Smooth distribution function estimation for lifetime distributions using Szasz–Mirakyan operators

Hanebeck, Ariane; Klar, Bernhard

doi:10.1007/s10463-020-00783-y

Cited by 3 publications

(5 citation statements)

References 22 publications

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“…In the literature, various asymptotic properties of Bernstein estimators with Poisson weights were also studied, namely: Gawronski & Stadtmüller (1980, 1981 studied the bias, variance and mean squared error for the density estimator, and Hanebeck & Klar (2020) studied the bias, variance, mean squared error, mean integrated squared error, asymptotic normality, uniform strong consistency and relative deficiency with respect to the empirical c.d.f. for the c.d.f.…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

“…As for the variance ofF S m,n , it was incorrectly stated in the Theorem 5 of (Hanebeck & Klar, 2020, arXiv v.1), so (3.28) below fixes their original statement and proof. The subsequent arXiv versions of Hanebeck & Klar (2020) were corrected following Theorem 3.5 and Lemma 3.6 of the present paper.…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

“…By the independence of the observations X i , a Taylor expansion for the c.d.f. F , and the asymptotic expression for the bias in Theorem 4 of Hanebeck & Klar (2020), we have…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

“…From Theorem 3.5, other asymptotic expressions can be (and were) derived such as the mean squared error in Gawronski & Stadtmüller (1980) and Hanebeck & Klar (2020). We can also optimize the bandwidth parameter m with respect to these expressions to implement a plug-in selection method, exactly as we would in the setting of traditional multivariate kernel estimators, see e.g.…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

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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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Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

“…By the independence of the observations X i , a Taylor expansion for the c.d.f. F , and the asymptotic expression for the bias in Theorem 4 of Hanebeck & Klar (2020), we have…”

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

Section: Asymptotic Properties Of Bernstein Estimators With Poisson Weightsmentioning

confidence: 99%

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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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“…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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