2013
DOI: 10.3182/20130703-3-fr-4038.00086
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Data-driven bandwidth choice for gamma kernel estimates of density derivatives on the positive semi-axis

Abstract: In some applications it is necessary to estimate derivatives of probability densities defined on the positive semi-axis. The quality of nonparametric estimates of the probability densities and their derivatives are strongly influenced by smoothing parameters (bandwidths). In this paper an expression for the optimal smoothing parameter of the gamma kernel estimate of the density derivative is obtained. For this parameter data-driven estimates based on methods called "rule of thumb" are constructed. The quality … Show more

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Cited by 6 publications
(3 citation statements)
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“…The substitution of b 0 into (1.8) yields an optimal M ISE with the rate of convergence O(n − 4 7 ). The unknown density and its second derivative in (1.10) were estimated by the rule of thumb method [12].…”
Section: Theoretical Backgroundmentioning
confidence: 99%
See 1 more Smart Citation
“…The substitution of b 0 into (1.8) yields an optimal M ISE with the rate of convergence O(n − 4 7 ). The unknown density and its second derivative in (1.10) were estimated by the rule of thumb method [12].…”
Section: Theoretical Backgroundmentioning
confidence: 99%
“…This corresponds to the optimal bandwidth of order n − 1 7 for symmetrical kernels. The estimation of the univariate density derivative using a gamma kernel estimator by independent data was proposed in [11], [12]. This allows us to achieve the optimal MISE of the same order n −4/7 with a bandwidth of order n − 2 7 .…”
Section: Introductionmentioning
confidence: 99%
“…Its properties were further investigated in Bouezmarni and Scaillet (2005), Hagmann and Scaillet (2007), Zhang (2010) and Malec and Schienle (2014). Asymmetric kernel density estimation remains an area of very active research, as the number of recent papers in the area evidences (Kuruwita et al, 2010, Jeon and Kim, 2013, Dobrovidov and Markovich, 2014, Igarashi and Kakizawa, 2014, Hirukawa and Sakudo, 2014, Funke and Kawka, 2015, Markovich, 2015, Hoffmann and Jones, 2015, Funke and Hirukawa, 2016, Igarashi, 2016, Markovich, 2016, Rosa and Nogueira, 2016, Balakrishna and Koul, 2017. Hirukawa and Sakudo (2015) describe a family of 'generalised Gamma kernels' which includes a variety of similar asymmetric kernels in an attempt to standardise those results.…”
Section: Introductionmentioning
confidence: 99%