, given a sequence P of n points in the plane that determine a simple polygonal chain consisting of n − 1 segments, we describe algorithms for selecting a subsequence Q ⊂ P (including the first and last points of P) that determines a second polygonal chain to approximate P, such that the number of crossings between the two polygonal chains is maximized, and the cardinality of Q is minimized among all such maximizing subsets of P. Our algorithms have respective running times \(O(n^2\sqrt{\log n})\) when P is monotonic and O(n 2log4/3 n) when P is any simple polyline.

The projection median as a weighted average

The projection median of a set

ROBUST NONPARAMETRIC SIMPLIFICATION OF POLYGONAL CHAINS

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Second-order nonlinear least squares estimation

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On estimating distribution functions using Bernstein polynomials

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On the boundary properties of Bernstein polynomial estimators of density and distribution functions

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