Nonparametric bivariate distribution estimation using Bernstein polynomials under right censoring

Dib, Kheireddine; Bouezmarni, Taoufik; Belalia, Mohamed; Kitouni, Abderrahim

doi:10.1080/03610926.2020.1734832

Cited by 5 publications

(2 citation statements)

References 18 publications

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“…Our goal is to prove several asymptotic properties for these five new estimators (bias, variance, mean squared error, mean integrated squared error and asymptotic normality) and compare their numerical performance against traditional 1 In the setting of Bernstein estimators, c.d.f. estimation on compact sets was tackled for example by Babu et al (2002), Leblanc (2009), Leblanc (2012a), Leblanc (2012b), Dutta (2016), Jmaei et al (2017), Erdogan et al (2019) and Wang et al (2019) in the univariate setting, and by Babu & Chaubey (2006), Belalia (2016), Dib et al (2020) and Ouimet (2020a,b) in the multivariate setting. In Hanebeck & Klar (2020), the authors introduced Bernstein estimators with Poisson weights (also called Szasz estimators) for the estimation of c.d.f.s that are supported on [0, ∞), see also Ouimet (2020d).…”

Section: Introductionmentioning

confidence: 99%

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

de Micheaux,

Ouimet

2020

Preprint

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In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line [0, ∞). They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.

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Section: Introductionmentioning

confidence: 99%

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

de Micheaux,

Ouimet

2020

Preprint

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“…(In the setting of Bernstein estimators, c.d.f. estimation on compact sets was tackled, for example, by Babu et al [42], Leblanc [43], Leblanc [44], Leblanc [45], Dutta [46], Jmaei et al [47], Erdo gan et al [48] and Wang et al [49] in the univariate setting, and by Babu and Chaubey [50], Belalia [51], Dib et al [52] and Ouimet [53,54] in the multivariate setting. In [55], the authors introduced Bernstein estimators with Poisson weights (also called Szasz estimators) for the estimation of c.d.f.s that are supported on [0, ∞), see also Ouimet [56]).…”

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confidence: 99%

A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

Micheaux

Ouimet

2021

Mathematics

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In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [0,∞), namely the Gamma, inverse Gamma, LogNormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum–Saunders and Weibull kernel c.d.f. estimators from Mombeni, Masouri and Akhoond. By using the same experimental design, we show that the LogNormal and Birnbaum–Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method from C. Tenreiro.

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On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

Ouimet

2021

Journal of Mathematical Analysis and Applications

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In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimators with Poisson weights on the positive half-line (also called Szasz estimators). The propagation of errors in the literature due to the incorrect estimate in Lemma 2 (iv) of Leblanc (2012a) is addressed in the Appendix.

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Nonparametric bivariate distribution estimation using Bernstein polynomials under right censoring

Cited by 5 publications

References 18 publications

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators

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